A Mechanical Quadrature Method for Solving Delay Volterra ...downloads.hindawi.com/journals/complexity/2019/4813802.pdf · A Mechanical Quadrature Method for Solving Delay Volterra

Research ArticleA Mechanical Quadrature Method for Solving Delay VolterraIntegral Equation with Weakly Singular Kernels

Li Zhang Jin Huang Yubin Pan and Xiaoxia Wen

School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 611731 China

Correspondence should be addressed to Jin Huang huangjin12345163com

Received 15 February 2019 Accepted 21 May 2019 Published 16 June 2019

Academic Editor Danilo Comminiello

Copyright copy 2019 Li Zhang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this work a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterraintegral equation with proportional delays An improved Gronwall inequality is testified and adopted to prove the existence anduniqueness of the solution of the original equation Then we study the convergence and the error estimation of the mechanicalquadrature method Moreover Richardson extrapolation based on the asymptotic expansion of error not only possesses a highaccuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm Numerical experimentsdemonstrate the efficiency and applicability of the proposed method

1 Introduction

In recent years Volterra integral equation with delay hasreceived a considerable amount of attention This paperconsiders the following weakly singular Volterra integralequations with proportional delays

119906 (119905) = 119892 (119905) + (119868119906) (119905) + (119868120579119906) (119905) 119905 isin 119868 = [0 119879] (1)

with

(119868119906) (119905) fl int1199050

1198701 (119905 119904) 119906 (119904) 119889119904(119868120579119906) (119905) fl int120579(119905)

01198702 (119905 119904) 119906 (119904) 119889119904

(2)

and

120579 (119905) fl 119902119905 0 lt 119902 lt 11198701 (119905 119904) fl 1199041205721198961 (119905 119904) 1198702 (119905 119904) fl 1199041205741198962 (119905 119904)

minus 1 lt 120572 lt 0 minus1 lt 120574 le 0(3)

where 1198961(119905 119904) and 1198962(119905 119904) are known continuous functionsdefined on the domains 119863 fl (119905 119904) 0 le 119904 le 119905 le 119879 and

119863120579 fl (119905 119904) 0 le 119904 le 120579(119905) 119905 isin 119868 = [0 119879] respec-tively 119892(119905) is a known function and 119906(119905) is an unknownfunction In practice the delay arguments are consistent withthe real phenomena which make the models more realisticfor simulationDelay integral equation and partial differentialequation have been widely used in many population growthand relevant phenomena in mathematical biology [1ndash3]

Numerical algorithms for implementing delay modelsshould be designed specially according to the nature of theequations There aremany numerical techniques for the delaydifferential equations [4ndash8] and integral-differential equa-tions [9ndash11] Simultaneously vast researchers also focusedtheir interests on the numerical techniques of delay integralequations with continuous kernels such as least squaresapproximation method [12] spectral method [13] Bernoulliwavelet method [14] and collocation method [15] Xie etal in [16] handled the Volterra integral equation with thedelay function vanishing at the initial point in the giveninterval they found that the iterated collocation solutionpossessed local superconvergence at the mesh points In [17]the authors adopted multistep method based on Hermitecollocation method and it turned out that the numericalmethod had uniform order 2119898+2119903with 119898 collocation pointsand 119903 previous time steps In [18] an hp-spectral collocationmethod was used for nonlinear Volterra integral equationswith vanishing variable delays

HindawiComplexityVolume 2019 Article ID 4813802 12 pageshttpsdoiorg10115520194813802

2 Complexity

There are a few researches on the delay integral equationwith weakly singular kernels such as [19] In this paperwe concentrate on (1) whose upper limit of integral is adelay function and the integral kernels are weakly singularfunctions which increase the computational complexity andthe theoretical difficulty To the best of our knowledge thereare no studies in (1) by the mechanical quadrature methodin recent years A further advantage of this method for (1)is that the error has an asymptotic expansion Thus we canimprove the accuracy order of approximation Simultane-ously the theoretical analysis is complete and the calculationis simplified

In this paper we firstly deduce the improved Gronwallinequality the existence and uniqueness of the solution for (1)are testified via the improved Gronwall inequality Then theequation is approximated by employing the floor techniqueto the delay argument 120579(119905) and by adopting the quadratureformula [20] to the weakly integrals Next the approximateequation for (1) is constructed by combining the mechanicalquadrature method and the interpolation technique thenwe use the iterative method for solving the approximateequationThe existence and uniqueness of the solution for theapproximate equation are testified via the discrete Gronwallinequality Finally we prove that the convergent order is119874(ℎ2+min(120572120574)) In order to achieve a higher accuracy order119874(ℎ2) the Richardson-ℎ2+min(120572120574) extrapolation based onthe asymptotic expansion of error is adopted moreover aposterior error estimate is realized conveniently

The layouts of this paper are as follows In Section 2 weprove the existence and uniqueness of the solution for (1)In Section 3 we introduce the quadrature method iterativemethod and interpolation technique In Section 4 the exis-tence and uniqueness of the solution for the approximateequation are discussed In Section 5 the convergence andthe error estimation are obtained to ensure the reliabilityof the method In Section 6 the asymptotic expansion ofthe error is achieved a higher accuracy order is realizedby extrapolation and a posterior error estimate is derivedIn Section 7 some numerical examples are demonstrated toillustrate the theoretical results Some concluding remarks areprovided in Section 8

2 The Existence and Uniqueness ofthe Solution for the Original Equation

In this section we will verify the existence and uniquenessof the solution for (1) We first prove the improved Gronwallinequality

Lemma 1 Suppose that 119892(119905) ℎ(119905) and 119906(119905) are nonnegativeintegrable functions 119905 isin [0 119879] 0 lt 119902 lt 1 and 119860 ge 0 basedon the inequality

119906 (119905) le 119860 + int1199050

119892 (119904) 119906 (119904) 119889119904 + int1199021199050

ℎ (119904) 119906 (119904) 119889119904 (4)

we have

119906 (119905) le 119860119890int1199050 (119892(119904)+ℎ(119904))119889119904 (5)

Proof Due to the fact that 0 lt 119902 lt 1 119902119905 lt 119905 and119906 (119905) le 119860 + int119905

0119892 (119904) 119906 (119904) 119889119904 + int119905

0ℎ (119904) 119906 (119904) 119889119904

le 119860 + int1199050

(119892 (119904) + ℎ (119904)) 119906 (119904) 119889119904(6)

Let 119867(119904) = 119892(119904) + ℎ(119904) we can deduce

119867 (119905) 119906 (119905)119860 + int1199050 119867 (119904) 119906 (119904) 119889119904 le 119867 (119905) (7)

Integrate on both sides

ln(119860 + int1199050

119867 (119904) 119906 (119904) 119889119904) minus ln119860 le int1199050

119867 (119904) 119889119904 (8)

and then

119860 + int1199050

119867 (119904) 119906 (119904) 119889119904 le 119860119890int1199050 119867(119904)119889119904 (9)

namely

119906 (119905) le 119860119890int1199050 119867(119904)119889119904 (10)

The proof of the Lemma 1 is completed

Theorem 2 Assume that 119896119894(119905 119904) (119894 = 1 2) are knowncontinuous functions defined on the domains 119863 fl (119905 119904) 0 le119904 le 119905 le 119879 and 119863120579 fl (119905 119904) 0 le 119904 le 120579(119905) 119905 isin 119868 = [0 119879]respectively then the solution of (1) is existent uniquely

Proof We construct the sequence 119906119896(119905) (119896 isin N) satisfying1199060 (119905) = 119906 (0) = 119892 (119905) 119906119896 (119905) = 119892 (119905) + int119905

01198701 (119905 119904) 119906119896minus1 (119904) 119889119904

+ int1199021199050

1198702 (119905 119904) 119906119896minus1 (119904) 119889119904(11)

where 1198701(119905 119904) and 1198702(119905 119904) are defined in (3) with 119896 isin N119896119894(119905 119904) (119894 = 1 2) are continuous functions then there existsa constant 119862 such that |119896119894(119905 119904)| le 119862 We have

10038161003816100381610038161199062 (119905) minus 1199061 (119905)1003816100381610038161003816le 119862 int1199050

119904120572 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 119889119904+ 119862 int1199021199050

119904120574 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 119889119904le 119862 int1199050

[119904120572 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816] 119889119904le 119862119886 int119905

0(119904120572 + 119904120574) 119889119904 le 2119862119886 int119905

0119904120583119889119904 = 2119862119886 119905120583+1120583 + 1

(12)

Complexity 3

with 119886 = max0le119905le119879|1199061(119905) minus 1199060(119905)| 120583 = min120572 120574 Now we candeduce

1003816100381610038161003816119906119896 (119905) minus 119906119896minus1 (119905)1003816100381610038161003816 le 119886 (2119862)119896minus1(119896 minus 1) (120583 + 1)119896minus1 119905

(119896minus1)(120583+1) (13)

By means of the mathematical induction when 119899 = 119896 + 1 weobtain

1003816100381610038161003816119906119896+1 (119905) minus 119906119896 (119905)1003816100381610038161003816 le 119862 int1199050

119904120572 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904+ 119862 int1199021199050

119904120574 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904le 119862 int1199050

[119904120572 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816+ 119904120574 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816] 119889119904 le 2119862 int119905

0119904120583 1003816100381610038161003816119906119896 (119904)

minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904 le 119886 (2119862)119896119896 (120583 + 1)119896 119905

119896(120583+1)

(14)

Next we prove that 119906119896(119905) is the basic sequence in 119862[0 119879]in fact

1003816100381610038161003816119906119896 (119905) minus 119906119896+119898 (119905)1003816100381610038161003816le 1003816100381610038161003816119906119896+1 (119905) minus 119906119896 (119905)1003816100381610038161003816 + 1003816100381610038161003816119906119896+2 (119905) minus 119906119896+1 (119905)1003816100381610038161003816 + sdot sdot sdot

+ 1003816100381610038161003816119906119896+119898 (119905) minus 119906119896+119898minus1 (119905)1003816100381610038161003816le 119886 (2119862)119896

119896 (120583 + 1)119896 119905119896(120583+1) + sdot sdot sdot

+ 119886 (2119862)119896+119898minus1(119896 + 119898 minus 1) (120583 + 1)119896+119898minus1 119905

(119896+119898minus1)(120583+1)

(15)

For sufficiently small 120598 gt 0 there exists a positive integer 119873such that when 119899 gt 119873 and any 119898 gt 0 we have

1003816100381610038161003816119906119899 (119905) minus 119906119899+119898 (119905)1003816100381610038161003816 le 120598 (16)

According to Cauchyrsquos test for convergence the sequence119906119899(119905) (119899 isin N) is convergent uniformly to 119906(119905) which is thesolution of (1)

Suppose that both 119906 and V are the solutions of (1) let |119908| =|119906 minus V| then we get

|119908 (119905)|= 10038161003816100381610038161003816100381610038161003816int119905

01199041205721198961 (119905 119904) 119908 (119904) 119889119904 + int119902119905

01199041205741198962 (119905 119904) 119908 (119904) 11988911990410038161003816100381610038161003816100381610038161003816

le int1199050

119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904+ int1199021199050

119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904

le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904(17)

Next we verify that 119904120572|1198961(119905 119904)| + s120574|1198962(119905 119904)| is integrableint1199050

119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 119889119904 le 119862 int1199050

(119904120572 + 119904120574) 119889119904le 119862 ( 1199051+1205721 + 120572 + 1199051+1205741 + 120574)

(18)

According to Lemma 1 we can derive that |119908(119905)| = 0 thesolution is unique

3 The Quadrature Method andthe Iterative Algorithm

Let the delay function 120579(119905) = 119905 minus 120591(119905) ge 0 119905 isin [0 119879] satisfythe following conditions [21]

(1) 120591(0) = 0 and 120591(119905) gt 0with 119905 isin (0 119879] (vanishing delay)(2) 120579(119905) le 1199021119905 on 119868 = [0 119879] for some 1199021 isin (0 1) and1205791015840(119905) ge 1199020 gt 0 119902119894 (119894 = 0 1) are constants(3) 120579(119905) isin 1198621(119868)For 0 lt 119902 lt 1 the special case is 120591(119905) = (1 minus 119902)119905 we get120579(119905) = 119905 minus 120591(119905) = 119902119905 (1) is a weakly integral equation with

vanishing delay In order to solve (1) the modified trapezoidquadrature formula is considered to deal with the integrals(119868119906)(119905) and (119868120579119906)(119905)

It is challenging because the integral upper limit is a delayfunction 120579(119905) and the integral kernels are weakly singular atthe endpoint

Lemma 3 (see [22]) Let 119868(119866) = int119887119886 119866(119909)119889119909 = int119887119886 (119909 minus119886)120572119892(119909)119889119909 minus1 lt 120572 lt 0 and 119892(119909) isin 1198622119898[119886 119887] then themodified trapezoidal rule is

119876119873 (119866) = minusℎ(1+120572)119892 (119886) 120577 (minus120572) + ℎ119873minus1sum119894=1

119866 (119909119894) + ℎ2 119866 (119887) (19)

where 120577 is the zeta function Further 119868(119866) has the followingasymptotic expansion

119864119873 (119866) = 119876119873 (119866) minus 119868 (119866)= 119898minus1sum119895=1

119861(2119895)(2119895)119866(2119895minus1) (119887) ℎ2119895

+ 2119898minus1sum119895=1

(minus1)119895 120577 (minus120572 minus 119895) 119892(119895) (119886) ℎ119895+120572+1119895

+ 119874 (ℎ2119898)

(20)

where 119861(2119895) 119895 = 1 119898 minus 1 are the Bernoulli numbers

4 Complexity

Lemma 4 Let 119906 isin 1198623[0 1] and 119911 = 120573119909 + (1 minus 120573)119910 with120573 isin [0 1] 119909 119910 isin [0 119879] then there is

119906 (119911) = 120573119906 (119909) + (1 minus 120573) 119906 (119910)minus 120573 (1 minus 120573)

2 (119909 minus 119910)2 11990610158401015840 (119911) + 119874 ((119909 minus 119910)3) (21)

Proof TheTaylor expansion of function 119906(119909) at the point 119911 is

119906 (119909) = 119906 (120573119909 + (1 minus 120573) 119909)= 119906 (120573119909 + (1 minus 120573) 119910 + (1 minus 120573) (119909 minus 119910))= 119906 (119911 + (1 minus 120573) (119909 minus 119910))= 119906 (119911) + (1 minus 120573) (119909 minus 119910) 1199061015840 (119911)

+ (1 minus 120573)22 (119909 minus 119910)2 11990610158401015840 (119911) + 119874 ((119909 minus 119910)3)

(22)

Similarly the Taylor expansion of function 119906(119910) at the point119911 is

119906 (119910) = 119906 (119911 minus 120573 (119909 minus 119910))= 119906 (119911) minus 120573 (119909 minus 119910) 1199061015840 (119911) + 12057322 (119909 minus 119910)2 11990610158401015840 (119911)

+ 119874 ((119909 minus 119910)3) (23)

Combining (22) with (23) we derive (21)

Now the concrete approximate scheme of (1) will beconstructed on the basis of the above lemmas We divide119868 isin [0 119879] into 119873 subintervals with the uniform step sizeℎ = 119879119873 119905119894 = 119894ℎ 119894 = 0 119873 Let 119905 be replaced by 119905119894 in(1) that is to say

119906 (119905119894) = 119892 (119905119894) + int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904+ int1199021199051198940

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + int119905119894

01199041205721198961 (119905119894 119904) 119906 (119904) 119889119904

+ int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904+ int119902119905119894119905[119902119894]

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + 1198681 + 1198682 + 1198683

(24)

where [119902119894] denotes the maximum integer less than 119902119894 It isobvious that 119906(119902119905119894) 0 lt 119902 lt 1 are not the node values andwe can turn 119906(119902119905119894) to the adjacent points 119906(119905[119902119894]) and 119906(119905[119902119894]+1)by the linear interpolation approximation with 119905[119902119894] le 119902119905119894 le119905[119902119894]+1 Then we have that 119902119905119894 = 120573119894119905[119902119894] + (1 minus 120573119894)119905[119902119894]+1 areestablished for 120573119894 isin [0 1] with 120573119894 = 1 + [119902119894] minus 119902119894 and

119906 (119902119905119894) = 120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1) (25)

Now we structure the quadrature algorithm by Lemma 3and the trapezoidal quadrature rule we obtain

1198681 = int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904 = ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906 (119905119896)+ ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906 (119905119894) minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 119906 (1199050)+ 1198641119894 (1199041205721198961 (119905 119904))

(26)

1198682 = int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904 = ℎ[119902119894]minus1sum119896=1

1199051205741198961198962 (119905119894 119905119896)sdot 119906 (119905119896) + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906 (119905[119902119894]) minus 120577 (minus120574)sdot ℎ1+1205741198962 (119905119894 1199050) 119906 (1199050) + 1198642119894 (1199041205741198962 (119905 119904))

(27)

1198683 = int119902119905119894119905[119902119894]

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906 (119905[119902119894]) + (119902119905119894)120574sdot 1198962 (119905119894 119902119905119894) 119906 (119902119905119894)] + 1198643119894(1199041205741198962 (119905 119904)= 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906 (119905[119902119894]) + (119902119905119894)120574sdot 1198962 (119905119894 119902119905119894) (120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1))]+ 1198643119894 (1199041205741198962 (119905 119904))

(28)

where 1198641119894 1198642119894 and 1198643119894 are error functions which arerespectively equal to

1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572 + 119900 (ℎ4+120572)= 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574 + 119900 (ℎ4+120574)= 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)

sdot (120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1)) + (119902119905119894 minus 119905[119902119894])212sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)

Complexity 5

sdot ℎ2 + (119902119905119894 minus 119905[119902119894])2 minus ℎ212

sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)sdot ℎ2 + 119874 (ℎ3)

(29)

where

1198791 (119905119894) = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120572 minus 1) 1198792 (119905119894) = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120574 minus 1) 1198793 (119905119894) = minus120573 (1 minus 120573)

2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)sdot (120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1)) + 112sdot int119902119905119894119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904

(30)

The discrete forms of (1) are obtained

119906 (1199050) = 119892 (1199050) (31)

119906 (119905119894) = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906 (119905119896) + ℎ2 119905120572119894 1198961 (119905119894 119905119894)sdot 119906 (119905119894) minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 119906 (1199050)+ ℎ[119902119894]minus1sum119896=1

1199051205741198961198962 (119905119894 119905119896) 119906 (119905119896) + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894])sdot 119906 (119905[119902119894]) minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 119906 (1199050)+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906 (119905[119902119894]) + (119902119905119894)120574sdot 1198962 (119905119894 119902119905119894) (120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1))]+ 1198641119894 + 1198642119894 + 1198643119894 119894 = 1 119873

(32)

Let 119906119894 be the approximate solution of 119906(119905119894) and ignore theerror function (32) becomes

1199060 = 119892 (1199050) 119906119894 = 119892 (119905119894) + ℎ 119894minus1sum

119896=11199051205721198961198961 (119905119894 119905119896) 119906119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894

minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 1199060 + ℎ[119902119894]minus1sum119896=1

1199051205741198961198962 (119905119894 119905119896) 119906119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060

+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(33)

The iterative algorithm is built to solve (33)

Iterative Algorithm

Step 1 Take sufficiently small 120598 gt 0 and set 1199060 = 119892(1199050) 119894 fl 1Step 2 Let 1199060119894 = 119906119894minus1 119898 fl 0 then compute 119906119898+1119894 (119894 le 119873) asfollows

119906119898+1119894 = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906119898119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119898119894


1199051205741198961198962 (119905119894 119905119896) 119906119898119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906119898[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906119898[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906119898[119902119894] + (1 minus 120573119894) 119906119898[119902119894]+1]

(34)

Step 3 If |119906119898+1119894 minus 119906119898119894 | le 120598 set 119906119894 fl 119906119898+1119894 and 119894 = 119894 + 1 andreturn to Step 2 else let 119898 = 119898 + 1 and return to Step 2

4 The Existence and Uniqueness ofthe Solution for the Approximate Equation

Now we prove the existence and uniqueness of the solutionfor the approximate equation We first introduce the follow-ing lemma

Lemma 5 (see [23]) Suppose that the sequence 119908119899 119899 =0 119873 satisfies

10038161003816100381610038161199081198991003816100381610038161003816 le ℎ 119899sum119896=1

119861119896 10038161003816100381610038161199081198961003816100381610038161003816 + 119860 0 le 119899 le 119873 (35)

where 119860 and 119861119896 (119896 = 1 119873) are nonnegative constants Letℎmax1le119896le119873119861119896 le 12 with ℎ = 1119873 then we can derive

max0le119899le119873

10038161003816100381610038161199081198991003816100381610038161003816 le 119860 exp(2ℎ 119873sum119896=1

119861119896) (36)

Theorem 6 Assume that ℎ is sufficiently small then the solu-tion of (34) is existent uniquely and the algorithm converges ata geometrical rate

Proof From the nature of the delay function 120579(119905) we discussthe existence and uniqueness of the solution for the approxi-mate equation under two situations

6 Complexity

First we prove that the solution of (34) is existent undertwo situations

(1) One situation is [119902119894] +1 = 119894 that is when 119894 lt 1(1minus119902)we can easily obtain

10038161003816100381610038161003816119906119898+1119894 minus 119906119898119894 10038161003816100381610038161003816 le ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816+ 119902119905119894 minus 119905[119902119894]2 (119902119905119894)120574 10038161003816100381610038161198962 (119905119894 119902119905119894)1003816100381610038161003816 (1 minus 120573119894)sdot 10038161003816100381610038161003816119906119898[119902119894]+1 minus 119906119898minus1[119902119894]+110038161003816100381610038161003816 le 119862ℎ2 (119905120572119894 + (119902119905119894)120574 (1 minus 120573119894))sdot 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816

(37)

Let 119862(ℎ2)(119905120572119894 + (119902119905119894)120574(1 minus 120573119894)) le 12 for a sufficiently small ℎthen |119906119898+1119894 minus 119906119898119894 | le (12)|119906119898119894 minus 119906119898minus1119894 | holds

(2) The other situation is [119902119894] + 1 lt 119894 that is when 119894 gt1(1 minus 119902) then10038161003816100381610038161003816119906119898+1119894 minus 119906119898119894 10038161003816100381610038161003816 le ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816

le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)

Let 119862(ℎ2)119905120572119894 le 12 for a sufficiently small ℎ then |119906119898+1119894 minus119906119898119894 | le (12)|119906119898119894 minus 119906119898minus1119894 | holdsWith the discussion of the above two situations one can

conclude that the iterative algorithm is convergent geomet-rically and the limit is the solution of (34) therefore thesolution of (34) is existent

Next we prove that the solution of (34) is unique If 119906119894and V119894 are solutions of (34) the difference can be representedas |119911119894| = |119906119894 minus V119894| 1 le 119894 le 119873 and100381610038161003816100381611991101003816100381610038161003816 = 0

10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1

ℎ119905120572119895 100381610038161003816100381610038161198961 (119905119894 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161199111198941003816100381610038161003816minus 120577 (minus120572) ℎ1+120572 10038161003816100381610038161198961 (119905119894 1199050)1003816100381610038161003816 100381610038161003816100381611991101003816100381610038161003816+ [119902119894]minus1sum119895=1

ℎ119905120574119895 100381610038161003816100381610038161198962 (119905119894 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2 119905120574[119902119894] 100381610038161003816100381610038161198962 (119905119894 119905[119902119894])10038161003816100381610038161003816sdot 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 minus 120577 (minus120574) ℎ1+120574 10038161003816100381610038161198962 (119905119894 1199050)1003816100381610038161003816 100381610038161003816100381611991101003816100381610038161003816+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894] 100381610038161003816100381610038161198962 (119905i 119905[119902119894])10038161003816100381610038161003816 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(39)

and |119896119894(119905 119904)| le 119862 (119894 = 1 2) because 119896119894(119905 119904) are continuous onbounded domains with 119902119905119894 minus 119905[119902119894] le ℎ then

100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2

sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119902119905119894)120574 (120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816] le 119862119894minus1sum119895=1

ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)

with 119905119894 = 119894ℎ 119894 = 1 119873 Let 120583 = min120572 120574(1) The first situation is [119902119894] + 1 = 119894 that is when 119894 lt1(1 minus 119902) Let ℎ119861119895 le 12 for a sufficiently small ℎ then (40)

can be written as follows

10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119905120583[119902119894] (120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + (1 minus 120573i) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816)]= 2119862[119902119894]minus1sum

119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7

(2) The second situation is [119902119894] + 1 lt 119894 that is when 119894 gt1(1 minus 119902) Let ℎ119861119895 le 12 for a sufficiently small ℎ then (40)can be written as

10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862

sdot ℎ2 [119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120583[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]= 2119862ℎ[119902119894]minus1sum

119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ119905120583[119902119894]+1 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+1)10038161003816100381610038161003816+ 119894minus1sum119895=[119902119894]+1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1

119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ119905120583[119902119894]+1 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816+ 119894minus1sum119895=[119902119894]+1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 119862119905120583[119902119894]+1 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = [119902119894] + 1119894minus1sum119895=[119902i]+1

119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)

Based on Lemma 5 |119911119894| = 0 with 119860 = 0 and the solution ofthe discrete equation (34) is unique The proof of Theorem 6is completed

5 The Error Estimation

In this section we give the error estimate between theapproximation solution and the exact solution of (1)

Theorem 7 Let 119906(119905) be the exact solution of (1) the kernelfunctions 1198701(119905 119904) = 1199041205721198961(119905 119904) 1198702(119905 119904) = 1199041205741198962(119905 119904) minus1 lt120572 120574 le 0 and the functions 1198961(119905 119904) and 1198962(119905 119904) are continuousin the domains 119863 fl (119905 119904) 0 le 119904 le 119905 le 119879 and 119863120579 fl(119905 119904) 0 le 119904 le 120579(119905) 119905 isin 119868 respectively en there is apositive constant 119887 independent of ℎ such that |119896119894(119905 119904)| le 119862 and|119890119894| = |119906(119905119894) minus 119906119894| (119894 = 0 119873) have the following estimation

max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)

Proof From (26) and (27) we have

1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)

By the trapezoidal formula and Lemma 4 we have

1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)

sdot (120573119894119906119905[119902119894] + (1 minus 120573119894) 119906119905[119902119894]+1) + (119902119905119894 minus 119905[119902119894])212sdot int119902119905119894119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)

sdot (120573119894119906119905[119902119894] + (1 minus 120573119894) 119906119905[119902119894]+1) + 112sdot int119902119905119894119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894 minus 120577 (minus120572)sdot ℎ1+1205721199060119896 (119905119894 1199050)

1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)

where 120573119894 = 1 + [119902119894] minus 119902119894 Subtracting (34) from (51) we get

100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1

ℎ119905120574119895 10038161003816100381610038161198962 (119905119894 119905119895)1003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120574[119902119894] 10038161003816100381610038161198962 (119905119894 119905 [119902119894])1003816100381610038161003816sdot 10038161003816100381610038161003816119890[119902119894]10038161003816100381610038161003816 + 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894] 100381610038161003816100381610038161198962 (119905119894 119905[119902119894])10038161003816100381610038161003816 119890[119902119894]+ (119902119905119894)120574 10038161003816100381610038161198962 (119905119894 119902119905119894)1003816100381610038161003816 (120573119894119890[119902119894] + (1 minus 120573119894) 119890[119902119894]+1]+ 1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120573 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)le 119862119894minus1sum119895=1

ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119890[119902119894]10038161003816100381610038161003816 + 119862119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]119890[119902119894]+ (119902119905119894)120574 (120573119894119890[119902119894] + (1 minus 120573119894) 119890[119902119894]+1] + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)

(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)

and the analysis is the same as (41) and (44) we have

10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)

From Lemma 5 there is a positive constant 119887 independent ofℎ such that

max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)

The proof of Theorem 7 is completed

6 Error Asymptotic Expansion andExtrapolation Algorithm

In this section we present the main theoretical result of theerror asymptotic expansions and the relevant extrapolationalgorithm

Theorem 8 Based on the conditions of eorem 7 there existcontinue functions 119894(119905) (119894 = 1 2 3) satisfying the asymptoticexpansion

119906119894 = 119906 (119905119894) + 1 (119905119894) ℎ2+120572 + 2 (119905119894)) ℎ2+120574 + 3 (119905119894) ℎ2+ 119874 (ℎ3+120583) minus 1 lt 120572 lt 0 minus1 lt 120574 le 0 (56)

Proof Suppose that 119896(119905) 119896 = 1 2 3 satisfies the auxiliarydelay equations

119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)

and 119896(119905119894) 119894 = 1 119873 satisfy the approximation equations

119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119896 (119905119895) + ℎ2sdot 119905120572119894 1198961 (119905119894 119905119894) 119906119894119896 (119905119894) minus 120577 (minus120572) ℎ1+1205721199060119896 (119905119894 1199050)sdot 119896 (1199050) + [119902119894]minus1sum

119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2

sdot 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]119896 (119905119902119894) minus 120577 (minus120574)sdot ℎ1+12057411990601198962 (119905119894 1199050) 119896 (1199050)+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894][119902119894] + (119902119905119894)120574sdot 1198962 (119905119894 119902119905119894) (120573119906119902119905119894[119902119894] + (1 minus 120573) 119906[119902119894]+1[119902119894]+1)]

(58)

Similar to the proof of Theorem 7 we obtain

max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894

minus 120577 (minus120572) ℎ1+1205721199060119896 (119905119894 1199050) 1198640 +[119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119864119895+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]119864[119902119894] minus 120577 (minus120574)sdot ℎ1+12057411990601198962 (119905119894 1199050) 1198640+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894][119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119906119902119905119894 + (1 minus 120573) 119906[119902119894]+1)] (120573119864[119902119894]+ (1 minus 120573) 119864[119902119894]+1)

(61)

From Lemma 1 there exists a constant 119889 such that

max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)

The asymptotic expansion is (56)

Based onTheorem 8 we adopt the Richardson extrapola-tion to improve the accuracy

Extrapolation Algorithm

Step 1 Assume that 120583 = min(120572 120574) = 120572 and halve the steplength to obtain

119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)

By combining (56) with (63) we get

1199061ℎ119894 = 22+120572119906ℎ2119894 minus 119906ℎ11989422+120572 minus 1= 119906 (119905119894) + 2 (119905119894) )ℎ2+120574 + 3 (119905119894) ℎ2 + 119874 (ℎ3+120572)

(64)

Step 2 We implement Richardson ℎ2+120574 extrapolation1199061ℎ2119894 = 119906 (119905119894) + 2 (119905119894) )(ℎ2 )2+120574 + 3 (119905119894) (ℎ2)2

+ 119874 ((ℎ2 )3+120572) (65)

Combining (64) with (65) we get

1199062ℎ119894 = 22+1205741199061ℎ2119894 minus 1199061ℎ11989422+120574 minus 1= 119906 (119905119894) + 3 (119905119894) ℎ2 + 119874 (ℎ3+120572)

(66)

Moreover a posterior asymptotic error estimate

10038161003816100381610038161003816119906ℎ2119894 minus 11990611989410038161003816100381610038161003816 = 100381610038161003816100381610038161003816100381610038161003816100381622+120572119906ℎ2119894 minus 119906ℎ11989422+120572 minus 1 minus 119906 (119894) + 119906ℎ119894 minus 119906ℎ211989422+120572 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816

22+120572119906ℎ2119894 minus 119906ℎ11989422+120572 minus 1 minus 119906 (119894)1003816100381610038161003816100381610038161003816100381610038161003816 + 1003816100381610038161003816100381610038161003816100381610038161003816119906ℎ119894 minus 119906ℎ211989422+120572 minus 1

1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)

The error 119906ℎ2119894 minus 119906(119894) is bounded by (119906ℎ119894 minus 119906ℎ2119894 )(22+120572 minus 1)which is essential to construct adaptable algorithms

7 Numerical Experiments

In this section three examples will be presented to show theefficiency of the quadrature method We design a set of gridson the interval 119868 the absolute error is denoted by

119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)

with 119894 = 1 119873 119906(119905119894) and 119906119894 are the exact solution and theapproximate solution at 119905 = 119905119894 respectively Set ℎ = 1119873 andthe convergence order is defined by

119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)

Example 1 Consider the following equation

119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)

where 119879 = 1 minus1 lt 120572 lt 0 and the analytical solution is119906(119905) = cos(119905)The numerical results at the point 119905 = 1 with the

partitions 119873 = 24 25 26 27 28 are addressed in Table 1 Bythe mechanical quadrature method and iterative method theobtained absolute errors 119890ℎ withmore refined partitions showa more accurate approximate solution and the convergentrate is adjacent to (32)-order Based on the Richardson ℎ2+120572extrapolation the errors are closer to the exact solution andthe convergence order is improved to 2-order The posteriorierror is also achieved The two kinds of convergence ordersare consistent with the theoretical analysis

10 Complexity

Table 1 Numerical results at point 119905 = 1 of Example 1

N 119890ℎ Rate R-ℎ2+120572 extrapolation Rate Posteriori errors24 927e-04 mdash mdash mdash mdash25 322e-04 2153 965e-06 mdash 331e-0426 112e-04 2152 224e-06 2210 114e-0427 393e-05 2151 532e-07 2208 399e-0528 138e-05 2151 128e-07 2206 139e-05

Table 2 Numerical results of Example 2

t 119890ℎ 119890ℎ2 119890ℎ4 Posteriori error02 389e-06 784e-07 177e-07 169e-0604 505e-06 101e-06 228e-07 220e-0606 567e-06 114e-06 255e-07 248e-0608 598e-06 119e-06 268e-07 261e-0610 607e-06 121e-06 271e-07 266e-06


t 119890ℎ 119890ℎ2 119890ℎ4 Posteriori errors02 143e-06 929e-07 718e-07 275e-0704 123e-05 452e-07 909e-07 648e-0606 479e-05 827e-06 523e-07 217e-0508 134e-04 290e-05 510e-06 574e-0510 325e-04 763e-05 116e-05 136e-04

Example 2 Consider the following delay Volterra integralequation with weakly singular kernel as

119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)

with 119902 = 09 119879 = 1 and the initial value 119906(0) = 119892(0) = 0 and119892(119905) is determined by the analytical solution 119906(119905) = 119905 + 1We denote the approximate solution by 119906ℎ with ℎ = 132

The error results at some interior points in the interval 119905 isin[0 1] with different partition are listed in Table 2 The com-parison of the exact solution and the approximate solutionwith partition 119873 = 23 is shown in Figure 1 It is obvious thatthis paper provides a high accuracy algorithm for the weaklysingular Volterra integral equation with proportional delays


119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)

with 119902 = 09 119879 = 1 and the initial value 119906(0) = 119892(0) = 0and 119892(119905) is determined by the analytical solution 119906(119905) = 119905 + 1

The absolute error |119890ℎ(119905)| = |119906(119905) minus 119906ℎ(119905)| 119905 isin [0 1]and the posteriori errors at some interior points are listedin Table 3 when 119873 = 25 26 27 which can observe that

the error results decay quickly with the increasing of 119873The absolute errors between the exact solution and theapproximate solution with partition 119873 = 22 23 are shown inFigure 2 which indicate that our algorithm is effective

8 Conclusion

In this paper we use the mechanical quadrature method andRomberg extrapolation for weakly Volterra integral equationwith proportional delays Most papers analyze the delayVolterra integral equation with continuous kernels the studyfor the delay Volterra integral equation with weakly singularkernels still faces a real challenge for the relevant researchersin both numerical computation and theoretical analysis Theimproved Gronwall inequality is adopted to prove the exis-tence and uniqueness of the solution of the original equationAt the same time the discrete Gronwall equality and iterativemethod are adopted to prove the existence and uniqueness ofthe solution of the approximate equation Moreover accord-ing to the error asymptotic expansion of the mechanicalquadrature method and the extrapolation a high order ofaccuracy can be achieved and a posterior error estimation canbe obtained Both the theoretical analysis and the numericalexamples show that the presented method is efficient

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact

Figure 1 Comparison of exact solution and approximate solution with 119873 = 23

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact


Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was supported by the financial support from theNational Natural Science Foundation of China (Grant no11371079)

References

[1] F Brauer ldquoConstant rate harvesting of populations governed byVolterra integral equationsrdquo Journal of Mathematical Analysisand Applications vol 56 no 1 pp 18ndash27 1976

[2] K L Cooke and J L Kaplan ldquoA periodicity threshold the-orem for epidemics and population growthrdquo MathematicalBiosciences vol 31 no 1-2 pp 87ndash104 1976

[3] P K Lamm and T L Scofield ldquoSequential predictor-correctormethods for the variable regularization of Volterra inverseproblemsrdquo Inverse Problems vol 16 no 2 pp 373ndash399 2000

12 Complexity

[4] H Brunner ldquoCollocation methods for Volterra integral andrelated functional differential equationsrdquo in Mathematics ofComputation vol 75 p 254 Cambridge University Press 2009

[5] G A Okeke and M Abbas ldquoA solution of delay differentialequations via Picard-Krasnoselskii hybrid iterative processrdquoArabian Journal of Mathematics vol 6 no 1 pp 21ndash29 2017

[6] N Bildik and SDeniz ldquoA new efficientmethod for solving delaydifferential equations and a comparison with other methodsrdquoe European Physical Journal Plus vol 132 no 1 p 51 2017

[7] F Mohammadi ldquoNumerical solution of systems of fractionaldelay differential equations using a new kind of wavelet basisrdquoComputational amp Applied Mathematics vol 37 no 4 pp 4122ndash4144 2018

[8] K Harriman P Houston B Senior and E Sli ldquohp-versiondiscontinuous Galerkin methods with interior penalty forpartial differential equations with nonnegative characteristicformrdquo in Recent Advances in Scientific Computing and PartialDifferential Equations vol 330 of Contemp Math pp 89ndash1192003

[9] C-T Sheng Z-Q Wang and B-Y Guo ldquoAn hp-spectralcollocation method for nonlinear Volterra functional integro-differential equations with delaysrdquo Applied Numerical Mathe-matics vol 105 pp 1ndash24 2016

[10] T Koto ldquoStability of Runge-Kutta methods for delay integro-differential equationsrdquo Journal of Computational and AppliedMathematics vol 145 no 2 pp 483ndash492 2002

[11] H Cai Y Chen and Y Huang ldquoA LegendrendashPetrovndashGalerkinmethod for solving Volterra integro-differential equations withproportional delaysrdquo International Journal of Computer Mathe-matics pp 1ndash15 2018

[12] MMosleh andMOtadi ldquoLeast squares approximationmethodfor the solution of Hammerstein-Volterra delay integral equa-tionsrdquo AppliedMathematics and Computation vol 258 pp 105ndash110 2015

[13] I Ali H Brunner and T Tang ldquoSpectral methods forpantograph-type differential and integral equations with mul-tiple delaysrdquo Frontiers of Mathematics in China vol 4 no 1 pp49ndash61 2009

[14] P K Sahu and S S Ray ldquoA new Bernoulli wavelet method foraccurate solutions of nonlinear fuzzy HammersteinndashVolterradelay integral equationsrdquo Fuzzy Sets and Systems vol 309 pp131ndash144 2017

[15] K Zhang and J Li ldquoCollocation methods for a class ofVolterra integral functional equations with multiple propor-tional delaysrdquo Advances in Applied Mathematics andMechanicsvol 4 no 5 pp 575ndash602 2012

[16] H Xie R Zhang and H Brunner ldquoCollocation methods forgeneral Volterra functional integral equations with vanishingdelaysrdquo SIAM Journal on Scientific Computing vol 33 no 6 pp3303ndash3332 2011

[17] P Darania and S Pishbin ldquoHigh-order collocation methodsfor nonlinear delay integral equationrdquo Journal of Computationaland Applied Mathematics vol 326 pp 284ndash295 2017

[18] Z-Q Wang and C-T Sheng ldquoAn hp-spectral collocationmethod for nonlinear Volterra integral equations with vanish-ing variable delaysrdquo Mathematics of Computation vol 85 no298 pp 635ndash666 2016

[19] Z Gu and Y Chen ldquoChebyshev spectral-collocation methodfor a class of weakly singular Volterra integral equations withproportional delayrdquo Journal of Numerical Mathematics vol 22no 4 pp 311ndash341 2014

[20] A Sidi and M Israeli ldquoQuadrature methods for periodicsingular and weakly singular fredholm integral equationsrdquoJournal of Scientific Computing vol 3 no 2 pp 201ndash231 1988

[21] H Brunner ldquoRecent advances in the numerical analysis ofVolterra functional differential equations with variable delaysrdquoJournal of Computational and AppliedMathematics vol 228 no2 pp 524ndash537 2009

[22] J Huang High Precision Algorithm for Multidimensional Singu-lar Integral Science Press 2017

[23] L Tao and H Yong ldquoA generalization of discrete Gronwallinequality and its application to weakly singular Volterraintegral equation of the second kindrdquo Journal of MathematicalAnalysis and Applications vol 282 no 1 pp 56ndash62 2003

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

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2 Complexity

There are a few researches on the delay integral equationwith weakly singular kernels such as [19] In this paperwe concentrate on (1) whose upper limit of integral is adelay function and the integral kernels are weakly singularfunctions which increase the computational complexity andthe theoretical difficulty To the best of our knowledge thereare no studies in (1) by the mechanical quadrature methodin recent years A further advantage of this method for (1)is that the error has an asymptotic expansion Thus we canimprove the accuracy order of approximation Simultane-ously the theoretical analysis is complete and the calculationis simplified

In this paper we firstly deduce the improved Gronwallinequality the existence and uniqueness of the solution for (1)are testified via the improved Gronwall inequality Then theequation is approximated by employing the floor techniqueto the delay argument 120579(119905) and by adopting the quadratureformula [20] to the weakly integrals Next the approximateequation for (1) is constructed by combining the mechanicalquadrature method and the interpolation technique thenwe use the iterative method for solving the approximateequationThe existence and uniqueness of the solution for theapproximate equation are testified via the discrete Gronwallinequality Finally we prove that the convergent order is119874(ℎ2+min(120572120574)) In order to achieve a higher accuracy order119874(ℎ2) the Richardson-ℎ2+min(120572120574) extrapolation based onthe asymptotic expansion of error is adopted moreover aposterior error estimate is realized conveniently

The layouts of this paper are as follows In Section 2 weprove the existence and uniqueness of the solution for (1)In Section 3 we introduce the quadrature method iterativemethod and interpolation technique In Section 4 the exis-tence and uniqueness of the solution for the approximateequation are discussed In Section 5 the convergence andthe error estimation are obtained to ensure the reliabilityof the method In Section 6 the asymptotic expansion ofthe error is achieved a higher accuracy order is realizedby extrapolation and a posterior error estimate is derivedIn Section 7 some numerical examples are demonstrated toillustrate the theoretical results Some concluding remarks areprovided in Section 8

2 The Existence and Uniqueness ofthe Solution for the Original Equation

In this section we will verify the existence and uniquenessof the solution for (1) We first prove the improved Gronwallinequality

Lemma 1 Suppose that 119892(119905) ℎ(119905) and 119906(119905) are nonnegativeintegrable functions 119905 isin [0 119879] 0 lt 119902 lt 1 and 119860 ge 0 basedon the inequality

119906 (119905) le 119860 + int1199050

119892 (119904) 119906 (119904) 119889119904 + int1199021199050

ℎ (119904) 119906 (119904) 119889119904 (4)

we have

119906 (119905) le 119860119890int1199050 (119892(119904)+ℎ(119904))119889119904 (5)

Proof Due to the fact that 0 lt 119902 lt 1 119902119905 lt 119905 and119906 (119905) le 119860 + int119905

0119892 (119904) 119906 (119904) 119889119904 + int119905

0ℎ (119904) 119906 (119904) 119889119904

le 119860 + int1199050

(119892 (119904) + ℎ (119904)) 119906 (119904) 119889119904(6)

Let 119867(119904) = 119892(119904) + ℎ(119904) we can deduce

119867 (119905) 119906 (119905)119860 + int1199050 119867 (119904) 119906 (119904) 119889119904 le 119867 (119905) (7)

Integrate on both sides

ln(119860 + int1199050

119867 (119904) 119906 (119904) 119889119904) minus ln119860 le int1199050

119867 (119904) 119889119904 (8)

and then

119860 + int1199050

119867 (119904) 119906 (119904) 119889119904 le 119860119890int1199050 119867(119904)119889119904 (9)

namely

119906 (119905) le 119860119890int1199050 119867(119904)119889119904 (10)

The proof of the Lemma 1 is completed

Theorem 2 Assume that 119896119894(119905 119904) (119894 = 1 2) are knowncontinuous functions defined on the domains 119863 fl (119905 119904) 0 le119904 le 119905 le 119879 and 119863120579 fl (119905 119904) 0 le 119904 le 120579(119905) 119905 isin 119868 = [0 119879]respectively then the solution of (1) is existent uniquely

Proof We construct the sequence 119906119896(119905) (119896 isin N) satisfying1199060 (119905) = 119906 (0) = 119892 (119905) 119906119896 (119905) = 119892 (119905) + int119905

01198701 (119905 119904) 119906119896minus1 (119904) 119889119904

+ int1199021199050

1198702 (119905 119904) 119906119896minus1 (119904) 119889119904(11)

where 1198701(119905 119904) and 1198702(119905 119904) are defined in (3) with 119896 isin N119896119894(119905 119904) (119894 = 1 2) are continuous functions then there existsa constant 119862 such that |119896119894(119905 119904)| le 119862 We have

10038161003816100381610038161199062 (119905) minus 1199061 (119905)1003816100381610038161003816le 119862 int1199050

119904120572 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 119889119904+ 119862 int1199021199050

119904120574 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 119889119904le 119862 int1199050

[119904120572 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161199061 (119904) minus 1199060 (119904)1003816100381610038161003816] 119889119904le 119862119886 int119905

0(119904120572 + 119904120574) 119889119904 le 2119862119886 int119905

0119904120583119889119904 = 2119862119886 119905120583+1120583 + 1

(12)

Complexity 3



(119896minus1)(120583+1) (13)


1003816100381610038161003816119906119896+1 (119905) minus 119906119896 (119905)1003816100381610038161003816 le 119862 int1199050

119904120572 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904+ 119862 int1199021199050



0119904120583 1003816100381610038161003816119906119896 (119904)

minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904 le 119886 (2119862)119896119896 (120583 + 1)119896 119905

119896(120583+1)

(14)



+ 1003816100381610038161003816119906119896+119898 (119905) minus 119906119896+119898minus1 (119905)1003816100381610038161003816le 119886 (2119862)119896

119896 (120583 + 1)119896 119905119896(120583+1) + sdot sdot sdot


(119896+119898minus1)(120583+1)

(15)


1003816100381610038161003816119906119899 (119905) minus 119906119899+119898 (119905)1003816100381610038161003816 le 120598 (16)



|119908 (119905)|= 10038161003816100381610038161003816100381610038161003816int119905

01199041205721198961 (119905 119904) 119908 (119904) 119889119904 + int119902119905

01199041205741198962 (119905 119904) 119908 (119904) 11988911990410038161003816100381610038161003816100381610038161003816

le int1199050

119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904+ int1199021199050

119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904

le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904(17)


119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 119889119904 le 119862 int1199050

(119904120572 + 119904120574) 119889119904le 119862 ( 1199051+1205721 + 120572 + 1199051+1205741 + 120574)

(18)









119866 (119909119894) + ℎ2 119866 (119887) (19)


119864119873 (119866) = 119876119873 (119866) minus 119868 (119866)= 119898minus1sum119895=1

119861(2119895)(2119895)119866(2119895minus1) (119887) ℎ2119895

+ 2119898minus1sum119895=1


+ 119874 (ℎ2119898)

(20)


4 Complexity



2 (119909 minus 119910)2 11990610158401015840 (119911) + 119874 ((119909 minus 119910)3) (21)




(22)



+ 119874 ((119909 minus 119910)3) (23)



119906 (119905119894) = 119892 (119905119894) + int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904+ int1199021199051198940

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + int119905119894

01199041205721198961 (119905119894 119904) 119906 (119904) 119889119904

+ int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904+ int119902119905119894119905[119902119894]

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + 1198681 + 1198682 + 1198683

(24)


119906 (119902119905119894) = 120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1) (25)


1198681 = int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904 = ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906 (119905119896)+ ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906 (119905119894) minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 119906 (1199050)+ 1198641119894 (1199041205721198961 (119905 119904))

(26)

1198682 = int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904 = ℎ[119902119894]minus1sum119896=1


(27)

1198683 = int119902119905119894119905[119902119894]


(28)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572 + 119900 (ℎ4+120572)= 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574 + 119900 (ℎ4+120574)= 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)

Complexity 5


sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)sdot ℎ2 + 119874 (ℎ3)

(29)

where



12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904

(30)


119906 (1199050) = 119892 (1199050) (31)

119906 (119905119894) = 119892 (119905119894) + ℎ 119894minus1sum119896=1



(32)


1199060 = 119892 (1199050) 119906119894 = 119892 (119905119894) + ℎ 119894minus1sum

119896=11199051205721198961198961 (119905119894 119905119896) 119906119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894


1199051205741198961198962 (119905119894 119905119896) 119906119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060

+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(33)


Iterative Algorithm


119906119898+1119894 = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906119898119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119898119894



(34)





10038161003816100381610038161199081198991003816100381610038161003816 le ℎ 119899sum119896=1

119861119896 10038161003816100381610038161199081198961003816100381610038161003816 + 119860 0 le 119899 le 119873 (35)


max0le119899le119873

10038161003816100381610038161199081198991003816100381610038161003816 le 119860 exp(2ℎ 119873sum119896=1

119861119896) (36)



6 Complexity




(37)



le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)




10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1



(39)


100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2


ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)



10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1


119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3



(119896minus1)(120583+1) (13)


1003816100381610038161003816119906119896+1 (119905) minus 119906119896 (119905)1003816100381610038161003816 le 119862 int1199050

119904120572 1003816100381610038161003816119906119896 (119904) minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904+ 119862 int1199021199050



0119904120583 1003816100381610038161003816119906119896 (119904)

minus 119906119896minus1 (119904)1003816100381610038161003816 119889119904 le 119886 (2119862)119896119896 (120583 + 1)119896 119905

119896(120583+1)

(14)



+ 1003816100381610038161003816119906119896+119898 (119905) minus 119906119896+119898minus1 (119905)1003816100381610038161003816le 119886 (2119862)119896

119896 (120583 + 1)119896 119905119896(120583+1) + sdot sdot sdot


(119896+119898minus1)(120583+1)

(15)


1003816100381610038161003816119906119899 (119905) minus 119906119899+119898 (119905)1003816100381610038161003816 le 120598 (16)



|119908 (119905)|= 10038161003816100381610038161003816100381610038161003816int119905

01199041205721198961 (119905 119904) 119908 (119904) 119889119904 + int119902119905

01199041205741198962 (119905 119904) 119908 (119904) 11988911990410038161003816100381610038161003816100381610038161003816

le int1199050

119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904+ int1199021199050

119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 |119908 (119904)| 119889119904

le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 |119908 (119904)| + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904le int1199050

(119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816) |119908 (119904)| 119889119904(17)


119904120572 10038161003816100381610038161198961 (119905 119904)1003816100381610038161003816 + 119904120574 10038161003816100381610038161198962 (119905 119904)1003816100381610038161003816 119889119904 le 119862 int1199050

(119904120572 + 119904120574) 119889119904le 119862 ( 1199051+1205721 + 120572 + 1199051+1205741 + 120574)

(18)









119866 (119909119894) + ℎ2 119866 (119887) (19)


119864119873 (119866) = 119876119873 (119866) minus 119868 (119866)= 119898minus1sum119895=1

119861(2119895)(2119895)119866(2119895minus1) (119887) ℎ2119895

+ 2119898minus1sum119895=1


+ 119874 (ℎ2119898)

(20)


4 Complexity



2 (119909 minus 119910)2 11990610158401015840 (119911) + 119874 ((119909 minus 119910)3) (21)




(22)



+ 119874 ((119909 minus 119910)3) (23)



119906 (119905119894) = 119892 (119905119894) + int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904+ int1199021199051198940

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + int119905119894

01199041205721198961 (119905119894 119904) 119906 (119904) 119889119904

+ int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904+ int119902119905119894119905[119902119894]

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + 1198681 + 1198682 + 1198683

(24)


119906 (119902119905119894) = 120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1) (25)


1198681 = int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904 = ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906 (119905119896)+ ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906 (119905119894) minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 119906 (1199050)+ 1198641119894 (1199041205721198961 (119905 119904))

(26)

1198682 = int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904 = ℎ[119902119894]minus1sum119896=1


(27)

1198683 = int119902119905119894119905[119902119894]


(28)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572 + 119900 (ℎ4+120572)= 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574 + 119900 (ℎ4+120574)= 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)

Complexity 5


sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)sdot ℎ2 + 119874 (ℎ3)

(29)

where



12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904

(30)


119906 (1199050) = 119892 (1199050) (31)

119906 (119905119894) = 119892 (119905119894) + ℎ 119894minus1sum119896=1



(32)


1199060 = 119892 (1199050) 119906119894 = 119892 (119905119894) + ℎ 119894minus1sum

119896=11199051205721198961198961 (119905119894 119905119896) 119906119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894


1199051205741198961198962 (119905119894 119905119896) 119906119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060

+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(33)


Iterative Algorithm


119906119898+1119894 = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906119898119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119898119894



(34)





10038161003816100381610038161199081198991003816100381610038161003816 le ℎ 119899sum119896=1

119861119896 10038161003816100381610038161199081198961003816100381610038161003816 + 119860 0 le 119899 le 119873 (35)


max0le119899le119873

10038161003816100381610038161199081198991003816100381610038161003816 le 119860 exp(2ℎ 119873sum119896=1

119861119896) (36)



6 Complexity




(37)



le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)




10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1



(39)


100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2


ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)



10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1


119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity



2 (119909 minus 119910)2 11990610158401015840 (119911) + 119874 ((119909 minus 119910)3) (21)




(22)



+ 119874 ((119909 minus 119910)3) (23)



119906 (119905119894) = 119892 (119905119894) + int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904+ int1199021199051198940

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + int119905119894

01199041205721198961 (119905119894 119904) 119906 (119904) 119889119904

+ int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904+ int119902119905119894119905[119902119894]

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904= 119892 (119905119894) + 1198681 + 1198682 + 1198683

(24)


119906 (119902119905119894) = 120573119894119906 (119905[119902119894]) + (1 minus 120573119894) 119906 (119905[119902119894]+1) (25)


1198681 = int1199051198940

1199041205721198961 (119905119894 119904) 119906 (119904) 119889119904 = ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906 (119905119896)+ ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906 (119905119894) minus 120577 (minus120572) ℎ1+1205721198961 (119905119894 1199050) 119906 (1199050)+ 1198641119894 (1199041205721198961 (119905 119904))

(26)

1198682 = int119905[119902119894]0

1199041205741198962 (119905119894 119904) 119906 (119904) 119889119904 = ℎ[119902119894]minus1sum119896=1


(27)

1198683 = int119902119905119894119905[119902119894]


(28)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572 + 119900 (ℎ4+120572)= 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=0 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574 + 119900 (ℎ4+120574)= 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)

Complexity 5


sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)sdot ℎ2 + 119874 (ℎ3)

(29)

where



12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904

(30)


119906 (1199050) = 119892 (1199050) (31)

119906 (119905119894) = 119892 (119905119894) + ℎ 119894minus1sum119896=1



(32)


1199060 = 119892 (1199050) 119906119894 = 119892 (119905119894) + ℎ 119894minus1sum

119896=11199051205721198961198961 (119905119894 119905119896) 119906119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894


1199051205741198961198962 (119905119894 119905119896) 119906119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060

+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(33)


Iterative Algorithm


119906119898+1119894 = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906119898119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119898119894



(34)





10038161003816100381610038161199081198991003816100381610038161003816 le ℎ 119899sum119896=1

119861119896 10038161003816100381610038161199081198961003816100381610038161003816 + 119860 0 le 119899 le 119873 (35)


max0le119899le119873

10038161003816100381610038161199081198991003816100381610038161003816 le 119860 exp(2ℎ 119873sum119896=1

119861119896) (36)



6 Complexity




(37)



le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)




10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1



(39)


100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2


ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)



10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1


119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5


sdot int119902119905119894119905[119902119894]

12059721205971199042 [1198962 (119905119894 119904) 119906 (119904) 119904120574] 119889119904 + 119874 (ℎ3) = 1198793 (119905119894)sdot ℎ2 + 119874 (ℎ3)

(29)

where



12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904

(30)


119906 (1199050) = 119892 (1199050) (31)

119906 (119905119894) = 119892 (119905119894) + ℎ 119894minus1sum119896=1



(32)


1199060 = 119892 (1199050) 119906119894 = 119892 (119905119894) + ℎ 119894minus1sum

119896=11199051205721198961198961 (119905119894 119905119896) 119906119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894


1199051205741198961198962 (119905119894 119905119896) 119906119896+ ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894] minus 120577 (minus120574) ℎ1+1205741198962 (119905119894 1199050) 1199060

+ 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(33)


Iterative Algorithm


119906119898+1119894 = 119892 (119905119894) + ℎ 119894minus1sum119896=1

1199051205721198961198961 (119905119894 119905119896) 119906119898119896 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119898119894



(34)





10038161003816100381610038161199081198991003816100381610038161003816 le ℎ 119899sum119896=1

119861119896 10038161003816100381610038161199081198961003816100381610038161003816 + 119860 0 le 119899 le 119873 (35)


max0le119899le119873

10038161003816100381610038161199081198991003816100381610038161003816 le 119860 exp(2ℎ 119873sum119896=1

119861119896) (36)



6 Complexity




(37)



le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)




10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1



(39)


100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2


ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)



10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1


119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity




(37)



le 119862ℎ2 119905120572119894 10038161003816100381610038161003816119906119898119894 minus 119906119898minus1119894 10038161003816100381610038161003816 (38)




10038161003816100381610038161199111198941003816100381610038161003816 = 119894minus1sum119895=1



(39)


100381610038161003816100381611991101003816100381610038161003816 = 010038161003816100381610038161199111198941003816100381610038161003816 le 119862119894minus1sum

119895=1ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum

119895=1ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2


ℎ119905120572119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161199111198941003816100381610038161003816 + [119902119894]minus1sum119895=1

ℎ119905120574119895119862 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + ℎ2sdot 119905120574[119902119894]119862 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 [119905120574[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119905120574[119902119894](120573119894 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (1 minus 120573119894) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816]

(40)



10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1


119895=1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816

+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(41)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ (119862ℎ2 119905120583119894 + 119862ℎ2 119905120583[119902119894] (1 minus 120573119894)) 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816

(42)

where

119861119895

=

2119862[119902119894]minus1sum119895=1

119905120583119895 119895 = 1 [119902119894] minus 12119862119905120583[119902119894] + 11986212119905120583[119902119894]120573119894 119895 = [119902119894] 11986212 119905120583119894 + 11986212119905120583[119902119894] (1 minus 120573119894) 119895 = 119894

(43)

Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7


10038161003816100381610038161199111198941003816100381610038161003816 le 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816+ 119862ℎ119905120583[119902119894]+1 10038161003816100381610038161003816119911[119902119894]+110038161003816100381610038161003816 + 119894minus1sum

119895=[119902119894]+1ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

+ 119862[119902119894]minus1sum119895=1

ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583[119902119894] 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816 + 119862


119895=1119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + (2119862ℎ119905120583[119902119894] + 119862ℎ2 119905120583[119902119894]120573119894) 10038161003816100381610038161003816119911[119902119894]10038161003816100381610038161003816


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(44)

namely

10038161003816100381610038161199111198941003816100381610038161003816 le 119894sum119895=1

119861119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816

= 2119862ℎ[119902119894]minus1sum119895=1


ℎ119905120583119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 + 119862ℎ2 119905120583119894 10038161003816100381610038161199111198941003816100381610038161003816

(45)

And

119861119895

=

2119862[119902119894]minus1sum119895=1


119905120583119895 119895 = [119902119894] + 1 119894 minus 111986212119905120583119894 119895 = 119894

(46)





max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (47)


1198641119894 = [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120572 minus 1) ℎ2+120572

+ [1198961 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120572 minus 2) ℎ3+120572+ 119900 (ℎ4+120572) = 1198791 (119905119894) ℎ2+120572 + 119874 (ℎ3+120572)

1198642119894 = [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=01 120577 (minus120574 minus 1) ℎ2+120574

+ [1198962 (119905119894 119904) 119906 (119904)]101584010038161003816100381610038161003816119904=02 120577 (minus120574 minus 2) ℎ3+120574+ 119900 (ℎ4+120574) = 1198792 (119905119894) ℎ2+120574 + 119874 (ℎ3+120574)

(48)


1198643119894 = minus120573 (1 minus 120573)2 ℎ211990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904 + 119874 (ℎ3) = 1198793 (119905119894) ℎ2

+ (119902119905119894 minus 119905[119902119894])2 minus ℎ212 int119902119905119894

119905[119902119894]

12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904+ 119874 (ℎ3) = 1198793 (119905119894) ℎ2 + 119874 (ℎ3)

(49)

where

1198793 (119905119894) = minus120573 (1 minus 120573)2 11990610158401015840 (119902119905119894) (119902119905119894)120574 1198962 (119905119894 119902119905119894)


12059721205971199042 1198962 (119905119894 119904) 119906 (119904) 119904120574119889119904(50)

8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

Then we have

119906 (119905119894) = 119892 (119905119894) + 1198761198731 + 1198761198732 + 1198761198733 + 1198791 (119905119894) ℎ2+120572+ 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ2)

1198761198731 = 119894minus1sum119895=1


1198761198732 = [119902119894]minus1sum119895=1

ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895 + ℎ2 119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]minus 120577 (minus120574) ℎ1+12057411990601198962 (119905119894 1199050)

1198761198733 = 119902119905119894 minus 119905[119902119894]2 [119905120574[119902119894]1198962 (119905119894 119905[119902119894]) 119906[119902119894]+ (119902119905119894)120574 1198962 (119905119894 119902119905119894) (120573119894119906[119902119894] + (1 minus 120573119894) 119906[119902119894]+1]

(51)


100381610038161003816100381611989001003816100381610038161003816 = 010038161003816100381610038161198901198941003816100381610038161003816 = 119894minus1sum

119895=1ℎ119905120572119895 100381610038161003816100381610038161198961 (119905i 119905119895)10038161003816100381610038161003816 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + ℎ2 119905120572119894 10038161003816100381610038161198961 (119905119894 119905119894)1003816100381610038161003816 10038161003816100381610038161198901198941003816100381610038161003816

+ [119902119894]minus1sum119895=1


ℎ119905120572119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 + 119862ℎ2 119905120572119894 10038161003816100381610038161198901198941003816100381610038161003816 + [119902119894]minus1sum119895=1


(52)

Let

119860= 100381610038161003816100381610038161198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894) ℎ2 + 119874 (ℎ3+120583)10038161003816100381610038161003816= 119874 (ℎ2+120583)

(53)


10038161003816100381610038161198901198941003816100381610038161003816 le 119860 + 119894sum119895=1

119861119895 1003816100381610038161003816100381611989011989510038161003816100381610038161003816 1 le 119894 le 119873 (54)


max1le119894le119873

10038161003816100381610038161198901198941003816100381610038161003816 le 119887ℎ2+120583 (55)







119896 (119905) = 119879119896 (119905) + int1199050

1199041205721198961 (119905 119904) 119906 (119904) 119896 (119904) 119889119904+ int1199021199050

1199041205741198962 (119905 119904) 119906 (119904) 119896 (119904) 119889119904(57)


119896 (119905119894) = 119879119896 (119905119894) + 119894minus1sum119895=1


119895=1ℎ1199051205741198951198962 (119905119894 119905119895) 119906119895119896 (119905119895) + 119896 (119905119895) ℎ2


(58)


max1le119894le119873

10038161003816100381610038161003816119896 (119905119894) minus 119879 (119905119894)10038161003816100381610038161003816 le 119862ℎ2+120583 (59)

Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9

Let

119864119894 = 119890119894 minus (1198791 (119905119894) ℎ2+120572 + 1198792 (119905119894) ℎ2+120574 + 1198793 (119905119894)) ℎ2 (60)

We have

119864119894 = 119894minus1sum119895=1

ℎ119905120572119895 1198961 (119905119894 119905119895) 119906119895119864119895 + ℎ2 119905120572119894 1198961 (119905119894 119905119894) 119906119894119864119894



(61)


max1le119894le119873

10038161003816100381610038161198641198941003816100381610038161003816 le 119889ℎ3+120583 (62)





119906ℎ2119894 = 119906 (119905119894) + 1 (119905119894) (ℎ2 )2+120572 + 2 (119905119894) )(ℎ2 )2+120574

+ 3 (119905119894) (ℎ2 )2 + 119874 ((ℎ2 )3+120572) (63)



(64)


+ 119874 ((ℎ2 )3+120572) (65)



(66)



1003816100381610038161003816100381610038161003816100381610038161003816le 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816100381610038161003816


1003816100381610038161003816100381610038161003816100381610038161003816+ 119874 (ℎ2)

(67)




119890ℎ (119905i) = 1003816100381610038161003816119906 (119905119894) minus 1199061198941003816100381610038161003816 (68)


119877119886119905119890 = log (119890ℎ119890ℎ2)log 2 (69)


119906 (119905) = 119892 (119905) minus int1199050

(119905 minus 119904)120572 119906 (119904) 119889119904 119905 isin [0 119879] (70)



10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity








119906 (119905) = 119892 (119905) minus int1199050

119904minus12119906 (119904) 119889119904 + int1199021199050

119904minus12119906 (119904) 119889119904119905 isin [0 119879]

(71)




119906 (119905) = 119892 (119905) minus int1199050

119904minus14119906 (119904) 119889119904 + int1199021199050

(1199052 + 119904) 119906 (119904) 119889119904119905 isin [0 119879]

(72)




8 Conclusion


Data Availability


Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=8exact


02 03 04 05 06 07 08 09 112

13

14

15

16

17

18

19

2

21

N=4N=8exact




Acknowledgments


References




12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity




























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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A Mechanical Quadrature Method for Solving Delay Volterra ...downloads.hindawi.com/journals/complexity/2019/4813802.pdf · A Mechanical Quadrature Method for Solving Delay Volterra