Research Article Linear Processes in Stochastic …downloads.hindawi.com/journals/tswj/2014/873624.pdfthe rediscovery by Gillespie [ ]ofKendall ssimulation method [ , ]. Poisson [

Research ArticleLinear Processes in Stochastic Population DynamicsTheory and Application to Insect Development

Hernaacuten G Solari1 and Mario A Natiello2

1 Departamento de Fısica FCEN-UBA and IFIBA-CONICET C1428EGA Buenos Aires Argentina2 Centre for Mathematical Sciences Lund University PO Box 118 221 00 Lund Sweden

Correspondence should be addressed to Mario A Natiello marionatiellomathlthse

Received 15 August 2013 Accepted 23 October 2013 Published 17 February 2014

Academic Editors S Casado W Fei and T Nguyen

Copyright copy 2014 H G Solari and M A Natiello This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We consider stochastic population processes (Markov jump processes) that develop as a consequence of the occurrence of randomevents at random time intervals The population is divided into subpopulations or compartments The events occur at rates thatdepend linearly on the number of individuals in the different described compartments The dynamics is presented in terms ofKolmogorov Forward Equation in the space of events and projected onto the space of populations when needed The generalproperties of the problem are discussed Solutions are obtained using a revised version of the Method of Characteristics Aftera few examples of exact solutions we systematically develop short-time approximations to the problem While the lowest orderapproximation matches the Poisson and multinomial heuristics previously proposed higher-order approximations are completelynew Further we analyse a model for insect development as a sequence of 119864 developmental stages regulated by rates that are linearin the implied subpopulations Transition to the next stage competes with death at all times The process ends at a predeterminedstage for example pupation or adult emergence In its simpler version all the stages are distributed with the same characteristictime

1 Introduction

Stochastic population dynamics deals with the (stochastic)time evolution of groups of similar individuals (called pop-ulations or subpopulations) immersed in a habitat The goalis to describe the evolution of population numbers and thenumber of events associated with changes in the populations

Most of the attention has been paied in the past to thepopulations considering events only by their extrinsic valueas jumps in populations However this perspective must berevised In any study of vector-transmitted diseases suchas in [1 2] the most relevant statistics corresponds to theinteractions between hosts and vectors Such interactionsare associated with events in the life cycle of the vector Inthe case of arthropods transmitting diseases to mammalsblood feeding to complete oogenesis is the opportunity totransmit pathogens In other problems the developmentof insects is experimentally described by the sequences oftransformations in their life cycle (such as moulting) [3ndash5] each transformation must be considered an event To

our knowledge explicit use of the event structure to studypopulation changes in Markov Jump processes has beenintroduced in [6] along with a short-time approximationbased on self-consistent Poisson processes

The present research originates in our efforts to modelvector-transmitted diseases such as Dengue starting from adescription of the life cycle of Aedes aegypti the vector [7ndash9]In modeling insect development in an aquatic phase (suchas the case of Aedes aegypti and Drosophila melanogaster) acrucial point is to represent accurately the statistics of theduration of the aquatic phase (also known as the time ofemergence statistics) as it determines the response of the pop-ulation to climatic events such as rains [10] A deterministicmodel for development was proposed in [11] as a sequentialprocess with associated linear rates The final sections of thispaper describe a stochastic development model

While [6] deals with the stochastic approximation ofgeneral problems (achieving amethodwith error 119900(11990532)) thispaper specifically addresses the problem of formulating andsolving population dynamics problems in event space with

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 873624 15 pageshttpdxdoiorg1011552014873624

2 The Scientific World Journal

linear rates that is rates that depend linearly on the pop-ulations (in the chemical literature they are known as first-order reactions [12]) We provide both exact solutions andgeneral approximation techniques with error 119900(119905

119899) Results

from [6] for the linear case are a modified version of one ofthe approximation schemes in this paper (Poisson approx-imation Section 5) The linear problem is mathematicallysimpler than the general case and a few general results canbe established before resorting to approximations

Linear processes are an important part of most biologicaldynamical descriptions Using an analogy from classicalpopulation dynamics linear (Malthusian) population growth[13] describes the evolution of a population at low densities(eg where the local environment is essentially unaffected bythe population growth and the individuals do not interferewith each other)More generally processes that consider onlyindividuals in a somewhat isolated basis can be satisfactorilydescribed with linear rates However whether to use lineardescriptions or not is ultimately determined by the nature ofthe process (or subprocess) For example finite environmentswill sooner or later force individuals to interact in growingpopulations and nonlinear rates become necessary such as inVerhulstrsquos description of population growth [14]

Perhaps the oldest application of linear processes topopulation dynamics regards birth and death processesHistorically according to Kendall [15] the first to proposebirth and death equations in population dynamics was Furry[16] for a system of physical origin Currently such anapproach is being used in several sciences linear stochasticprocesses have been proposed for the development of tumors[17 18] drug delivery and cell replication are also described inthese terms [19] Concerning the tumors themodel proposedin [17 18] is based on linear individual processes a detailedtreatment of which would require a paper of its own Theaccumulation of individual processes is finally approximatedwith a ldquotunnelling processrdquo corresponding to a maturationcascade that can be easily described with the tools of thispaper (see (71)) Concerning the question of drug deliverywe will come back in detail to [19] in Section 7

A substantial use of population dynamics driven by jumpprocesses has been done also in chemistry particularly afterthe rediscovery by Gillespie [20] of Kendallrsquos simulationmethod [21 22] Poisson [23 24] binomial [25 26] andmultinomial [27] heuristics for the short-time integrals havebeen proposed There is however an important differencebetween chemistry and biology concerning the issues of thispaper On the one hand linear rates are seldom the case inchemical reactions except perhaps in the case of irreversibledecompositions or isomeric recombination Attention to lin-ear rates in chemistry is relatively recent [12] when comparedto over 200 years of Malthusian population growth Onthe other hand detailed event statistics does not appearto be fundamental either highly diluted chemical reactionson a test tube may easily generate 10

14 events of eachtype Consequently attention to event based descriptionsappears to be absent from the chemistry literature Thismight however change in the future since the technicalpossibilities given by for example ldquoin-cell chemistryrdquo andldquonanochemistryrdquo allow dealingwith a lower number of events

where tiny differences in event count could be relevant for thegeneral outcome

In themathematical literature attention to jumpprocessesstarts with Kolmogorovrsquos foundational work [28] and itsfurther elaboration by Feller [29] A substantial effort torelate the stochastic description to deterministic equationswas performed by Kurtz [30ndash34] However attention hasalways focused on the dynamics in population space ratherthan event space

This work focuses on the event statistics associated withlinear processes Even for cases where the detailed eventstatistics is not of central interest such an approach offers analternative to the more traditional formulation in populationspace

Further we discuss in detail the stochastic version ofa linear developmental model Specifically we deal with asubprocess in the life cycle of insects that admits both a lineardescription and an exact solution namely the developmentof immature stages which is a highly individual subprocessFrom egg form to adult form the development of insectsgoes through several transformations at different levels Forexample at the most visible level insects undergoing com-plete metamorphosis evolve in the sequence egg larva pupaand adult Further different substages can be recognised byinspection in the development of larvae (instars) and evenmore stages when observed by other methods [35] Theimmature individual may die along the process or otherwisecomplete it and exit as adult

In Section 2 we formulate the dynamical populationproblem in event space In Section 21 we present the Kol-mogorov Forward Equations (for Markov jump processes)for the most general linear systems of population dynamicsin the form appropriated to probabilities and generatingfunctions The general solution of these equations will bewritten using the Theorem of the Characteristics reformu-lated with respect to the standard geometrical presentation[36] into a dynamical presentation We will analyse thegeneral structure of the equations and in particular theirfirst integrals of motion and the projection onto populationspace (Sections 2 and 3) Section 4 contains two classicalexamples In Section 5 Poisson and multinomial approx-imations (both with error 119900(119905)) are derived and a higher-order approximation with error 119900(119905

119899) (with 119899 as an arbi-

trary positive integer) is elaborated The larval developmentmodel is developed in Sections 6 and 7 treating the twogeneral cases We also revisit in Section 7 the question ofdrug delivery introduced in [19] in terms of the presentapproach Concluding remarks are left for the final Sec-tion

2 Mathematical Formulation

The main tools of stochastic population dynamics are a listof (nonnegative integer) populations 119883

119895 where 119895 = 1 sdot sdot sdot 119873

a list of events 120572 = 1 sdot sdot sdot 119864 an (integer) incidence matrix 120575120572

119895

describing how event 120572 modifies population 119895 and a list oftransition rates 119882

120572(119883

1 119883

119899 119905) describing the probability

rate per unit time of occurrence of each event

The Scientific World Journal 3

An important feature of the present approach is that thedynamical description will be handled in event-space In thissense the ldquostaterdquo of the system is given by the array n(119904 119905) =

(1198991 119899

119864)119905indicating how may events of each class have

occurred from time 119904 up to time 119905 Hence event-space is inthis setup a nonnegative integer lattice of dimension 119864 Theconnection with the actual population values at time 119905 is (119904denotes the initial condition)

119883119895(119905) = 119883

119895(119904) + sum

120572

120575120572

119895119899120572(119904 119905) (1)

Definition 1 (linear independency) One says that the popula-tions are linearly dependent if there is a vector u = 0 such thatfor all 119905 ge 119904 one has sum

119895119906119895119883

119895(119905) = 0 Otherwise one says that

the populations are linearly independent and abbreviate it asLI

In similar form we say that the populations have linearlydependent increments if there is u = 0 so that for all 1199051015840 gt 119905 ge 119904

it holds that sum119895119906119895(119883

119895(1199051015840) minus 119883

119895(119905)) = 0

The populations are said to have linearly independentincrements (abbreviated LII) if they do not have linearlydependent increments

Along this work we will assume the following

Assumption 2 The populations have linearly independentincrements that is all involved populations are necessary toprovide a complete description of the problem none of themcan be expressed as a linear combination of the remainingones for all times

Lemma 3 (a) If the populations have linearly independentincrements then the populations are linearly independent

(b) The matrix 120575 has no nonzero vectors u such thatsum

119895120575120572

119895119906119895= 0 for all 120572 if and only if the populations have linearly

independent increments

Proof (a) The assertion is equivalent to the following if thepopulations are linearly dependent then they have linearlydependent increments which is immediate after the defini-tion of linearly dependent increments

(b) Assume that for all 120572 there exists u = 0 such thatsum

119895119906119895120575120572

119895= 0 then

sum

119895

119906119895(119883

119895(119905

1015840) minus 119883

119895(119905)) = sum

119895

119906119895(sum

120572

120575120572

119895119899120572(119905

1015840 119905)) = 0 (2)

and the populations have linearly dependent incrementsConversely assume that the populations have linearly

dependent increments and select time intervals such that119899120573(1199051015840

120573 119905

120573) = 1 and 119899

120572(1199051015840

120573 119905

120573) = 0 for 120572 = 120573 then we have that

sum

119895

119906119895120575120572

119895= 0 forall120572 (3)

Statement (b) is the negative statement of these facts

Strictly speaking the last step in the above proof takesfor granted that there exist time intervals where only one

event occurs This is assured by the next assumption (thirditem) Also the initial conditions and the problem descriptionshould be such that all defined events are relevant for thedynamics (eg the event ldquocontagionrdquo is irrelevant for anepidemic problem with zero infectives as initial condition)

21 Dynamical EquationmdashBasic Assumptions Populationnumbers evolve in ldquojumpsrdquo given by the occurrence of eachevent (birth death etc) In the sequel we will use Latinindices 119894 119895 119896 and 119898 for populations and Greek indices 120572120573 and 120574 for events

The ultimate goal of stochastic population dynamicsis to calculate the probabilities 119875(119899

1 119899

119864 119904 119905) of having

1198991 119899

119864events in each class up to time 119905 given an initial

condition 119883119895(119904)

The stochastic dynamics we are interested in correspondto a Density Dependent Markov Jump Process [37 38] For asufficiently small time interval ℎ we consider the following

Assumption 4 For each event

(i) event occurrences in disjoint time intervals are inde-pendent

(ii) the Chapman-Kolmogorov equation [28 29] holds

(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882

120572(119883 119905)ℎ + 119900(ℎ)

(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882

120572(119883 119905)ℎ + 119900(ℎ)

(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2

Here ℎ is the elapsed time since the time 119905 and 119899120572are the

events of type 120572 accumulated up to 119905

It goes without saying that probabilities 119875(sdot sdot sdot ) areassumed to be differentiable functions of time 119882

120572(119883 119905)

denotes the probability rate for event 120572 which is assumedin the following lemmas to be a smooth function of thepopulations not necessarily linear (yet) Two well-knowngeneral results describe the dynamical evolution under theseassumptions

Lemma 5 (see Theorem 1 in [29]) Under Assumptions 2and 4 the waiting time to the next event is exponentiallydistributed

Theorem6 (see [28] andTheorem 1 in [29]) Under Assump-tions 2 and 4 the dynamics of the process in the space of eventsobeys the Kolmogorov Forward Equation namely

(1198991 119899

119864 119904 119905)

= sum

120572

119875 ( 119899120572minus 1 119904 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119904 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(4)

Proofs of these results are given in Appendix A


22 Linear Rates We will assume that the transition ratesdependmdashin a linear fashionmdashon the populations only Todistinguish from general rates (denoted above by the letter119882) we will use the letter 119877 to denote linear transition ratesHence

119877120572= sum

119896

119903119896

120572119883

119896(119905) (5)

Since rates are nonnegative we have that the proportionalitycoefficient between event 120572 and subpopulation 119896 satisfies119903119896

120572ge 0 These proportionality coefficients convey the

environmental information and as such they may be time-dependent since the environment varieswith timeWheneverpossible we do not write this time dependency explicitly tolighten the notation

Lemma 7 For the case of linear rates population space 119883119896ge

0 is positively invariant under the time evolution given by (1)and the previous assumptions if and only if 120575120572

119894ge minus1 and 120575

120572

119894=

minus1 only if 119877120572= 119903

119894

120572119883

119894(119905) (no sum)

Proof Theprobability of occurrence of event120572 in a small timeinterval ℎ is

ℎ119877120572+ 119900 (ℎ) = ℎsum

119896

119903119896

120572119883

119896(119905) + 119900 (ℎ) (6)

After an occurrence population 119883119896is modified to 119883

119896+ 120575

120572

119896

Population space is positively invariant under time evolutionif and only if the probability of occurrence of event 120572 is zerofor those population values119883

119896such that119883

119896+ 120575

120572

119896lt 0 (we are

just saying that events pushing the populations to negativevalues do not exist) Since this should hold for arbitrary ℎ wehave that 119877

120572= 0

Assume that 120575120572119896

le minus1 Then 119877120572

= 0 for 0 le 119883119896

lt minus120575120572

119896

Letting 119883119896

= 0 we realise that sum119895 = 119896

119903119895

120572119883

119895(119905) = 0 regardless

of the values of 119883119895 and hence it holds that 119903119895

120572= 0 for 119895 = 119896

and therefore 119877120572

= 119903119896

120572119883

119896(no sum) This proves the second

statement If 120575120572119896

lt minus1 we have that 119877120572

= 0 also for 119883119896

= 1Hence 119903

119896

120572= 0 as well 119877

120572equiv 0 and the event 120572 can be ignored

since it does not influence the dynamics

3 Kolmogorov Forward Equation

Let us recast the dynamical equation given by Theorem 6 interms of the generating function [39] in event-space definedas

Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899

1 119899

119864 119904 119905) (7)

For the case of linear rates we define

1198770

120572= sum

119896

119903119896

120572119883

119896 (119904)

119865120573

120572= sum

119896

119903119896

120572120575120573

119896

(8)

As a consequence

119877120572= 119877

0

120572+ sum

120573

119865120573

120572119899120573 (9)

Using Theorem 6 to rewrite 120597Ψ120597119905 and using further 119899120573Ψ =

119911120573(120597Ψ120597119911

120573) (a matter of replacing in the definition of Ψ) we

obtain

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

(119911120572minus 1)(119877

0

120572Ψ + sum

120573

119865120573

120572119911120573120597Ψ120597119911

120573) (10)

Since the generating function reflects a probability distri-bution we have the probability condition Ψ(1 1 119904 119905) =

1 Further the solutions of (10) can be uniquely trans-lated into those given by Theorem 6 for correspondinginitial conditions 119875(119898

1 119898

119864 119904 119904) = 1 corresponding

to Ψ(1199111 119911

119864 119904 119904) = 119911

1198981

11199111198982

2sdot sdot sdot 119911

119898119864

119864 where the integers

1198981 119898

119864denote howmany events of each class had already

occurred at 119905 = 119904 The natural initial condition becomesΨ(119911

1 119911

119864 119904 119904) = 1 corresponding to 119875(0 0 119904 119904) = 1

31 The Method of Characteristics Revisited

Lemma 8 Equation (10) with 1198770

120572= 0 and with the initial

condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904)) (11)

where 120601(119911 119905 120591) is the flow of the following ODE with initialcondition 119909(0) = 119911 119904(0) = 119905

119889119909120573

119889120591= sum

120572

119865120573

120572(119909

120572minus 1) 119909

120573equiv 119887

120573(119909 119904)

119889119904

119889120591= minus1

(12)

Proof It is convenient to rename 119904 as 119904 = 119909119864+1

introduce119887119864+1

= minus1 and recall that 119887120573 depends only on the extendedvariable array 119909 However when necessary we will writeexplicitly 119909(0) as (119911 119905) We have that

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

119889120601120573(119911 119905 119905 minus 119904)

119889119905

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (119887120573(120601 (119911 119905 119905 minus 119904)) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

(13)

Further consider the monodromy matrix [40]

119872120573

120574=

120597120601120573(119909 120591)

120597119909120574

(14)


119872 transforms the initial velocity into the final velocity hence

sum

120574

119872120573

120574119887120574(119911 119905) = 119887

120573(120601 (119911 119905 120591)) (15)

Thus we get

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

(

119864+1

sum

120574

119872120573

120574119887120574(119911 119905) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (

119864

sum

120574

119872120573

120574119887120574(119911 119905) minus 119872

120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864

sum

120574

120597Ψ

120597119911120574

119887120574(119911 119905)

(16)

From the definition of monodromy matrix above it followsthat

minus119872120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

= 0 (17)

and by (11) and the chain rule

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

119872120573

120574= sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

120597120601120573(119911 119905 119905 minus 119904)

120597119911120574

=120597Ψ

120597119911120574

(18)



condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904))Φ

1(119911

1 119911

119864 119904 119905)

(19)

where Φ0(120601(119911 119905 119905 minus 119904)) is a solution of the homogeneous

problem with 1198770

120572= 0 and

Φ1(119911

1 119911

119864 119904 119905)

= expint

119905

119904

(sum

120572

1198770

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

(20)

Proof The proof is nothing more than an application of themethod of the variation of the constants The details are asfollows

Introducing the proposed solution Ψ into (10) and usingLemma 8 to eliminateΦ

0we obtain thatΦ

1satisfies (10) with

initial condition Φ1(119911 119904 119904) = 1 Computing directly from

(20) we have that

120597Φ1

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

1198770

120572(120601

120572(119911 119905 0) minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573

120597120601120573(119911 119905 119905 minus 120591)

120597119905

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1

+ sum

120573

120597Φ1

120597120601120573(119887

120573(120601 (119911 119905 119905 minus 120591)) +

120597120601120573(119911 119906 119905 minus 120591)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

119872120573

120574119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

120597120601120573

120597119911120574

119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120574

120597Φ1

120597119911120574

119887120574(119911 119905)

(21)

after retracing our steps from the previous Lemma

The result presented in Lemma 9 was constructed usingthe method of variation of constants which provides analternative (constructive) proof

32 Basic Properties of the Solution

Definition 10 A nonzero vector v such that sum120572120575120572

119895V120572

= 0for 119895 = 1 119873 is called a structural zero of 119865 Other zeroeigenvalues of 119865 are called nonstructural

Lemma 11 (a)There are structural zeroes if and only if119864 gt 119873(b) The only structurally stable zero eigenvalues of the matrix119865 are the structural zeroes

Proof (a) The ldquoif rdquo part is immediate since dim(ker(120575)) ge

119864minus119873The ldquoonly if rdquo part follows from the assumption of inde-pendent population increments (and therefore independentpopulations) For (b) put 119865 = r120575 If 119865119908 = 0 for 119908 nonzeroand 120575119908 = 119901 = 0 then r119901 = 0 This relation is not structurallystable since by adding 119888 = 0 arbitrarily small to all diagonalelements of r we can assure r1015840119901 = 0 for this modified r1015840 andfor any 119901 = 0

Lemma 12 First integrals of motion for the characteristicequations Let v be a structural zero of 119865 then

prod

120572

(119909120572(119905))

V120572

= 119862 (22)

is a constant of motion for the characteristic equations (12)


Proof Rewrite the first equation (12) as (indices renamed)

119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572

120573) = 0 the expression in the lemma is the

exponential of the integral obtained from

sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)

Given relationship (1) between initial conditions eventsand populations from the probability generating function inevent space

Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)

a corresponding generating function in population space canbe computed namely

Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883

119896 be shorthand for (119899


119864) (119883

1 119883

119873)

Lemma 13 (projection lemma) Equation 119911120572

= prod119873

119896=1119910120575120572

119896

119896

realises the transformation from event space to populationspace Moreover

Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)

Proof Taking logarithms on 119911 and 119910 it is clear that wheneverthere exist structural zeroes the transformation is noninvert-ible In this sense we call it a projection Concerning thestatement we have

(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)

The statement is proved using (1) and rearranging the sumsas

119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)

where the prime denotes sum over 119899120572 such that 119883

0

119895+

sum120572120575120572

119895119899120572= 119883

119895

Corollary 14 The integrals of motion of Lemma 12 projectunder the projection Lemma to 119862 = 1

Proof The above substitution operated on prod119911V120572

120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation

Assumption 15 There exist integers119898120573120573 = 1 119864 such that

(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)

This assumption amounts to considering that there exists away to mimic the initial condition on the rates with the samematrix 119865 involved in the time evolution

Theorem 16 Under Assumptions 2 4 and 15 equation(10) with natural initial condition Ψ(119911

1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911

119864) = 1 has the solution

Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601

120573(119911 119905 119905 minus 119904) in terms of the solutions

of (12) of Lemma 8 Hence 119891(119911 119904 119905) satisfies (after eliminatingthe auxiliary time 120591)

119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)

Proof The differences between 119891( ) and 120601( ) are (a) factoringout the initial condition 120601

120573(119911 119905 0) = 119911

120573and (b) rearranging

the time arguments since the argument in120601( ) is natural to thead hoc autonomous ODE (12) while the argument in 119891( ) isnatural to the (nonautonomous) PDE (10)

The proof is just a matter of rewriting the previousequations under the present assumptions Because of thenatural initial condition the desired solution to (10) isgiven by (20) in Lemma 9 which can be rewritten usingAssumption 15 as (we use 119911 as shorthand for 119911

1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)


Recalling that we defined 119911120572119891120572(119911 119904 119905) = 120601(119911 119905 119905 minus 119904) the

parenthesis can be recognised as the formal solution of (34)integrated from 119905 to 119904

Corollary 17 For v a structural zero of Lemma 12 thegenerating function is unmodified upon the substitution119898

120572rarr

119898120572+ 119896V

120572(119896 isin R)

Remark 18 Having structural zeroes it is amatter of choice toaddress the problem (a) using the variables 119891

120572and imposing

the additional constraints given by structural zeroes or (b)shifting to population coordinates where these constraints areautomatically incorporated Because of Assumption 2 andLemma 12 the transformation mapping one description tothe other corresponds to

119891120572= prod

119895

119908120575120572

119895

119895 (36)

However option (b) proves to bemore efficientwhen it comesto deal with approximate solutions (see Section 5)

4 Examples of Exact Solutions

41 Example I Pure Death Process In the case 119873 = 119864 = 1120575 = minus1 we have 119903 gt 0 119865 = minus119903(119904) and 119898 = minus119872 lt

0 (119872 is the initial population value) Equation (34) reads119889119891119889119904 = 119891119903(119911119891 minus 1) to be integrated in [119905 119904] with 119891(119911 119905 119905) =

1 Hence for constant death rate 119903 we obtain 119891(119911 119904 119905) =

[119911 + (1 minus 119911)119890minus119903(119905minus119904)

]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)

+ 119911 (1 minus 119890minus119903(119905minus119904)

)]119872

(37)

42 Example II Linear Birth andDeath Process Wehave119873 =

1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903

1denotes the birth rate

and 1199032denotes the death rate 119883

0= 119898

1minus 119898

2is the initial

population There exists one structural zero with vector v =

(1 1)119879 while the corresponding constant of motion results in

the relations 11991111199112= 119862 and119891

11198912= 1The differential equation

(34) for 1198911(to be integrated in [119905 119904]) becomes

1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)

with solution (for119862 = 1whichwill be the case after projectionand assuming time-independent birth rates)

1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032

minus 1199031)(119905 minus 119904) Finally Ψ(119911 119904 119905) =

(1198911(119911 119904 119905))

1198981minus1198982 which after projecting in population space

using Lemma 13 and Corollary 14 gives (cf [41])

Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)

Remark 19 A pure death linear process corresponds to abinomial distribution a pure birth process corresponds to anegative binomial and a birth-death process corresponds toa combination of both as independent processes

5 Consistent Approximations

When (34) of Theorem 16 cannot be solved exactly we haveto rely on approximate solutions It is desirable to workwith consistent approximations namely those where basicproperties of the problem are fulfilled exactly rather than ldquoupto an error of size 120598rdquo For example since our solutions shouldbe probabilities we want the coefficients of Ψ(119911

1 119911

119864 119905)

to be always nonnegative and sum up to one Ideally wewant approximations to be constrained to satisfy Lemma 12and to preserve positive invariance in population space (nojumps into negative populations) Whenever consistency isnot automatic it is mandatory to analyse the approximatesolution before jumping into conclusions

51 Poisson Approximation Let us replace 119891120573

equiv 1 in theRHS of (34) (this amounts to propose that 119891

120573is constant and

equal to its initial condition) The solution of the modifiedequation will be a good approximation to the exact solutionfor sufficiently short times such that 119891 is not significantlydifferent from one

For constant rates 119903120572the rhs of (34) does not depend on

time The approximate solution reads 119891120572(119911 119904 119905) = exp((119905 minus

119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)

In other words the generating function is approximatedas a product of individual Poisson generating functions

This approximation has an error of 119900(119905) Simple as it looksthis approach gives a good probability generating functionthat respects structural zeroes but it does not guaranteepositive invariance a Poisson jump could throw the systeminto negative populations However the probability of sucha jump is also 119900(119905) Fortunately there is a natural way toimpose positive invariance and also a natural way to improvethe error up to 119900(119905)

32 in the broader framework of generaltransition rates [6] In this way the approximation can besafely used with error control to address problems that resistexact computation one way or the other

52 Consistent Multinomial Approximations A systematicapproach to obtain a chain of approximations of increasingaccuracy to (34) can be devised via Picard iterations Firstly itis convenient to recast the problem in population coordinates(cf Remark 18) so that the structural zeroes are automaticallytaken into account regardless of other properties of the


adopted approximation From 119891120572

= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908

119896and (34) can be recast as

sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)

Because of the linear independence of population increments(and hence of populations) Lemma 3(b) the counterpart of(34) in population coordinates reads

119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)

Remark 20 In the case of linearly dependent populationincrements (34) still holds while it is possible to elaboratea more general form in terms of the 119908

119895instead of (43) This

is however outside the scope of this paper

Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)

119895 which is so far an exact result Let us produce

some approximated values for the 119908119896rsquos and correspondingly

for Ψ(1199111 119911

119864 119904 119905)

To avoid further notational complications wewill assumein the sequel that 119903

119896

120573is time independent Hence 119908

119896will

depend only on the time difference 119905 minus 119904 Equation (43) canbe recast as

119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)

now to be integrated along the interval [0 119905 minus 119904] We will evenset 119904 = 0 in what follows

Remark 21 For the case of time-independent proportionalityfactors 119903119896

120573 a correspondingmodification can be introduced in

(34) namely overall change of sign integration in [0 119905 minus 119904]and setting 119904 = 0 to reduce the notational burden

Equations (44) satisfy the Picard-Lindelof theorem sincethe RHS is a Lipschitz function Then the Picard iterationscheme converges to the (unique) solution Taking advantage

of the initial condition the 119899th order truncated version of thePicard iterations reads

119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)

where [sdot sdot sdot ]119898denote the Maclaurin polynomial of order119898 in

119905

Lemma 22 For 119905 isin [0 ℎ] ℎ gt 0 being sufficiently smalland for each order of approximation 119899 ge 0 119908(119899)

119896(119905) in (44)

is a polynomial generating function for one individual of thepopulation that is it has the following properties

(a) 119908(119899)

119896(119905) is a polynomial of degree 119899 in 119911

1 119911

119864 where

the coefficient of 11991111989911 119911

119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+

sdot sdot sdot + 119899119864

(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)

(d) the coefficients of 119908(119899)

119896(119905) regarded as a polynomial in

119911120572 are nonnegative functions of time

Proof (a) The statement holds for 119899 = 0 and 119899 = 1Assuming that it holds up to order 119899 minus 1 we have that[119908

(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1

is also a polynomial of degree119899 minus 1 in 119911

1 119911

119864with time-dependent coefficients of type

119874(119905119901) since each power 119901 of 119905 carries along a power 119901 in 119911

120572

(as well as lower 119911-powers) A Picard iteration step gives

119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)

The first sum contributes with one time integration and one119911-power which exactly preserves the polynomial structureand the order of the coefficientsThe result is a polynomial ofdegree 119899 in 119911

1 119911

119864 where the coefficients are polynomials

of highest degree at most 119899 in ℎ each one of lowest order119874(ℎ

119901) (as in the statement) The second integral does not


alter these properties since it generates again a polynomial ofdegree 119899minus1 in 119911

1 119911

119864 with time-dependent coefficients of

order119874(ℎ119901+1

) or higher up to ℎ119899 (they are just the coefficients

of 119908(119899minus1)

119896(119905) integrated once in time)

(b) Letting 1199111= 119911

2= sdot sdot sdot = 119911

119864= 1 Picard iteration gives

trivially 119908(119899)

119896(119905) = 1 to all orders again by induction

(c) The statement holds for 119899 = 1 Assuming it holds upto 119899 minus 1 we compute

119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)

Both expressions in the first integral have the sameMaclaurinpolynomial up to order (119899 minus 2) Hence the difference in thefirst integral contains only terms of order119874(119905

119899minus1) This is also

the case for the second integral After integration we obtainΔ(119899)

119896(ℎ) = 119874(ℎ

119899)

(d) The statement holds for 119899 = 0 and 119899 = 1 and ingeneral for the zero-order coefficient because it is unity for119905 = 0 as a consequence of the initial condition For thehigher-order coefficients we use induction again If all 119911-coefficients in 119908

(119899minus1)

119896(119905) are nonnegative then the same holds

automatically for 119908(119899)

119896(ℎ) up to order 119899 minus 1 since the 119874(ℎ

119899)

contribution adding to each inherited coefficient from theprevious step does not alter its sign for ℎ sufficiently small Itremains to show that the coefficients corresponding to 119901 = 119899

are nonnegative These contributions arise from the (119899 minus 1)thorder of

119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)

after time integration If all 120575120573119895are positive then the expres-

sion above is a product of 119911-polynomials with nonnegative

coefficients and the statement follows Suppose that some120575120573

119895= minus1 Then by Lemma 7 119903119896

120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895

= minus 1 for 119895 = 119896 that is they are nonnegative andhence the expression has only nonnegative coefficients

521 Basic Properties of the First- and Second-OrderApproximations Let us analyse the generating functionΨ(119911

1 119911

119864 119905) = prod

119895119908

119883119895(0)

119895 The simplest nontrivial

applications of the approximation scheme are the first- andsecond-order approximations Expanding explicitly up tosecond-order we get

119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)

We begin by noticing that the first-order approximation isobtained by neglecting all terms in ℎ

2 in the above equationWhenever 119908

119895is linear in 119911 which is always the case for the

first-order approximation the resulting generating functionΨ corresponds to the product of (independent) multinomialdistributions for the events affecting each subpopulation

Let us now consider Ψ(1199111 119911

119864 119905) computed with the

second-order approximated 119908119895rsquos to perform one simulation

step of size ℎCounting powers of 119911

120574on each 119908

119896we have (a) the

probability of no events occurring in time-step ℎ

119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)

(b) the probability of only one (120573) event occurring in time-step ℎ

119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)

and (c) the probability of another event (120572) occurring afterthe first one

119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)


522 Simulation Algorithm A simulation step can be oper-ated in the following way Let

sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)

be the probability of one (120573) event occurring alone orfollowed by a second event It follows that

119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)

The second factor above is the conditional probability119875119896(ℎ 120572120573) for the second event being 120572 given that there was

a first (120573) eventThe algorithm proceeds in two steps First a multino-

mial deviate is computed with probabilities 119875119896(ℎ 119868 120573) (the

probability of no event occurring is still 119875119896(ℎ 0)) This gives

the number 119899120573of occurrences of each event In the second

step for each 119899120573we compute the deviates for the occurrence

of a second event or no more events with the multinomialgiven by the array of probabilities 119875

119896(ℎ 120572120573) defined above

(the probability of no second event occurring is here 1 minus

(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))

6 Implementation Developmental Cascadeswithout Mortality

In this and the following Sections we apply the tools devel-oped up to now to the description of a subprocess in insectdevelopment namely the development of immature larvaewhich is in principle a highly individual subprocess Weassume that the evolution of an insect from egg to pupaoccurs via a sequence of 119864 gt 0 maturation stages Biologistsrecognise some substages by inspection in the developmentof larvae (instars) and even more stages when observed byother methods [35] The immature individual may die alongthe process or otherwise complete it and exit as pupa oradult The actual value of apparent maturation stages 119864 willdepend on environmental and experimental conditions andwill ultimately be specified when analying experimental datain Section 71

Empirical evidence based on existing and new experi-mental results as well as biological insight produced by thepresent description will be the subject of an independentwork [42]We discuss however a couple of examples from theliterature in Section 71

The events for this process are maturation events pro-moting individuals from one subpopulation into to the nextand death events for all subpopulations In other wordseach subpopulation represents a given (intermediate) level ofdevelopment and maturation events correspond to progressin the development Development at level 119895 promotes oneindividual to level 119895 + 1 and the event rate is linear in the119895-subpopulation Hence in 119877

120572= sum

119895119903119896

120572119883

119895 the matrix 119903

is square and diagonal for 0 le 120572 le 119864 minus 1 (we haveexactly 119864 + 1 subpopulations counted from 0 to 119864) All ofthe environmental influence is at this point incorporatedthrough 119903

Moreover level 119864 is the matured pupa and we may set119903119864

= 0 since at pupation stage the present mechanism endsand new processes take place Subpopulation 119864 is the exitpoint of the dynamics and we assume it to be determined bythe stochastic evolution of the immature stages 0 le 120572 le 119864minus1

In the same spirit the action of each event on thepopulations modifies just the actual population and the nextone in the chain Hence

120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)

Following Section 3 [1198770

120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)

Finally letting m119879= minus119872(1 1 1) we can write 119877

0

120572=

sum120573119865120573

120572119898

120573 In this framework the index pair (119895 120572) is highly

interconnected and we can achieve a complete descriptionusing just one index We use 0 le 119895 le 119864 minus 1 throughout thissection

The procedure of Section 3 gives Ψ(119911 119905) = prod119895(119891

119895)119898119895 =

prod119895(119891

minus119872

119895) while the dynamical system for 119891

119895reads recalling

the recasting in Remark 21119891119895= 119891

119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1

(minus119903119864minus1

(119911119864minus1

119891119864minus1

minus 1)) The overall initialcondition is 119891

119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882

119895+1 Rewriting the

equations120579119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1

= minus119903119864minus1

(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)

with initial condition 120579119895(0) = 0 for all 119895With this notation the

generating function reads Ψ(119911 119905) = 119882119872

0 and we formulate

hence an ODE for the 119882119895rsquos

119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1

minus 119882119864minus1

) 119882119864minus1

(0) = 1

(61)


By defining the auxiliary quantity 119882119864(119905) = 1 the solution

for the 119882119895-differential equations can be written in recursive

form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)

In Appendix B we give the details of the explicit solutionof this problem in terms of Laplace transforms Finally weobtain

1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1

prod119864minus119896

119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)

The solution looksmore compact when all 119903119896are assumed

to take the common value 119903

1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)

7 Developmental Cascades with Mortality

We add to the previous description 119864 subpopulation specificdeath events 119876

120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We

do not consider death of the pupa as an event for the samereasons as before the pupa subpopulation 119883

119864leaves the

present framework of study We have actually event pairsfor each subpopulation and hence the analysis can still bedone with just one index 120572 = 0 119864 minus 1 since eachevent is population specific The calculations get howevermore involved It is practical to organise the events as fol-lows 119876

0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1

(development) 120572 lt 119864

119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)

Recall that the indices in119865 run from 0 to 2119864minus1 It is also prac-tical to order the variables 119891

120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)

Also for notational convenience we will assume that thereexist 119902

119864= 119903

119864= 0 when necessary Defining

119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)

we have (recall again Remark 21 and note that 119867119864equiv 0)

119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)

The overall initial condition is still 119892120572(0) = 1 = 119891

120572(0)

Since there are more events than subpopulations accord-ing to Definition 1 and Corollary 14 119865 has structural zeroescorresponding to constants of motion in the associateddynamical system The zero eigenvectors of 119865 are 119881

119879

120572=

(0 minus1 1 1 0 ) nonzero in positions 2120572 2120572 + 1 and2120572 + 2 for 120572 = 0 119864 minus 1 (for the last one recall that thereis no position ldquo2119864rdquo) Definitions similar to those used in theprevious section will be useful as well namely 120579

120572= log119892

120572

and 120601120572

= log119891120572 changing 119867

120572and the initial conditions

correspondingly Hence

120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)

The structural zeroes and constants of motion allow solvingall equations related to the 120601-variables and only a chain of120579-equations is left very much in the spirit of the previoussection m119879

= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872

0 We now compute 119892

0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1

120572 0 le 120572 le 119864minus1 Defining the auxiliary

variable119882119864(119905) = 1 the dynamical equations for 120579 in terms of

119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)

Again we have that Ψ(119911 119910 119905) = 119882119872

0 Specialising for the

case where all 119903 and all 119902 coefficients are equal the solutionreads

1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)

71 Examples from the Literature In [19] Section 3 the drugdelivery process by oral ingestion of a medicine consisting ofmicrospheres containing the active principle is discussedTheprocess is modeled considering that microspheres enter thestomach (119864 = 0) and either dissolve (passing subsequentlyto the circulatory system) or proceed to the next stage ofthe GI tract namely the duodenum Both processes areassumed to obey linear transition rates In the same waythe undissolved duodenal microspheres pass to the nextintestinal stagemdashthe jejunummdashand later to the ilium Theremaining undissolved microspheres entering the colon areeliminated without further dissolutionThis is a neat exampleof a cascade where dissolution corresponds to mortality in(71) while maturation corresponds to progress through thedifferent stages Remaining particles exit the system uponarrival to the fifth and last stage (119864 = 4) Using the data listedin page 239 of [19] for the dissolution and passage coefficientsthe results of that paper can be directly read from (71) Theprobabilities of staying or dissolving at a given stage functionof time appear as coefficients of 119882

0for different powers of

119911119896and 119910

119896 As an example we compute the probabilities 119901

119894(119905)

of permanence at a given stage (as a function of time) usingthe 120582- and 120583-notation from page 239 in [19] (correspondingin (71) to 119903 and 119902 resp) Define first 119875(119904) = prod

119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3

for each integer 119898 isin [0 119904] and finally 119877(119904) = 120582119904+ 120583

119904

119904 = 0 3 With these ingredients the probabilities are1199010(119905) = exp(minus119905119877

0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))

8 Concluding Remarks

The description of stochastic population dynamics at eventlevel has several advantages with respect to the descriptionat population level While the projection onto populationspace is straightforward the description of the number ofevents carries biological information that can be lost in theprojection From a practical mathematical point of view thedescription in terms of events is more regular and makesroom for a general discussion since the number of eventsalways increases by one

When individual populationsrsquo processes are consideredthe event rates become linear and in addition if the result ofthe event is to decrease one (sub)population then they canonly be proportional to the decreased population

The probability generating function for the number ofevents 119899

120573 conditioned to an initial state described by

population values 119883119894(0) has been obtained as a function

of the solutions of a set of ODE known as the equations ofthe characteristics but presented in a form oriented towardsnumerical calculus rather than the usual presentation ori-ented towards geometrical methods

We have introduced short-time approximations that tothe lowest order are in coincidence with previously consid-ered heuristic proposals Our presentation is systematic andcan be carried out up to any desired order In particularwe have shown how the second-order approximation can beimplemented

The general solution represents independent individualsidentically distributedwithin their (sub)population compart-ment

Additionally we believe that obtaining general solutionsof the linear problem is a sound starting point when themore complicated approximations for nonlinear rates areconsidered

Further we have solved a stochastic individual develop-mental model that is a model that at the population levelis described by rates that are linear in the (sub)populationsThe model not only describes just the populations but alsodescribes the events that change the populations

Appendices

A Proofs of Traditional Results

The original version of these proofs (in slightly differentflavours) can be found on page 428-9 (equations (48) and(50)) in [28] and equations (29) and (30) on page 498 in [29]


A1 Proof of Lemma 5

Proof We assume the population and number of occurredevents up to time 119905 to be known Let 120591 = 119905

1minus 119905 ge 0 be

sufficiently small It is clear that if no events occur duringthe time interval 120591 then 119883(119904) = 119883(119905) 119905 le 119904 le 119905 + 120591

For 0 le ℎ le 120591 the Chapman-Kolmogorov equationimplies that

119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)

Note that since 119883 is constant along the time interval inconsideration the rates 119882(119883 119905) are well-defined Hence119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)

Given the natural initial condition 119875(119899 = 0 119905) = 1 thesolution to this equation is

119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)

A2 Proof of Theorem 6

Proof Again the Chapman-Kolmogorov equation gives

119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)

Repeating the previous procedure taking the limit ℎ rarr 0we finally obtain Kolmogorov Forward Equation

(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)

B Solution via Laplace Transform

After multiplying (62) by the Heaviside function 119867(119905) werealise that the recursion involves a convolution Hence it ispractical to use Laplace transforms Indeed defining 119866

119896(119904) =

L(119867(119905)119882119896(119905)) the equation can be rewritten as

119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)

since convolution product becomes standard product aftertransforming After some manipulation the recursion issolved by induction

1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)

Inverse Laplace transform leads us back to (63) Note as wellthat when all 119903

119896are taken to be equal the corresponding119866

0(119904)

still gives the Laplace transform of the desired solution whilethe inverse transform involves integrals related to theGammafunction (see (64))

The case with mortality is similar although slightly morecomplicated Equation (70) reads after Laplace transform asabove (0 le 119896 lt 119864)

119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)

Again inverse Laplace transform leads to (71)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Jorg Schmeling and Victor Ufnarovskifor a valuable suggestion Hernan G Solari acknowledgessupport from Universidad de Buenos Aires under Grant20020100100734 Mario A Natiello acknowledges grantsfromVetenskapsradet andKungliga Fysiografiska Sallskapet


References

[1] D A Focks DGHaile E Daniels andGAMount ldquoDynamiclife table model for Aedes aegypti (Diptera Culcidae) analysisof the literature and model developmentrdquo Journal of MedicalEntomology vol 30 no 6 pp 1003ndash1018 1993

[2] M Otero and H G Solari ldquoStochastic eco-epidemiologicalmodel of dengue disease transmission by Aedes aegyptimosquitordquoMathematical Biosciences vol 223 no 1 pp 32ndash462010

[3] R Amalberti ldquoThe paradoxes of almost totally safe transporta-tion systemsrdquo Safety Science vol 37 pp 109ndash126 2001

[4] T R E Southwood G Murdie M Yasuno R J Tonn andP M Reader ldquoStudies on the life budget of Aedes aegypti inWat Samphaya BangkokThailandrdquoBulletin of theWorldHealthOrganization vol 46 no 2 pp 211ndash226 1972

[5] L M Rueda K J Patel R C Axtell and R E StinnerldquoTemperature-dependent development and survival rates ofCulex quinquefasciatus and Aedes aegypti (Diptera Culicidae)rdquoJournal of Medical Entomology vol 27 no 5 pp 892ndash898 1990

[6] H G Solari and M A Natiello ldquoStochastic population dynam-ics the Poisson approximationrdquo Physical Review E vol 67 no3 part 1 Article ID 031918 2003

[7] M Otero H G Solari and N Schweigmann ldquoA stochasticpopulation dynamics model for Aedes aegypti formulationand application to a city with temperate climaterdquo Bulletin ofMathematical Biology vol 68 no 8 pp 1945ndash1974 2006

[8] M Otero N Schweigmann and H G Solari ldquoA stochasticspatial dynamical model for Aedes aegyptirdquo Bulletin of Mathe-matical Biology vol 70 no 5 pp 1297ndash1325 2008

[9] M Otero D H Barmak C O Dorso H G Solari andM A Natiello ldquoModeling dengue outbreaksrdquo MathematicalBiosciences vol 232 no 2 pp 87ndash95 2011

[10] V R Aznar M J Otero M S de Majo S Fischer and H GSolari ldquoModelling the complex hatching and development ofAedes aegypti in temperated climatesrdquo Ecological Modelling vol253 pp 44ndash55 2013

[11] T J Manetsch ldquoTime-varying distributed delays and their usein aggregative models of large systemsrdquo IEEE Transactions onSystems Man and Cybernetics vol 6 no 8 pp 547ndash553 1976

[12] C Gadgil C H Lee andH G Othmer ldquoA stochastic analysis offirst-order reaction networksrdquo Bulletin ofMathematical Biologyvol 67 no 5 pp 901ndash946 2005

[13] T Malthus ldquoAn Essay on the Principle of PopulationrdquoElectronic Scholarly Publishing Project London UK 1798httpwwwesporg

[14] L M Spencer and P S M Spencer Competence atWorkModelsfor Superior performance John Wiley amp Sons 2008

[15] D G Kendall ldquoStochastic processes and population growthrdquoJournal of the Royal Statistical Society B vol 11 pp 230ndash2821949

[16] W H Furry ldquoOn fluctuation phenomena in the passage of highenergy electrons through leadrdquo Physical Review vol 52 no 6pp 569ndash581 1937

[17] M A Nowak N L Komarova A Sengupta et al ldquoThe role ofchromosomal instability in tumor initiationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 99 no 25 pp 16226ndash16231 2002

[18] Y Iwasa F Michor and M A Nowak ldquoStochastic tunnels inevolutionary dynamicsrdquo Genetics vol 166 no 3 pp 1571ndash15792004

[19] K K Yin H Yang P Daoutidis and G G Yin ldquoSimula-tion of population dynamics using continuous-time finite-stateMarkov chainsrdquo Computers and Chemical Engineering vol 27no 2 pp 235ndash249 2003

[20] D TGillespie ldquoExact stochastic simulation of coupled chemicalreactionsrdquo Journal of Physical Chemistry vol 81 no 25 pp2340ndash2361 1977

[21] D G Kendall ldquoAn artificial realization of a simple ldquobirth-and-deathrdquo processrdquo Journal of the Royal Statistical Society B vol 12pp 116ndash119 1950

[22] M S Bartlett ldquoDeterministic and stochastic models for recur-rent epidemicsrdquo in Proceedings of the 3rd Berkeley Symposiumon Mathematical Statisics and Probability Statistics 1956

[23] D T Gillespie ldquoApproximate accelerated stochastic simulationof chemically reacting systemsrdquo Journal of Chemical Physics vol115 no 4 pp 1716ndash1733 2001

[24] D T Gillespie and L R Petzold ldquoImproved lead-size selectionfor accelerated stochastic simulationrdquo Journal of ChemicalPhysics vol 119 no 16 pp 8229ndash8234 2003

[25] T Tian and K Burrage ldquoBinomial leap methods for simulatingstochastic chemical kineticsrdquo Journal of Chemical Physics vol121 no 21 pp 10356ndash10364 2004

[26] X PengW Zhou andYWang ldquoEfficient binomial leapmethodfor simulating chemical kineticsrdquo Journal of Chemical Physicsvol 126 no 22 Article ID 224109 2007

[27] M F Pettigrew andH Resat ldquoMultinomial tau-leaping methodfor stochastic kinetic simulationsrdquo Journal of Chemical Physicsvol 126 no 8 Article ID 084101 2007

[28] A Kolmogoroff ldquoUber die analytischen methoden in derwahrscheinlichkeitsrechnungrdquo Mathematische Annalen vol104 pp 415ndash458 1931

[29] W Feller ldquoOn the integro-differential equations of purelydiscontinuousMarkoff processesrdquo Transactions of the AmericanMathematical Society vol 48 no 3 pp 488ndash515 1940

[30] T G Kurtz ldquoSolutions of ordinary differential equations aslimits of pure jump Markov processesrdquo Journal of AppliedProbability vol 7 pp 49ndash58 1970

[31] T G Kurtz ldquoLimit theorems for sequences of jump pro-cesses approximating ordinary differential equationsrdquo Journalof Applied Probability vol 8 pp 344ndash356 1971

[32] T G Kurtz ldquoLimit theorems and diffusion approximations fordensity dependentMarkov chainsrdquoMathematical ProgrammingStudies vol 5 article 67 1976

[33] T G Kurtz ldquoStrong approximation theorems for density depen-dentMarkov chainsrdquo Stochastic Processes and their Applicationsvol 6 no 3 pp 223ndash240 1978

[34] T G Kurtz ldquoApproximation of discontinuous processes by con-tinuous processesrdquo in Stochastic Nonlinear Systems in PhysicsChemistry and Biology L Arnold and R Lefever Eds SpringerBerlin Germany 1981

[35] B S Heming Insect Development and Evolution Cornell Uni-versity Press Ithaca NY USA 2003

[36] R Courant and D Hilbert Methods of Mathematical PhysicsInterscience New York NY USA 1953

[37] S N Ethier and T G KurtzMarkov Processes John Wiley andSons New York NY USA 1986

[38] RDurrettEssentials of Stochastic Processes SpringerNewYorkNY USA 2001

[39] A Renyi Calculo de Probabilidades Editorial Reverte MadridEspana 1976


[40] H G Solari M A Natiello and B G Mindlin NonlinearDynamics A Two-Way Trip from Physics to Math Institute ofPhysics Bristol UK 1996

[41] N T J BaileyThe Elements of Stochastic Processes with Applica-tions to the Natural Sciences Wiley New York NY USA 1964

[42] V R Aznar M S DeMajo S Fischer D Francisco M NatielloandH G Solari ldquoAmodel for the developmental times ofAedes(Stegomyia) aegypti (and other insects) as a function of theavailable foodrdquo preprint

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


linear rates that is rates that depend linearly on the pop-ulations (in the chemical literature they are known as first-order reactions [12]) We provide both exact solutions andgeneral approximation techniques with error 119900(119905

119899) Results

from [6] for the linear case are a modified version of one ofthe approximation schemes in this paper (Poisson approx-imation Section 5) The linear problem is mathematicallysimpler than the general case and a few general results canbe established before resorting to approximations

Linear processes are an important part of most biologicaldynamical descriptions Using an analogy from classicalpopulation dynamics linear (Malthusian) population growth[13] describes the evolution of a population at low densities(eg where the local environment is essentially unaffected bythe population growth and the individuals do not interferewith each other)More generally processes that consider onlyindividuals in a somewhat isolated basis can be satisfactorilydescribed with linear rates However whether to use lineardescriptions or not is ultimately determined by the nature ofthe process (or subprocess) For example finite environmentswill sooner or later force individuals to interact in growingpopulations and nonlinear rates become necessary such as inVerhulstrsquos description of population growth [14]

Perhaps the oldest application of linear processes topopulation dynamics regards birth and death processesHistorically according to Kendall [15] the first to proposebirth and death equations in population dynamics was Furry[16] for a system of physical origin Currently such anapproach is being used in several sciences linear stochasticprocesses have been proposed for the development of tumors[17 18] drug delivery and cell replication are also described inthese terms [19] Concerning the tumors themodel proposedin [17 18] is based on linear individual processes a detailedtreatment of which would require a paper of its own Theaccumulation of individual processes is finally approximatedwith a ldquotunnelling processrdquo corresponding to a maturationcascade that can be easily described with the tools of thispaper (see (71)) Concerning the question of drug deliverywe will come back in detail to [19] in Section 7

A substantial use of population dynamics driven by jumpprocesses has been done also in chemistry particularly afterthe rediscovery by Gillespie [20] of Kendallrsquos simulationmethod [21 22] Poisson [23 24] binomial [25 26] andmultinomial [27] heuristics for the short-time integrals havebeen proposed There is however an important differencebetween chemistry and biology concerning the issues of thispaper On the one hand linear rates are seldom the case inchemical reactions except perhaps in the case of irreversibledecompositions or isomeric recombination Attention to lin-ear rates in chemistry is relatively recent [12] when comparedto over 200 years of Malthusian population growth Onthe other hand detailed event statistics does not appearto be fundamental either highly diluted chemical reactionson a test tube may easily generate 10

14 events of eachtype Consequently attention to event based descriptionsappears to be absent from the chemistry literature Thismight however change in the future since the technicalpossibilities given by for example ldquoin-cell chemistryrdquo andldquonanochemistryrdquo allow dealingwith a lower number of events

where tiny differences in event count could be relevant for thegeneral outcome

In themathematical literature attention to jumpprocessesstarts with Kolmogorovrsquos foundational work [28] and itsfurther elaboration by Feller [29] A substantial effort torelate the stochastic description to deterministic equationswas performed by Kurtz [30ndash34] However attention hasalways focused on the dynamics in population space ratherthan event space

This work focuses on the event statistics associated withlinear processes Even for cases where the detailed eventstatistics is not of central interest such an approach offers analternative to the more traditional formulation in populationspace

Further we discuss in detail the stochastic version ofa linear developmental model Specifically we deal with asubprocess in the life cycle of insects that admits both a lineardescription and an exact solution namely the developmentof immature stages which is a highly individual subprocessFrom egg form to adult form the development of insectsgoes through several transformations at different levels Forexample at the most visible level insects undergoing com-plete metamorphosis evolve in the sequence egg larva pupaand adult Further different substages can be recognised byinspection in the development of larvae (instars) and evenmore stages when observed by other methods [35] Theimmature individual may die along the process or otherwisecomplete it and exit as adult

In Section 2 we formulate the dynamical populationproblem in event space In Section 21 we present the Kol-mogorov Forward Equations (for Markov jump processes)for the most general linear systems of population dynamicsin the form appropriated to probabilities and generatingfunctions The general solution of these equations will bewritten using the Theorem of the Characteristics reformu-lated with respect to the standard geometrical presentation[36] into a dynamical presentation We will analyse thegeneral structure of the equations and in particular theirfirst integrals of motion and the projection onto populationspace (Sections 2 and 3) Section 4 contains two classicalexamples In Section 5 Poisson and multinomial approx-imations (both with error 119900(119905)) are derived and a higher-order approximation with error 119900(119905

119899) (with 119899 as an arbi-

trary positive integer) is elaborated The larval developmentmodel is developed in Sections 6 and 7 treating the twogeneral cases We also revisit in Section 7 the question ofdrug delivery introduced in [19] in terms of the presentapproach Concluding remarks are left for the final Sec-tion

2 Mathematical Formulation

The main tools of stochastic population dynamics are a listof (nonnegative integer) populations 119883

119895 where 119895 = 1 sdot sdot sdot 119873

a list of events 120572 = 1 sdot sdot sdot 119864 an (integer) incidence matrix 120575120572

119895

describing how event 120572 modifies population 119895 and a list oftransition rates 119882

120572(119883

1 119883

119899 119905) describing the probability

rate per unit time of occurrence of each event



(1198991 119899



119883119895(119905) = 119883

119895(119904) + sum

120572

120575120572

119895119899120572(119904 119905) (1)


119895119906119895119883




it holds that sum119895119906119895(119883

119895(1199051015840) minus 119883

119895(119905)) = 0






119895120575120572





119895119906119895120575120572

119895= 0 then

sum

119895

119906119895(119883

119895(119905

1015840) minus 119883

119895(119905)) = sum

119895

119906119895(sum

120572

120575120572

119895119899120572(119905

1015840 119905)) = 0 (2)



120573 119905

120573) = 1 and 119899

120572(1199051015840

120573 119905


sum

119895

119906119895120575120572

119895= 0 forall120572 (3)






1 119899

119864 119904 119905) of having

1198991 119899


condition 119883119895(119904)





(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882

120572(119883 119905)ℎ + 119900(ℎ)

(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882

120572(119883 119905)ℎ + 119900(ℎ)

(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2




120572(119883 119905)




(1198991 119899

119864 119904 119905)

= sum

120572

119875 ( 119899120572minus 1 119904 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119904 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(4)




119877120572= sum

119896

119903119896

120572119883

119896(119905) (5)






119894ge minus1 and 120575

120572

119894=

minus1 only if 119877120572= 119903

119894

120572119883

119894(119905) (no sum)


ℎ119877120572+ 119900 (ℎ) = ℎsum

119896

119903119896

120572119883

119896(119905) + 119900 (ℎ) (6)


119896+ 120575

120572

119896


119896such that119883

119896+ 120575

120572

119896lt 0 (we are


120572= 0

Assume that 120575120572119896


= 0 for 0 le 119883119896

lt minus120575120572

119896

Letting 119883119896


119903119895

120572119883

119895(119905) = 0 regardless


120572= 0 for 119895 = 119896


= 119903119896

120572119883


statement If 120575120572119896


= 0 also for 119883119896

= 1Hence 119903

119896

120572= 0 as well 119877





Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899

1 119899

119864 119904 119905) (7)


1198770

120572= sum

119896

119903119896

120572119883

119896 (119904)

119865120573

120572= sum

119896

119903119896

120572120575120573

119896

(8)

As a consequence

119877120572= 119877

0

120572+ sum

120573

119865120573

120572119899120573 (9)


119911120573(120597Ψ120597119911


obtain

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

(119911120572minus 1)(119877

0

120572Ψ + sum

120573

119865120573

120572119911120573120597Ψ120597119911

120573) (10)



1 119898

119864 119904 119904) = 1 corresponding

to Ψ(1199111 119911

119864 119904 119904) = 119911

1198981

11199111198982


119898119864


1198981 119898



1 119911





condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904)) (11)


119889119909120573

119889120591= sum

120572

119865120573

120572(119909

120572minus 1) 119909

120573equiv 119887

120573(119909 119904)

119889119904

119889120591= minus1

(12)


introduce119887119864+1


120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

119889120601120573(119911 119905 119905 minus 119904)

119889119905

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (119887120573(120601 (119911 119905 119905 minus 119904)) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

(13)


119872120573

120574=

120597120601120573(119909 120591)

120597119909120574

(14)



sum

120574

119872120573

120574119887120574(119911 119905) = 119887

120573(120601 (119911 119905 120591)) (15)

Thus we get

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

(

119864+1

sum

120574

119872120573

120574119887120574(119911 119905) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (

119864

sum

120574

119872120573

120574119887120574(119911 119905) minus 119872

120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864

sum

120574

120597Ψ

120597119911120574

119887120574(119911 119905)

(16)


minus119872120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

= 0 (17)


sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

119872120573

120574= sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

120597120601120573(119911 119905 119905 minus 119904)

120597119911120574

=120597Ψ

120597119911120574

(18)



condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904))Φ

1(119911

1 119911

119864 119904 119905)

(19)



120572= 0 and

Φ1(119911

1 119911

119864 119904 119905)

= expint

119905

119904

(sum

120572

1198770

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

(20)



0we obtain thatΦ



(20) we have that

120597Φ1

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

1198770

120572(120601

120572(119911 119905 0) minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573

120597120601120573(119911 119905 119905 minus 120591)

120597119905

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1

+ sum

120573

120597Φ1

120597120601120573(119887

120573(120601 (119911 119905 119905 minus 120591)) +

120597120601120573(119911 119906 119905 minus 120591)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

119872120573

120574119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

120597120601120573

120597119911120574

119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120574

120597Φ1

120597119911120574

119887120574(119911 119905)

(21)





119895V120572






prod

120572

(119909120572(119905))

V120572

= 119862 (22)




119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572



sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)


Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)


Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883



119864) (119883

1 119883

119873)


= prod119873

119896=1119910120575120572

119896

119896


Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)


(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)


119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)


0

119895+

sum120572120575120572

119895119899120572= 119883

119895



120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation


(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)



1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911


Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601



119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)


120573(119911 119905 0) = 119911




1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)





120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1198991 119899



119883119895(119905) = 119883

119895(119904) + sum

120572

120575120572

119895119899120572(119904 119905) (1)


119895119906119895119883




it holds that sum119895119906119895(119883

119895(1199051015840) minus 119883

119895(119905)) = 0






119895120575120572





119895119906119895120575120572

119895= 0 then

sum

119895

119906119895(119883

119895(119905

1015840) minus 119883

119895(119905)) = sum

119895

119906119895(sum

120572

120575120572

119895119899120572(119905

1015840 119905)) = 0 (2)



120573 119905

120573) = 1 and 119899

120572(1199051015840

120573 119905


sum

119895

119906119895120575120572

119895= 0 forall120572 (3)






1 119899

119864 119904 119905) of having

1198991 119899


condition 119883119895(119904)





(iii) 119875(119899120572+ 1 119905 + ℎ 119905) = 119882

120572(119883 119905)ℎ + 119900(ℎ)

(iv) 119875(119899120572 119905 + ℎ 119905) = 1 minus 119882

120572(119883 119905)ℎ + 119900(ℎ)

(v) 119875(119899120572(ℎ) + 119896 119905 + ℎ 119905) = 119900(ℎ) 119896 ge 2




120572(119883 119905)




(1198991 119899

119864 119904 119905)

= sum

120572

119875 ( 119899120572minus 1 119904 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119904 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(4)




119877120572= sum

119896

119903119896

120572119883

119896(119905) (5)






119894ge minus1 and 120575

120572

119894=

minus1 only if 119877120572= 119903

119894

120572119883

119894(119905) (no sum)


ℎ119877120572+ 119900 (ℎ) = ℎsum

119896

119903119896

120572119883

119896(119905) + 119900 (ℎ) (6)


119896+ 120575

120572

119896


119896such that119883

119896+ 120575

120572

119896lt 0 (we are


120572= 0

Assume that 120575120572119896


= 0 for 0 le 119883119896

lt minus120575120572

119896

Letting 119883119896


119903119895

120572119883

119895(119905) = 0 regardless


120572= 0 for 119895 = 119896


= 119903119896

120572119883


statement If 120575120572119896


= 0 also for 119883119896

= 1Hence 119903

119896

120572= 0 as well 119877





Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899

1 119899

119864 119904 119905) (7)


1198770

120572= sum

119896

119903119896

120572119883

119896 (119904)

119865120573

120572= sum

119896

119903119896

120572120575120573

119896

(8)

As a consequence

119877120572= 119877

0

120572+ sum

120573

119865120573

120572119899120573 (9)


119911120573(120597Ψ120597119911


obtain

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

(119911120572minus 1)(119877

0

120572Ψ + sum

120573

119865120573

120572119911120573120597Ψ120597119911

120573) (10)



1 119898

119864 119904 119904) = 1 corresponding

to Ψ(1199111 119911

119864 119904 119904) = 119911

1198981

11199111198982


119898119864


1198981 119898



1 119911





condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904)) (11)


119889119909120573

119889120591= sum

120572

119865120573

120572(119909

120572minus 1) 119909

120573equiv 119887

120573(119909 119904)

119889119904

119889120591= minus1

(12)


introduce119887119864+1


120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

119889120601120573(119911 119905 119905 minus 119904)

119889119905

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (119887120573(120601 (119911 119905 119905 minus 119904)) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

(13)


119872120573

120574=

120597120601120573(119909 120591)

120597119909120574

(14)



sum

120574

119872120573

120574119887120574(119911 119905) = 119887

120573(120601 (119911 119905 120591)) (15)

Thus we get

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

(

119864+1

sum

120574

119872120573

120574119887120574(119911 119905) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (

119864

sum

120574

119872120573

120574119887120574(119911 119905) minus 119872

120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864

sum

120574

120597Ψ

120597119911120574

119887120574(119911 119905)

(16)


minus119872120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

= 0 (17)


sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

119872120573

120574= sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

120597120601120573(119911 119905 119905 minus 119904)

120597119911120574

=120597Ψ

120597119911120574

(18)



condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904))Φ

1(119911

1 119911

119864 119904 119905)

(19)



120572= 0 and

Φ1(119911

1 119911

119864 119904 119905)

= expint

119905

119904

(sum

120572

1198770

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

(20)



0we obtain thatΦ



(20) we have that

120597Φ1

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

1198770

120572(120601

120572(119911 119905 0) minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573

120597120601120573(119911 119905 119905 minus 120591)

120597119905

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1

+ sum

120573

120597Φ1

120597120601120573(119887

120573(120601 (119911 119905 119905 minus 120591)) +

120597120601120573(119911 119906 119905 minus 120591)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

119872120573

120574119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

120597120601120573

120597119911120574

119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120574

120597Φ1

120597119911120574

119887120574(119911 119905)

(21)





119895V120572






prod

120572

(119909120572(119905))

V120572

= 119862 (22)




119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572



sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)


Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)


Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883



119864) (119883

1 119883

119873)


= prod119873

119896=1119910120575120572

119896

119896


Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)


(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)


119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)


0

119895+

sum120572120575120572

119895119899120572= 119883

119895



120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation


(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)



1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911


Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601



119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)


120573(119911 119905 0) = 119911




1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)





120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877120572= sum

119896

119903119896

120572119883

119896(119905) (5)






119894ge minus1 and 120575

120572

119894=

minus1 only if 119877120572= 119903

119894

120572119883

119894(119905) (no sum)


ℎ119877120572+ 119900 (ℎ) = ℎsum

119896

119903119896

120572119883

119896(119905) + 119900 (ℎ) (6)


119896+ 120575

120572

119896


119896such that119883

119896+ 120575

120572

119896lt 0 (we are


120572= 0

Assume that 120575120572119896


= 0 for 0 le 119883119896

lt minus120575120572

119896

Letting 119883119896


119903119895

120572119883

119895(119905) = 0 regardless


120572= 0 for 119895 = 119896


= 119903119896

120572119883


statement If 120575120572119896


= 0 also for 119883119896

= 1Hence 119903

119896

120572= 0 as well 119877





Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899

1 119899

119864 119904 119905) (7)


1198770

120572= sum

119896

119903119896

120572119883

119896 (119904)

119865120573

120572= sum

119896

119903119896

120572120575120573

119896

(8)

As a consequence

119877120572= 119877

0

120572+ sum

120573

119865120573

120572119899120573 (9)


119911120573(120597Ψ120597119911


obtain

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

(119911120572minus 1)(119877

0

120572Ψ + sum

120573

119865120573

120572119911120573120597Ψ120597119911

120573) (10)



1 119898

119864 119904 119904) = 1 corresponding

to Ψ(1199111 119911

119864 119904 119904) = 119911

1198981

11199111198982


119898119864


1198981 119898



1 119911





condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904)) (11)


119889119909120573

119889120591= sum

120572

119865120573

120572(119909

120572minus 1) 119909

120573equiv 119887

120573(119909 119904)

119889119904

119889120591= minus1

(12)


introduce119887119864+1


120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

119889120601120573(119911 119905 119905 minus 119904)

119889119905

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (119887120573(120601 (119911 119905 119905 minus 119904)) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

(13)


119872120573

120574=

120597120601120573(119909 120591)

120597119909120574

(14)



sum

120574

119872120573

120574119887120574(119911 119905) = 119887

120573(120601 (119911 119905 120591)) (15)

Thus we get

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

(

119864+1

sum

120574

119872120573

120574119887120574(119911 119905) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (

119864

sum

120574

119872120573

120574119887120574(119911 119905) minus 119872

120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864

sum

120574

120597Ψ

120597119911120574

119887120574(119911 119905)

(16)


minus119872120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

= 0 (17)


sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

119872120573

120574= sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

120597120601120573(119911 119905 119905 minus 119904)

120597119911120574

=120597Ψ

120597119911120574

(18)



condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904))Φ

1(119911

1 119911

119864 119904 119905)

(19)



120572= 0 and

Φ1(119911

1 119911

119864 119904 119905)

= expint

119905

119904

(sum

120572

1198770

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

(20)



0we obtain thatΦ



(20) we have that

120597Φ1

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

1198770

120572(120601

120572(119911 119905 0) minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573

120597120601120573(119911 119905 119905 minus 120591)

120597119905

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1

+ sum

120573

120597Φ1

120597120601120573(119887

120573(120601 (119911 119905 119905 minus 120591)) +

120597120601120573(119911 119906 119905 minus 120591)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

119872120573

120574119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

120597120601120573

120597119911120574

119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120574

120597Φ1

120597119911120574

119887120574(119911 119905)

(21)





119895V120572






prod

120572

(119909120572(119905))

V120572

= 119862 (22)




119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572



sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)


Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)


Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883



119864) (119883

1 119883

119873)


= prod119873

119896=1119910120575120572

119896

119896


Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)


(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)


119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)


0

119895+

sum120572120575120572

119895119899120572= 119883

119895



120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation


(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)



1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911


Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601



119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)


120573(119911 119905 0) = 119911




1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)





120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

120574

119872120573

120574119887120574(119911 119905) = 119887

120573(120601 (119911 119905 120591)) (15)

Thus we get

120597Ψ

120597119905

10038161003816100381610038161003816100381610038161003816119904

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

(

119864+1

sum

120574

119872120573

120574119887120574(119911 119905) +

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864+1

sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816120601(119909119905minus119904)

times (

119864

sum

120574

119872120573

120574119887120574(119911 119905) minus 119872

120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

=

119864

sum

120574

120597Ψ

120597119911120574

119887120574(119911 119905)

(16)


minus119872120573

119864+1+

120597120601120573(119911 119906 119905 minus 119904)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

= 0 (17)


sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

119872120573

120574= sum

120573

120597Φ0

120597119909120573

1003816100381610038161003816100381610038161003816100381610038161003816119909=120601(119909119905minus119904)

120597120601120573(119911 119905 119905 minus 119904)

120597119911120574

=120597Ψ

120597119911120574

(18)



condition Ψ(1199111 119911

119864 119904 119904) = Φ

0(119911

1 119911

119864) admits the

solution

Ψ (1199111 119911

119864 119904 119905) = Φ

0(120601 (119911 119905 119905 minus 119904))Φ

1(119911

1 119911

119864 119904 119905)

(19)



120572= 0 and

Φ1(119911

1 119911

119864 119904 119905)

= expint

119905

119904

(sum

120572

1198770

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

(20)



0we obtain thatΦ



(20) we have that

120597Φ1

120597119905

10038161003816100381610038161003816100381610038161003816119904

= sum

120572

1198770

120572(120601

120572(119911 119905 0) minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573

120597120601120573(119911 119905 119905 minus 120591)

120597119905

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1

+ sum

120573

120597Φ1

120597120601120573(119887

120573(120601 (119911 119905 119905 minus 120591)) +

120597120601120573(119911 119906 119905 minus 120591)

120597119906

100381610038161003816100381610038161003816100381610038161003816119906=119905

)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

119872120573

120574119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120573

120597Φ1

120597120601120573sum

120574

120597120601120573

120597119911120574

119887120574(119911 119905)

= sum

120572

1198770

120572(119911

120572minus 1)Φ

1+ sum

120574

120597Φ1

120597119911120574

119887120574(119911 119905)

(21)





119895V120572






prod

120572

(119909120572(119905))

V120572

= 119862 (22)




119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572



sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)


Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)


Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883



119864) (119883

1 119883

119873)


= prod119873

119896=1119910120575120572

119896

119896


Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)


(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)


119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)


0

119895+

sum120572120575120572

119895119899120572= 119883

119895



120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation


(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)



1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911


Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601



119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)


120573(119911 119905 0) = 119911




1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)





120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889

119889119905ln119909

120572= sum

120573

119865120572

120573(119909

120573minus 1) (23)

Then

sum

120572

V120572

119889

119889119905ln119909

120572= sum

120573

sum

120572

(V120572119865120572

120573) (119909

120573minus 1) = 0 (24)

Since sum120572(V

120572119865120572



sum

120572

V120572

119889

119889119905ln119909

120572= 0 (25)


Ψ (1199111 119911

119864 119904 119905) = sum

119899120572

(prod

120572

119911119899120572

120572)119875 (119899


119864 119904 119905) (26)


Ψ (1199101 119910

119873 119904 119905) = sum

119883119896

(prod

119896

119910119883119896

119896)119875 (119883

1 119883

119873 119904 119905)

(27)

letting 119899120572 119883



119864) (119883

1 119883

119873)


= prod119873

119896=1119910120575120572

119896

119896


Ψ (1199101 119910

119873 119904 119905) = (prod

119896

1199101198830

119896

119896)Ψ(119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896)Ψ(

119873

prod

119896=1

1199101205751

119896

119896

119873

prod

119896=1

119910120575119864

119896

119896 119904 119905)

(28)


(prod

119896

1199101198830

119896

119896)Ψ (119911

1 119911

119864 119904 119905)

= (prod

119896

1199101198830

119896

119896) sum

119899120572

(prod

120572

(

119873

prod

119896=1

119910120575120572

119896

119896)

119899120572

)119875 (119899120572 119904 119905)

= sum

119899120572

(

119873

prod

119896=1

1199101198830

119896+sum120572120575120572

119896119899120572

119896)119875 (119899

120572 119904 119905)

(29)


119875 (1198831 119883

119873 119904 119905) = sum

119899120572

1015840

119875 (119899120572 119904 119905) (30)


0

119895+

sum120572120575120572

119895119899120572= 119883

119895



120572= 119862 gives

119862 =

119873

prod

119896=1

119910(sum120572V120572120575120572

119896)

119896equiv

119873

prod

119896=1

1199100

119896= 1 (31)

33 Reduced Equation


(cf (9))

1198770

120572= sum

120573

119865120573

120572119898

120573 (32)



1 119911

119864 119904 119904 ) =

Φ0(119911

1 119911


Ψ (1199111 119911

119864 119904 119905) = prod

120573

119891119898120573

120573 (33)

where 119911120573119891120573(119911 119904 119905) = 120601



119889 log (119891120573 (119911 119904 119905))

119889119904

= minussum

120572

119865120573

120572(119904) (119911

120572119891120572(119911 119904 119905) minus 1) 119891

120573(119911 119905 119905) = 1

(34)


120573(119911 119905 0) = 119911




1 119911

119864)

Ψ (119911 119904 119905) = Φ1 (119911 119904 119905)

= prod

120573

(expint

119905

119904

sum

120572

119865120573

120572(120591) (120601

120572(119911 119905 119905 minus 120591) minus 1) 119889120591)

119898120573

(35)





120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120572rarr

119898120572+ 119896V

120572(119896 isin R)


120572and imposing


119891120572= prod

119895

119908120575120572

119895

119895 (36)







]minus1

and

Ψ (119911 119904 119905) = [119890minus119903(119905minus119904)


)]119872

(37)


1 119864 = 2 and 1205751

1= minus120575

2

1= 1 while 119903



0= 119898

1minus 119898

2is the initial






1198891198911

119889119904= (119903

111991111198912

1minus (119903

1+ 119903

2) 119891

1+

1198621199032

1199111

) 1198911 (119911 119905 119905) = 1

(38)


1198911(119911 119904 119905) =

1

1199111

(1199032minus 119903

11199111)119882 minus 119903

2(1 minus 119911

1)

(1199032minus 119903

11199111)119882 minus 119903

1(1 minus 119911

1) (39)

where 119882 = exp(1199032


(1198911(119911 119904 119905))



Ψ (119910 119904 119905) = [(119903

2minus 119903

1119910)119882 minus 119903

2(1 minus 119910)

(1199032minus 119903

1119910)119882 minus 119903

1(1 minus 119910)

]

1198981minus1198982

(40)




1 119911

119864 119905)








119904)sum120573119865120572

120573(119911

120573minus 1)) and

Ψ (1199111 119911

119864 119904 119905) =

119864

prod

120572

119890119898120572(119905minus119904)sum

120573119865120572

120573(119911120573minus1)

= 119890(119905minus119904)sum

1205731198770

120573(119911120573minus1)

(41)







= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= prod119895119908

120575120572

119895

119895we obtain

119891120572119891

120572= sum

119896120575120572

119896119896119908


sum

119896

120575120572

119896(

119896

119908119896

+ sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1)) = 0

119908119896 (119911 119905 119905) = 1

(42)


119896= minus119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896(119911 119905 119905) = 1

(43)




Since sum120572120575120572

119895119898

120572= 119883

119895(119904) we obtain Ψ(119911

1 119911

119864 119904 119905) =

prod119895119908

119883119895(119904)



for Ψ(1199111 119911

119864 119904 119905)


119896


119896will


119896= 119908

119896sum

120573

119903119896

120573(119911

120573(prod

119895

119908120575120573

119895

119895) minus 1) 119908

119896 (119911 0) = 1

(44)







119908(0)

119896(ℎ) = 1

119908(1)

119896(ℎ) = 1 + int

ℎ

0

119908(0)

119896(119905)sum

120573

119903119896

120573(119911

120573prod

119895

(119908(0)

119895(119905))

120575120573

119895

minus 1)119889119905

= 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1)

119908(119899)

119896(ℎ) = 1 + int

ℎ

0

119889119905[

[

119908(119899minus1)

119896(119905)

timessum

120573

119903119896

120573(119911

120573prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

minus 1)]

]119899minus1

(45)


119905


119896(119905) in (44)


(a) 119908(119899)


1 119911

119864 where


119899119864

119864is 119874(119905

119901) with 119901 = 119899

1+


(b) 119908(119899)

119896(1 1 119905) = 1

(c) Δ(119899)

119896(ℎ) = 119908

(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ) = 119874(ℎ

119899)





(119899minus1)

119896(119905)prod

119895(119908

(119899minus1)

119895(119905))

120575120573

119895 ]119899minus1


1 119911



120572


119908(119899)

119896(119905) = 1 + sum

120573

119903119896

120573119911120573int

ℎ

0

[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

119908(119899minus1)

119896(119905) 119889119905

(46)


1 119911






1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 119911


order119874(ℎ119901+1


of 119908(119899minus1)


(b) Letting 1199111= 119911



trivially 119908(119899)



119908(119899)

119896(ℎ) minus 119908

(119899minus1)

119896(ℎ)

= sum

120573

119903119896

120573119911120573

times int

ℎ

0

([(119908(119899minus2)

119896(119905) + Δ

(119899minus1)

119896(119905))

times prod

119895

(119908(119899minus2)

119895(119905) + Δ

(119899minus1)

119895(119905))

120575120573

119895

]

119899minus1

minus[

[

119908(119899minus2)

119896(119905)prod

119895

(119908(119899minus2)

119895(119905))

120575120573

119895]

]119899minus2

)119889119905

minus (sum

120573

119903119896

120573)int

ℎ

0

(119908(119899minus1)

119896(119905) minus 119908

(119899minus2)

119896(119905)) 119889119905

(47)




119896(ℎ) = 119874(ℎ

119899)


(119899minus1)




119899)



119903119896

120573119911120573[

[

119908(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895]

]119899minus1

(48)





120573= 0 for 119896 = 119895 Hence

119903119896

120573119911120573119908

(119899minus1)

119896(119905)prod

119895

(119908(119899minus1)

119895(119905))

120575120573

119895

=

0 119895 = 119896

119903119896

120573119911120573prod

119895 = 119896

(119908(119899minus1)

119895(119905))

120575120573

119895

119895 = 119896

(49)

However 120575120573119895



1 119911

119864 119905) = prod

119895119908

119883119895(0)



119908119896(ℎ) = 1 + ℎsum

120573

119903119896

120573(119911

120573minus 1) +

ℎ2

2(sum

120573

119903119896

120573(119911

120573minus 1))

times (sum

120574

119903119896

120574(119911

120574minus 1))

+ℎ2

2sum

120573

119903119896

120573119911120573sum

120574

119865120573

120574(119911

120574minus 1)

(50)









120574on each 119908



119875119896(ℎ 0) = 1 minus ℎ(sum

120573

119903119896

120573) +

ℎ2

2(sum

120573

119903119896

120573)

2

(51)


119875119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus ℎ

2sum

120572

119903119896

120572minus

ℎ2

2sum

120572

119865120573

120572 (52)


119875119896(ℎ 119868119868 120573 120572) =

ℎ2

2119903119896

120573(119903

119896

120572+ 119865

120573

120572) (53)



sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

120572

(119875119896(ℎ 119868119868 120573 120572)) + 119875

119896(ℎ 119868 120573) = 119903

119896

120573ℎ minus

ℎ2

2sum

120572

119903119896

120572

= 119875119896(ℎ 119868 120573)

(54)


119875119896(ℎ 119868119868 120573 120572) = 119875

119896(ℎ 119868 120573)

ℎ

2

(119903120572

119896+ 119865

120573

120572)

(1 minus (ℎ2)sum120572119903119896120572)+ 119900 (ℎ

2)

(55)








119896(ℎ 120572120573) defined above


(ℎ2)(sum120572(119903

120572

119896+ 119865

120573

120572)(1 minus (ℎ2)sum

120572119903119896

120572)))





120572= sum

119895119903119896

120572119883

119895 the matrix 119903





120575120572

119895=

minus1 120572 = 119895

+1 120572 = 119895 minus 1(56)


120572]119879

= (1199030119872 0 0) and

119865120573

120572=

minus119903120572 120573 = 120572

+119903120572 120573 = 120572 minus 1

0 otherwise(57)


0

120572=

sum120573119865120573

120572119898




119895)119898119895 =

prod119895(119891

minus119872




119895(minus119903

119895(119911

119895119891119895minus 1) + 119903

119895+1(119911

119895+1119891119895+1

minus 1))

119895 = 0 119864 minus 2

(58)

and 119891119864minus1

= 119891119864minus1


(119911119864minus1

119891119864minus1


119895(0) = 1 Let 120579

119895= log(119891

119895) and

120595119895=

119864minus1

sum

120573=119895

120579120573 119882

119895= exp (minus120595

119895) (59)

Note that 119882119895119891119895

= 119882119895

sdot exp(120579119895) = 119882



119895(119911

119895exp (120579

119895) minus 1) + 119903

119895+1(119911

119895+1exp (120579

119895+1) minus 1)

119895 lt 119864 minus 1

119895= minus119903

119895(119911

119895exp (120579

119895) minus 1) = 120579

119895+

119895+1 119895 lt 119864 minus 1

120579119864minus1

= 119864minus1


(119911119864minus1

exp (120579119864minus1

) minus 1)

(60)



0 and we formulate


119895+ 119903

119895119882

119895= 119903

119895119911119895119882

119895+1 119882

119895 (0) = 1 119895 = 0 119864 minus 2

119864minus1

= 119903119864minus1

(119911119864minus1


) 119882119864minus1

(0) = 1

(61)




form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












form

119882119895(119905) = 119890

minus119903119895119905+ 119911

119895119903119895int

119905

0

119890119903119895(119904minus119905)

119882119895+1

(119904) 119889119904 119895 = 0 119864 minus 1

(62)


1198820(119905) = (

119864minus1

prod

119898=0

119911119898)(1 minus

119864minus1

sum

119896=0

exp (minus119903119896119905)

119864minus1

prod

119895=0119895 = 119896

119903119895

119903119895minus 119903

119896

)

+

119864

sum

119896=1


119895=0119911119895

119911119864minus119896

119903119864minus119896

119864minus119896

sum

119898=0

119903119898exp (minus119903

119898119905)

119864minus119896

prod

119897=0119897 =119898

119903119897

119903119897minus 119903

119898

(63)



1198820(119905) =

prod119864minus1

119898=0119911119898

(119864 minus 1)int

119903119905

0

exp (minus119904) 119904119864minus1

119889119904

+

119864

sum

119896=1


119898=0119911119898

119911119864minus119896

(119903119905)119864minus119896 exp (minus119903119905)

(119864 minus 119896)

(64)



120572= 119902

120572119883

120572 where 120572 = 0 119864 minus 1 We


119864leaves the


0 119877

0 119876

1 119877

1 119876

119864minus1 and 119877

119864minus1 Then ((119876 119877)

0)119879

=

(1199020119872 119903

0119872 0 0) We have

1205752120572

120572= minus1 (death) 120572 lt 119864

1205752120572+1

120572= minus1

1205752120572+1

120572+1= +1


119865120573

2120572=

minus119902120572 120573 = 2120572

minus119902120572 120573 = 2120572 + 1

119902120572 120573 = 2120572 minus 1

119865120573

2120572+1=

minus119903120572 120573 = 2120572 + 1

minus119903120572 120573 = 2120572

119903120572 120573 = 2120572 minus 1

120572 lt 119864

(65)


120572and the 119911

120572in pairs (119892 119891) and

(119911 119910) as (1198920 119891

0 119892

119864minus1 119891

119864minus1) and (119911

0 119910

0 119911

119864minus1 119910

119864minus1)


119864= 119903


119867120572(119892

120572 119891

120572) = 119902

120572(119911

120572119892120572minus 1) + 119903

120572(119910

120572119891120572minus 1) (66)


119892120572= minus119867

120572(119892

120572 119891

120572) 119892

120572

119891120572= (minus119867

120572(119892

120572 119891

120572) + 119867

120572+1(119892

120572+1 119891

120572+1)) 119891

120572

120572 = 0 119864 minus 2

119892119864minus1

119892119864minus1

=

119891119864minus1

119891119864minus1


(119892119864minus1

119891119864minus1

)

(67)


120572(0)


119879

120572=


120572= log119892

120572

and 120601120572

= log119891120572 changing 119867



120601120572= 120579

120572minus 120579

120572+1997904rArr 120601

120572= 120579

120572minus 120579

120572+1 120572 = 0 119864 minus 2

120579120572= minus119867

120572(120579

120572 120579

120572minus 120579

120572+1) 120572 = 0 119864 minus 2

120601119864minus1

= 120579119864minus1


(120579119864minus1

120579119864minus1

)

(68)


= minus119872(1 0 0 0) satisfies 1198770

= 119865119898 FinallyΨ(119911 119910 119905) = 119892

minus119872


0(119905)

Let119882120572= 119890

minus120579120572 = 119892

minus1



119882 read

120572+ (119902

120572+ 119903

120572)119882

120572= 119902

120572119911120572+ 119903

120572119910120572119882

120572+1 119882

120572 (0) = 1

(69)

with solution

119882120572(119905) = 119890

minus(119902120572+119903120572)119905

+ int

119905

0

119890minus(119902120572+119903120572)(119905minus119904)

(119902120572119911120572+ 119903

120572119910120572119882

120572+1(119904)) 119889119904

(70)



1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198820 (119905) =

119864minus1

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

minus

119864minus1

sum

119898=0

119903119898119910119898119890minus(119902119898+119903119898)119905

119902119898

+ 119903119898

times

119864minus1

prod

119896=0119896 =119898

119903119896119910119896

119902119896minus 119902

119898+ 119903

119896minus 119903

119898

+

119864

sum

119896=1

1

[119903119910]119864minus119896

times [

[

119864minus119896

prod

119898=0

119903119898119910119898

119902119898

+ 119903119898

[119902119911]119864minus119896

+

119864minus119896

sum

119898=0

(1 minus[119902119911]

119864minus119896

119902119898

+ 119903119898

)

times 119903119898119910119898119890minus(119902119898+119903119898)119905

times

119864minus119896

prod

119895=0119895 =119898

119903119895119910119895

119902119895minus 119902

119898+ 119903

119895minus 119903

119898

]

]

(71)




1198820(119905) =

prod119864minus1

119898=0119903119910

119898

(119864 minus 1)int

119905

0

119890minus(119902+119903)119909

119909119864minus1

119889119909

+

119864minus1

sum

119896=0

prod119896

119898=0119903119910

119898

119903119910119896119896

[119905119896119890minus(119902+119903)119905

+ 119902119911119896int

119905

0

119890minus(119902+119903)119909

119909119896119889119909]

(72)



119911119896and 119910


119894(119905)


119904

119898=0120582119898 119904 =

0 1 2 119876(119904119898) = prod119904

119896=0119896 =119898(120582

119896minus 120582

119898+ 120583

119896minus 120583

119898) 119904 = 1 2 3


119904


0) while for 119894 = 1 2 3 we have 119901

119894(119905) =

119875(119894 minus 1)sum119894

119898=0(exp(minus119905119877

119898)119876(119894 119898))












Appendices




A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A1 Proof of Lemma 5





119875 (119899 = 0 119905 + ℎ) = 119875 (119899 = 0 119905) (1 minus sum

120572

119882120572(119883 119905) ℎ) + 119900 (ℎ)

(A1)


ℎ= minus119875 (119899 = 0 119905)

times (sum

120572

119882120572 (119883 119905)) +

119900 (ℎ)

ℎ

(A2)

and finally

limℎrarr0

119875 (119899 = 0 119905 + ℎ) minus 119875 (119899 = 0 119905)

ℎ

= (119899 = 0 119905) = minus119875 (119899 = 0 119905) (sum

120572

119882120572(119883 119905))

(A3)


119875 (119899 = 0 119905 + 120591) = 119890minus120591sum120572119882120572(119883119905)

(A4)



119875 (1198991 119899

119864 119905 + ℎ)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905)) ℎ

+ 119875 (1198991 119899

119864 119905) (1 minus sum

120572

119882120572(119883 (119899

1 119899

119864 119905)) ℎ)

+ 119900 (ℎ)

(A5)


(1198991 119899

119864 119905)

= sum

120572

119875 ( 119899120572minus 1 119905)119882

120572(119883 ( 119899

120572minus 1 119905))

minus 119875 (1198991 119899

119864 119905) (sum

120572

119882120572(119883 (119899

1 119899

119864 119905)))

(A6)



119896(119904) =


119866119896(119904)=

1

119904 + 119903119896

(1 + 119911119896119903119896119866119896+1

(119904)) 119866119864(119904)=

1

119904 0 le 119896 lt 119864

(B1)


1198660(119904) =

1

119904

119864minus1

prod

119898=0

(119911119898119903119898

119904 + 119903119898

) +

119864

sum

119896=1

(1

119911119864minus119896

119903119864minus119896

119864minus119896

prod

119898=0

(119911119898119903119898

119904 + 119903119898

))

(B2)



0(119904)



119866119896(119904) =

1

119904 + 119902119896+ 119903

119896

(1 +119902119896119911119896

119904+ 119903

119896119910119896119866119896+1

(119904))

119866119864(119904) =

1

119904

(B3)

with solution

1198660(119904) =

1

119904

119864minus1

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

+

119864

sum

119896=1

[119904 + 119902119911

119904119903119910]

119864minus119896

119864minus119896

prod

119898=0

119903119898119910119898

119904 + 119902119898

+ 119903119898

(B4)




Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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