Research ArticleOn the Agaciro Equation via the Scope of Green Function

Abdon Atangana1 and Innocent Rusagara2

1 Institute for Groundwater Studies Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa

2 School of Computational and Applied Mathematics University of Witwatersrand Private Bag 3 WitsJohannesburg 2050 South Africa

Correspondence should be addressed to Innocent Rusagara rusagara2002yahoofr

Received 18 October 2013 Accepted 20 November 2013 Published 21 January 2014

Academic Editor Muhammet Kurulay

Copyright copy 2014 A Atangana and I RusagaraThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We have undertaken an investigation of a kind of third-order equation called Agaciro equation within the folder of both integerand fractional order derivative In the way of deriving the general exact solution of this equation we employed the philosophy ofthe Green function together with some integral transform operators and special functions including but not limited to the LaplaceFourier and Mellin transform We presented some examples of exact solution of this class of third-order equations for integer andfractional order derivative It is important to point out that the value of Agaciro equation can be extended to describe assortedphenomenon in sciences

1 Introduction

An importantmethod used in the field of partial and ordinarylinear is Green functionmethod It is important to rememberthat the construction of the Green function is an art and dif-ficult exercise because in the way of construction one mustfirst make sure of the existence and the uniqueness of thisfunction which is a whole topic in mathematics Once theuniqueness and the existence are insured one needs to havea knowledge of methods of solving either partial differentialequations or ordinary differential equations again this is awhole topic on its own in mathematics Within the scope offractional calculus one further needs a clear knowledge ofspecial function andother useful integral transformoperatorsincluding but not limited to Laplace transform Fouriertransform and Mellin transform This method was recentlyextended to the scope of partial and ordinary differentialequations with noninteger order derivative [1 2] It should bementioned that the finding of theGreen function in the folderof the fractional calculus is very difficult because this involvesat the same time some special functions and also someintegral transform for instance the Mellin the Laplace and

the Fourier transform operators It is perhaps important topresent a brief history regarding the Green function method

In mathematical sciences and related disciplines a Greenfunction is the desired answer of an inhomogeneous differ-ential equation defined on a domain with particular preli-minary conditions or boundary conditions By means of thesuperposition theory the convolution of a Green functionwith a subjective function119891(119909) on that domain is the solutionto the inhomogeneous differential equation for the unknownfunction 119891(119909) These classes of functions are named after theBritishmathematicianGeorgeGreen who first developed theconcept in the year 1830 [3 4] In the contemporary studyof linear partial differential equations Greenrsquos functions arerevised for the most part from the standpoint of the originalsolutions as a replacement is significant to also draw attentionto the fact that under many-folders theories the conceptis employed in physics more importantly in quantum fieldtheory aerodynamics aeroacoustics electrodynamics andstatistical field theory to downgrade to an assortment of typesof correlations functions still those that do not able-bodiedmathematical explanation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 201796 8 pageshttpdxdoiorg1011552014201796

2 Mathematical Problems in Engineering

In this present work we will present the discussion under-pinning the construction of Green function of a novel equa-tion called ldquoAgacirordquo equation presented as

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 le 1

(1)

We will examine two cases the case where the orders of thederivative are integer numbers meaning 120572 = 120573 = 120574 = 1 andthe case where 0 lt 120572 120573 120574 lt 1

2 Some Useful Information aboutFractional Derivative

With the purpose to provide lodgings to readers that are notin the field of fractional calculus we dedicate this subdivisionto the symposium supporting the fundamental principleof the fractional calculus But we will much stress on theproperties of the Caputo fractional derivative since it will beused throughout the remainder of the paper

Definition 1 (Riemann-Liouville integral [5ndash10]) The Rie-mann-Liouville integral gives the orthodox outward appear-ance of fractional calculus The theory for intermittent func-tions consequently including the ldquoboundary conditionrdquo ofrepeating after a period is the Weyl integral The Riemann-Liouville integral of order 120572 of a function 119891(119909) is given by

119886119863minus120572

119905119891 (119909) =

119886119868120572

119905119891 (119909) =

1

Γ (120572)int119909

119886

(119909 minus 119905)120572minus1119891 (119905) 119889119905 (2)

Definition 2 (Caputo fractional derivative) There is onemoreoption for working out the definition of fractional derivationCaputo established it in 1967 in his paper [11] Quite theopposite of the Riemann Liouville fractional derivative whengetting to the bottom of differential equations by meansof Caputorsquos definition it is not indispensable to describethe fractional order initial conditions Caputorsquos definition isillustrated as follows

119862

0119863120572

119905119891 (119909) =

1

Γ (119899 minus 120572)int119909

119886

(119909 minus 119905)119899minus120572minus1

119889119899

119889119905119899119891 (119905) 119889119905 (3)

The above definitions are frequently used in pure and appliedmathematics

Definition 3 The Laplace transform is an extensively usedintegral transform in the midst of numerous applicationsin physics and engineering The Laplace transform of thefunction 119891 is defined as follows

L (119891 (119909)) (119904) = intinfin

0

119890minus119904119909119891 (119909) 119889119909 (4)

Let us observe the Laplace transform of the fractional deri-vative with Caputo

L ( 1198620119863120572

119905119891 (119909)) (119904)

= 119904120572119865 (119904) minus119899minus1

sum119896=0

119904120572minus119896minus1119891(119896) (0) (119899 minus 1 lt 120572 le 119899)

(5)

The above uses the usual initial conditions or values of thefunctions

Another useful property of the Caputo derivative is thefollowing

119886119863minus120572

119905[ 1198620119863120572

119905119891 (119909)]

= 119891 (119909) minus119899minus1

sum119895=1

119891119895(0)

Γ (120572 minus 119895 + 1)119909119895 (119899 minus 1 lt 120572 le 119899)

(6)

where Γ is the gamma function defined as

Γ (119909) = intinfin

0

119905119909minus1119890minus119905119889119905 (7)

The Fourier transform of a function 119891(119909) of real variable 119905 isdefined as

(F119891) (119901) = intinfin

minusinfin

119890minus119894119909119901119891 (119909) 119889119909 (8)

The Fourier transform of a fractional derivative of order 120572 is

F [ 1198620119863120572

119905119891 (119909)] = (119894119901)

120572

F (119891) (119901) (9)

The Mellin transform of a function 119891(119905) of a real variable 119905 isinR+ is defined as

(M119891) (119901) = intinfin

0

119905119901minus1119891 (119905) 119889119905 (10)

3 Construction of Green Function

We will devote this section to the symposium supporting theconstruction of the Green function of the nonhomogeneousAgaciro equation (1) this will be achieved via some well-known integral operator transform The general equationunder analysis here is given as

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905)

minus 120597120573+120574119909119910

119877 (119909 119910 119905) minus 120597120574+120572

119910119905119877 (119909 119910 119905)

minus 120597120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905)

minus 119877 (119909 119910 119905) = 119891 (119909 119910 119905) 0 lt 120572 120573 120574 le 1

(11)

31 Green Function for Agaciro Equation Wewill present thegeneral solution of the Agaciro equation when 120572 = 120573 =120574 = 1 To solve this equation we make use of two integral

Mathematical Problems in Engineering 3

transforms in 119905-direction we apply the Laplace transformand in 119909-119910-direction we apply the Fourier transform

Then applying the Laplace transform on both sides of (1)we obtain the following equation

1199041205972119909119910119877119897(119909 119910 119904) minus 1205972

119909119910119877 (119909 119910 0) + 1199041205971

119909119877119897(119909 119910 119904)

minus 1205971119909119877 (119909 119910 0) minus 1205972

119909119910119877119897(119909 119910 119904) minus 1199041205971

119910119877119897(119909 119910 119904)

+ 1205971119910119877 (119909 119910 0) minus 1199041205971

119909119877119897(119909 119910 119904) + 1205971

119909119877 (119909 119910 0)

+ 119904119877119897(119909 119910 119904) minus 119877 (119909 119910 0) + 1205971

119909119877119897(119909 119910 119904)

+ 1205971119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(12)

Now if we apply the double Fourier transformonboth sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904 (119894119901) (119894119902) 119877119897119891119891

(119901 119902 119904) minus (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ 119904 (119894119901) 119877119897119891119891

(119901 119902 119904) minus (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119901) (119894119902) 119877119891119891

(119901 119902 119904) minus 119904 (119894119902) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119891119891

(119901 119902 0) minus 119904 (119894119901) 119877119897119891119891

(119901 119902 119904)

+ 119904119877119897119891119891

(119901 119902 119904) minus 119877119891119891

(119901 119902 0) + (119894119901) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(13)

The above equation can be converted as follows119877119897119891119891

(119901 119902 119904)

=1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119902) 119877119891119891

(119901 119902 0) minus 119877119891119891

(119901 119902 0))

(14)where 119877

119897119891119891(119901 119902 119904) = L(F(F(119877(119909 119910 119905)))) 119877

119891119891(119901 119902 0) =

F(F(119877(119909 119910 0)))The general Green function associated with the Agaciro

equation is given as119866 (119909 119910 119905)

=Lminus1119904

(Fminus1119901

(Fminus1119902

times (1 times (119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904

+ (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1)minus1

)))

(15)

To find the Green function it will perhaps be important torevert the denominator of the above to the suitable form thatwill allow the inversion of the two operators thus one can seethat the denominator can be factorised as follows

1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

=1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1)

(16)

With this new version in hand it is possible for us to concludethat the Green function for the Agaciro equation is given as

119866 (119909 119910 119905) = exp (119909 + 119910 + 119905) (17)For the sake of simplicity let us put

1198661(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901) (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198662(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198663(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198664(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904minus1) (119894119901 minus 1) (119894119902 minus 1))))

(18)Then we observe that

F (F (1198661(119909 119910 119905))) = (119894119901) (119894119902)F (F (119866 (119909 119910 119905)))

(19)Therefore the following relationship can be established

1198661(119909 119910 119905) = 120597

119909120597119910[119866 (119909 119910 119905)]

1198662(119909 119910 119905) = 120597

119910[119866 (119909 119910 119905)]

1198663(119909 119910 119905) = 120597

119909[119866 (119909 119910 119905)]

1198664(119909 119910 119905) = minus 119866 (119909 119910 119905)

(20)

With this material in hand it is now possible for us to con-struct the exact solution of the Agaciro equation employingthe convolution theorem

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+4

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(21)

The above solution is the exact solution to Agaciro equationWe will in the next subsection present some examples of

exact solution of Agaciro equation

32 Exact Analytical Solution of Some Agaciro Equations

Example 4 Let us consider the following nonhomogeneousAgaciro equation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 1205971

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=4 exp [minus119905 minus 1199092] cos [119910]

radic120587minus 2 exp [minus119905] cos [119910] erf [119909]

+4 exp [minus119905 minus 1199092] sin [119910]

radic120587minus 2 exp [minus119905] sin [119910] erf [119909]

(22)

4 Mathematical Problems in Engineering

with initial condition119877 (119909 119910 0) = cos (119910) erf (119909) (23)

To derive the exact solution to (22) and (23) we apply onboth sides of (22) the double Fourier-Laplace operator aspresented earlier and then we obtain the desired Greenfunction

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (24)With the above Green function in hand and making useof the initial condition equation (23) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883 119889119884 119889119879

= cos (119910) exp (minus119905) erf [119909]

(25)

This is the exact solution of the Agaciro equation(22)

Example 5 Let us now consider the nonhomogeneous Aga-ciro equation given as follows

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= minus4 exp [minus119905 minus 1199092 + 119910]

radic120587119910+ 2

exp [119910 minus 119905] erf [119909]119910

+4 exp [minus119905 minus 1199092] expIntegralEi [119910]

radic120587

minus 2 exp [minus119905] erf [119909] expIntegralEi [119910]

(26)

with initial condition119877 (119909 119910 0) = expIntegralEi [119910] erf [119909] (27)

where expIntegralEi [119910] is the exponential integral defined as

expIntegralEi [119910] = minusintinfin

minus119910

119890minus119909

119909119889119909 (28)

and erf[119909] is the error function defined as

erf [119909] = 2

radic120587int119909

0

119890minus1199102

119889119910 (29)

To obtain the exact solution to (26) and (27) we apply on bothsides of (22) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (30)

With the above Green function in hand and making useof the initial condition equation (27) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= expIntegralEi [119910] erf [119909] exp [minus119905]

(31)

This is the exact solution of the nonhomogeneous Agaciroequation (26)

Example 6 Let us consider the following nonlinear Agaciroequation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=2 exp [minus119905 + 119910]Bessel119869 [1 119909]

119910

(32)

together with initial condition119877 (119909 119910 0) = Bessel119869 [1 119909] expIntegralEi [119910] (33)

where Bessel119869[1 119909] is the Bessel function of first kind and isdefined as

Bessel119869 [120572 119909] =infin

sum119898=0

(minus1)119898

119898Γ (119898 + 120572 + 1)(119909

2)2119898+120572

(34)

To obtain the exact solution to (32) and (33) we apply on bothsides of (33) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (35)With the above Green function in hand and making useof the initial condition equation (21) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= Bessel119869 [1 119909] expIntegralEi [119910] exp [minus119905]

(36)

This is the exact solution of the nonhomogeneous Agaciroequation (32)

Mathematical Problems in Engineering 5

252015105

10

5

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

24

68

10

Time

Agaciro

x-distance

Figure 1 Exact solution of Agaciro equation for 119910 = 0

252015105

105

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

2143

658 9

710

Time

Agaciro

y-distance

Figure 2 Exact solution of Agaciro equation (22) for 119909 = 10

33 Analytical Results of the Agaciro Equations We presentin this section the numerical results of the nonhomogeneousAgaciro equation as function of time and space The numer-ical results have been depicted in Figures 1 2 3 and 4

4 Green Function of Space-Time FractionalAgaciro Equation

Wedevote this section to the symposium supporting the con-struction of the Green function for the space-time fractionalAgaciro equation (1)

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 lt 1

(37)

The construction of this Green function will be achievedvia the application of the double Fourier-Laplace operator asfollows Therefore applying the Laplace on both sides of (1)we arrive at the following equation

119904120572120597120573+120574

119909119910119877119897(119909 119910 119904) minus 120597120573+120574

119909119910119877119897(119909 119910 0) + 119904120572120597120573

119909119877119897(119909 119910 119904)

minus 120597120573119909119877119897(119909 119910 0) minus 120597120573+120574

119909119910119877119897(119909 119910 119904) minus 119904120572120597120574

119910119877119897(119909 119910 119904)

2000

minus2000

10

50

10

5

0

0

minus5minus10

Agaciro

y-distance

x-distance

Figure 3 Exact solution of Agaciro equation (26) for 119905 = 10

1000

10

500

0

10

5

5

00

minus5

minus10

minus5

minus10

Agaciro

y-distance

x-distance

Figure 4 Exact solution of Agaciro equation (32) for 119905 = 10

+ 120597120574119910119877119897(119909 119910 0) minus 119904120572120597120573

119909119877119897(119909 119910 119904) + 120597120573

119909119877119897(119909 119910 0)

+ 119904120572119877119897(119909 119910 119904) minus 119877

119897(119909 119910 0) + 120597120572

119909119877119897(119909 119910 119904)

+ 120597120573119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(38)

Now if we affect the double Fourier transformon both sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904120572(119894119901)120573

(119894119902)120574119877119897119891119891

(119901 119902 119904) minus 119904120572minus1 (119894119901)120573

(119894119902)120574

119877119891119891

(119901 119902 0)

+ 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904) minus 119904120572minus1(119894119901)120573

119877119891119891

(119901 119902 0)

minus (119894119901)120573

(119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119904120572(119894119902)120574

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119902)120574

119877119891119891

(119901 119902 0) minus 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119901)120573

119877119891119891

(119901 119902 0) + 119904120572119877119897119891119891

(119901 119902 119904)

minus 119904120572minus1119877119891119891

(119901 119902 0) + (119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ (119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(39)

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

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Mathematical Problems in Engineering

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2 Mathematical Problems in Engineering

In this present work we will present the discussion under-pinning the construction of Green function of a novel equa-tion called ldquoAgacirordquo equation presented as

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 le 1

(1)

We will examine two cases the case where the orders of thederivative are integer numbers meaning 120572 = 120573 = 120574 = 1 andthe case where 0 lt 120572 120573 120574 lt 1

2 Some Useful Information aboutFractional Derivative

With the purpose to provide lodgings to readers that are notin the field of fractional calculus we dedicate this subdivisionto the symposium supporting the fundamental principleof the fractional calculus But we will much stress on theproperties of the Caputo fractional derivative since it will beused throughout the remainder of the paper

Definition 1 (Riemann-Liouville integral [5ndash10]) The Rie-mann-Liouville integral gives the orthodox outward appear-ance of fractional calculus The theory for intermittent func-tions consequently including the ldquoboundary conditionrdquo ofrepeating after a period is the Weyl integral The Riemann-Liouville integral of order 120572 of a function 119891(119909) is given by

119886119863minus120572

119905119891 (119909) =

119886119868120572

119905119891 (119909) =

1

Γ (120572)int119909

119886

(119909 minus 119905)120572minus1119891 (119905) 119889119905 (2)

Definition 2 (Caputo fractional derivative) There is onemoreoption for working out the definition of fractional derivationCaputo established it in 1967 in his paper [11] Quite theopposite of the Riemann Liouville fractional derivative whengetting to the bottom of differential equations by meansof Caputorsquos definition it is not indispensable to describethe fractional order initial conditions Caputorsquos definition isillustrated as follows

119862

0119863120572

119905119891 (119909) =

1

Γ (119899 minus 120572)int119909

119886

(119909 minus 119905)119899minus120572minus1

119889119899

119889119905119899119891 (119905) 119889119905 (3)

The above definitions are frequently used in pure and appliedmathematics

Definition 3 The Laplace transform is an extensively usedintegral transform in the midst of numerous applicationsin physics and engineering The Laplace transform of thefunction 119891 is defined as follows

L (119891 (119909)) (119904) = intinfin

0

119890minus119904119909119891 (119909) 119889119909 (4)

Let us observe the Laplace transform of the fractional deri-vative with Caputo

L ( 1198620119863120572

119905119891 (119909)) (119904)

= 119904120572119865 (119904) minus119899minus1

sum119896=0

119904120572minus119896minus1119891(119896) (0) (119899 minus 1 lt 120572 le 119899)

(5)

The above uses the usual initial conditions or values of thefunctions

Another useful property of the Caputo derivative is thefollowing

119886119863minus120572

119905[ 1198620119863120572

119905119891 (119909)]

= 119891 (119909) minus119899minus1

sum119895=1

119891119895(0)

Γ (120572 minus 119895 + 1)119909119895 (119899 minus 1 lt 120572 le 119899)

(6)

where Γ is the gamma function defined as

Γ (119909) = intinfin

0

119905119909minus1119890minus119905119889119905 (7)

The Fourier transform of a function 119891(119909) of real variable 119905 isdefined as

(F119891) (119901) = intinfin

minusinfin

119890minus119894119909119901119891 (119909) 119889119909 (8)

The Fourier transform of a fractional derivative of order 120572 is

F [ 1198620119863120572

119905119891 (119909)] = (119894119901)

120572

F (119891) (119901) (9)

The Mellin transform of a function 119891(119905) of a real variable 119905 isinR+ is defined as

(M119891) (119901) = intinfin

0

119905119901minus1119891 (119905) 119889119905 (10)

3 Construction of Green Function

We will devote this section to the symposium supporting theconstruction of the Green function of the nonhomogeneousAgaciro equation (1) this will be achieved via some well-known integral operator transform The general equationunder analysis here is given as

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905)

minus 120597120573+120574119909119910

119877 (119909 119910 119905) minus 120597120574+120572

119910119905119877 (119909 119910 119905)

minus 120597120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905)

minus 119877 (119909 119910 119905) = 119891 (119909 119910 119905) 0 lt 120572 120573 120574 le 1

(11)

31 Green Function for Agaciro Equation Wewill present thegeneral solution of the Agaciro equation when 120572 = 120573 =120574 = 1 To solve this equation we make use of two integral

Mathematical Problems in Engineering 3

transforms in 119905-direction we apply the Laplace transformand in 119909-119910-direction we apply the Fourier transform

Then applying the Laplace transform on both sides of (1)we obtain the following equation

1199041205972119909119910119877119897(119909 119910 119904) minus 1205972

119909119910119877 (119909 119910 0) + 1199041205971

119909119877119897(119909 119910 119904)

minus 1205971119909119877 (119909 119910 0) minus 1205972

119909119910119877119897(119909 119910 119904) minus 1199041205971

119910119877119897(119909 119910 119904)

+ 1205971119910119877 (119909 119910 0) minus 1199041205971

119909119877119897(119909 119910 119904) + 1205971

119909119877 (119909 119910 0)

+ 119904119877119897(119909 119910 119904) minus 119877 (119909 119910 0) + 1205971

119909119877119897(119909 119910 119904)

+ 1205971119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(12)

Now if we apply the double Fourier transformonboth sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904 (119894119901) (119894119902) 119877119897119891119891

(119901 119902 119904) minus (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ 119904 (119894119901) 119877119897119891119891

(119901 119902 119904) minus (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119901) (119894119902) 119877119891119891

(119901 119902 119904) minus 119904 (119894119902) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119891119891

(119901 119902 0) minus 119904 (119894119901) 119877119897119891119891

(119901 119902 119904)

+ 119904119877119897119891119891

(119901 119902 119904) minus 119877119891119891

(119901 119902 0) + (119894119901) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(13)

The above equation can be converted as follows119877119897119891119891

(119901 119902 119904)

=1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119902) 119877119891119891

(119901 119902 0) minus 119877119891119891

(119901 119902 0))

(14)where 119877

119897119891119891(119901 119902 119904) = L(F(F(119877(119909 119910 119905)))) 119877

119891119891(119901 119902 0) =

F(F(119877(119909 119910 0)))The general Green function associated with the Agaciro

equation is given as119866 (119909 119910 119905)

=Lminus1119904

(Fminus1119901

(Fminus1119902

times (1 times (119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904

+ (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1)minus1

)))

(15)

To find the Green function it will perhaps be important torevert the denominator of the above to the suitable form thatwill allow the inversion of the two operators thus one can seethat the denominator can be factorised as follows

1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

=1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1)

(16)

With this new version in hand it is possible for us to concludethat the Green function for the Agaciro equation is given as

119866 (119909 119910 119905) = exp (119909 + 119910 + 119905) (17)For the sake of simplicity let us put

1198661(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901) (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198662(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198663(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198664(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904minus1) (119894119901 minus 1) (119894119902 minus 1))))

(18)Then we observe that

F (F (1198661(119909 119910 119905))) = (119894119901) (119894119902)F (F (119866 (119909 119910 119905)))

(19)Therefore the following relationship can be established

1198661(119909 119910 119905) = 120597

119909120597119910[119866 (119909 119910 119905)]

1198662(119909 119910 119905) = 120597

119910[119866 (119909 119910 119905)]

1198663(119909 119910 119905) = 120597

119909[119866 (119909 119910 119905)]

1198664(119909 119910 119905) = minus 119866 (119909 119910 119905)

(20)

With this material in hand it is now possible for us to con-struct the exact solution of the Agaciro equation employingthe convolution theorem

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+4

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(21)

The above solution is the exact solution to Agaciro equationWe will in the next subsection present some examples of

exact solution of Agaciro equation

32 Exact Analytical Solution of Some Agaciro Equations

Example 4 Let us consider the following nonhomogeneousAgaciro equation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 1205971

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=4 exp [minus119905 minus 1199092] cos [119910]

radic120587minus 2 exp [minus119905] cos [119910] erf [119909]

+4 exp [minus119905 minus 1199092] sin [119910]

radic120587minus 2 exp [minus119905] sin [119910] erf [119909]

(22)

4 Mathematical Problems in Engineering

with initial condition119877 (119909 119910 0) = cos (119910) erf (119909) (23)

To derive the exact solution to (22) and (23) we apply onboth sides of (22) the double Fourier-Laplace operator aspresented earlier and then we obtain the desired Greenfunction

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (24)With the above Green function in hand and making useof the initial condition equation (23) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883 119889119884 119889119879

= cos (119910) exp (minus119905) erf [119909]

(25)

This is the exact solution of the Agaciro equation(22)

Example 5 Let us now consider the nonhomogeneous Aga-ciro equation given as follows

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= minus4 exp [minus119905 minus 1199092 + 119910]

radic120587119910+ 2

exp [119910 minus 119905] erf [119909]119910

+4 exp [minus119905 minus 1199092] expIntegralEi [119910]

radic120587

minus 2 exp [minus119905] erf [119909] expIntegralEi [119910]

(26)

with initial condition119877 (119909 119910 0) = expIntegralEi [119910] erf [119909] (27)

where expIntegralEi [119910] is the exponential integral defined as

expIntegralEi [119910] = minusintinfin

minus119910

119890minus119909

119909119889119909 (28)

and erf[119909] is the error function defined as

erf [119909] = 2

radic120587int119909

0

119890minus1199102

119889119910 (29)

To obtain the exact solution to (26) and (27) we apply on bothsides of (22) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (30)

With the above Green function in hand and making useof the initial condition equation (27) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= expIntegralEi [119910] erf [119909] exp [minus119905]

(31)

This is the exact solution of the nonhomogeneous Agaciroequation (26)

Example 6 Let us consider the following nonlinear Agaciroequation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=2 exp [minus119905 + 119910]Bessel119869 [1 119909]

119910

(32)

together with initial condition119877 (119909 119910 0) = Bessel119869 [1 119909] expIntegralEi [119910] (33)

where Bessel119869[1 119909] is the Bessel function of first kind and isdefined as

Bessel119869 [120572 119909] =infin

sum119898=0

(minus1)119898

119898Γ (119898 + 120572 + 1)(119909

2)2119898+120572

(34)

To obtain the exact solution to (32) and (33) we apply on bothsides of (33) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (35)With the above Green function in hand and making useof the initial condition equation (21) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= Bessel119869 [1 119909] expIntegralEi [119910] exp [minus119905]

(36)

This is the exact solution of the nonhomogeneous Agaciroequation (32)

Mathematical Problems in Engineering 5

252015105

10

5

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

24

68

10

Time

Agaciro

x-distance

Figure 1 Exact solution of Agaciro equation for 119910 = 0

252015105

105

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

2143

658 9

710

Time

Agaciro

y-distance

Figure 2 Exact solution of Agaciro equation (22) for 119909 = 10

33 Analytical Results of the Agaciro Equations We presentin this section the numerical results of the nonhomogeneousAgaciro equation as function of time and space The numer-ical results have been depicted in Figures 1 2 3 and 4

4 Green Function of Space-Time FractionalAgaciro Equation

Wedevote this section to the symposium supporting the con-struction of the Green function for the space-time fractionalAgaciro equation (1)

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 lt 1

(37)

The construction of this Green function will be achievedvia the application of the double Fourier-Laplace operator asfollows Therefore applying the Laplace on both sides of (1)we arrive at the following equation

119904120572120597120573+120574

119909119910119877119897(119909 119910 119904) minus 120597120573+120574

119909119910119877119897(119909 119910 0) + 119904120572120597120573

119909119877119897(119909 119910 119904)

minus 120597120573119909119877119897(119909 119910 0) minus 120597120573+120574

119909119910119877119897(119909 119910 119904) minus 119904120572120597120574

119910119877119897(119909 119910 119904)

2000

minus2000

10

50

10

5

0

0

minus5minus10

Agaciro

y-distance

x-distance

Figure 3 Exact solution of Agaciro equation (26) for 119905 = 10

1000

10

500

0

10

5

5

00

minus5

minus10

minus5

minus10

Agaciro

y-distance

x-distance

Figure 4 Exact solution of Agaciro equation (32) for 119905 = 10

+ 120597120574119910119877119897(119909 119910 0) minus 119904120572120597120573

119909119877119897(119909 119910 119904) + 120597120573

119909119877119897(119909 119910 0)

+ 119904120572119877119897(119909 119910 119904) minus 119877

119897(119909 119910 0) + 120597120572

119909119877119897(119909 119910 119904)

+ 120597120573119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(38)

Now if we affect the double Fourier transformon both sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904120572(119894119901)120573

(119894119902)120574119877119897119891119891

(119901 119902 119904) minus 119904120572minus1 (119894119901)120573

(119894119902)120574

119877119891119891

(119901 119902 0)

+ 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904) minus 119904120572minus1(119894119901)120573

119877119891119891

(119901 119902 0)

minus (119894119901)120573

(119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119904120572(119894119902)120574

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119902)120574

119877119891119891

(119901 119902 0) minus 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119901)120573

119877119891119891

(119901 119902 0) + 119904120572119877119897119891119891

(119901 119902 119904)

minus 119904120572minus1119877119891119891

(119901 119902 0) + (119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ (119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(39)

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

transforms in 119905-direction we apply the Laplace transformand in 119909-119910-direction we apply the Fourier transform

Then applying the Laplace transform on both sides of (1)we obtain the following equation

1199041205972119909119910119877119897(119909 119910 119904) minus 1205972

119909119910119877 (119909 119910 0) + 1199041205971

119909119877119897(119909 119910 119904)

minus 1205971119909119877 (119909 119910 0) minus 1205972

119909119910119877119897(119909 119910 119904) minus 1199041205971

119910119877119897(119909 119910 119904)

+ 1205971119910119877 (119909 119910 0) minus 1199041205971

119909119877119897(119909 119910 119904) + 1205971

119909119877 (119909 119910 0)

+ 119904119877119897(119909 119910 119904) minus 119877 (119909 119910 0) + 1205971

119909119877119897(119909 119910 119904)

+ 1205971119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(12)

Now if we apply the double Fourier transformonboth sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904 (119894119901) (119894119902) 119877119897119891119891

(119901 119902 119904) minus (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ 119904 (119894119901) 119877119897119891119891

(119901 119902 119904) minus (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119901) (119894119902) 119877119891119891

(119901 119902 119904) minus 119904 (119894119902) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119891119891

(119901 119902 0) minus 119904 (119894119901) 119877119897119891119891

(119901 119902 119904)

+ 119904119877119897119891119891

(119901 119902 119904) minus 119877119891119891

(119901 119902 0) + (119894119901) 119877119897119891119891

(119901 119902 119904)

+ (119894119902) 119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(13)

The above equation can be converted as follows119877119897119891119891

(119901 119902 119904)

=1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901) (119894119902) 119877119891119891

(119901 119902 0)

+ (119894119901) 119877119891119891

(119901 119902 0)

minus (119894119902) 119877119891119891

(119901 119902 0) minus 119877119891119891

(119901 119902 0))

(14)where 119877

119897119891119891(119901 119902 119904) = L(F(F(119877(119909 119910 119905)))) 119877

119891119891(119901 119902 0) =

F(F(119877(119909 119910 0)))The general Green function associated with the Agaciro

equation is given as119866 (119909 119910 119905)

=Lminus1119904

(Fminus1119901

(Fminus1119902

times (1 times (119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904

+ (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1)minus1

)))

(15)

To find the Green function it will perhaps be important torevert the denominator of the above to the suitable form thatwill allow the inversion of the two operators thus one can seethat the denominator can be factorised as follows

1

119904 (119894119901) (119894119902) minus 119904 (119894119901) minus 119904 (119894119902) + 119904 + (119894119901) minus (119894119902) (119894119901) + (119894119902) minus 1

=1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1)

(16)

With this new version in hand it is possible for us to concludethat the Green function for the Agaciro equation is given as

119866 (119909 119910 119905) = exp (119909 + 119910 + 119905) (17)For the sake of simplicity let us put

1198661(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901) (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198662(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus (119894119902)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198663(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

((119894119901)

(119904minus1) (119894119901minus1) (119894119902 minus 1))))

1198664(119909 119910 119905)=Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904minus1) (119894119901 minus 1) (119894119902 minus 1))))

(18)Then we observe that

F (F (1198661(119909 119910 119905))) = (119894119901) (119894119902)F (F (119866 (119909 119910 119905)))

(19)Therefore the following relationship can be established

1198661(119909 119910 119905) = 120597

119909120597119910[119866 (119909 119910 119905)]

1198662(119909 119910 119905) = 120597

119910[119866 (119909 119910 119905)]

1198663(119909 119910 119905) = 120597

119909[119866 (119909 119910 119905)]

1198664(119909 119910 119905) = minus 119866 (119909 119910 119905)

(20)

With this material in hand it is now possible for us to con-struct the exact solution of the Agaciro equation employingthe convolution theorem

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+4

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(21)

The above solution is the exact solution to Agaciro equationWe will in the next subsection present some examples of

exact solution of Agaciro equation

32 Exact Analytical Solution of Some Agaciro Equations

Example 4 Let us consider the following nonhomogeneousAgaciro equation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 1205971

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=4 exp [minus119905 minus 1199092] cos [119910]

radic120587minus 2 exp [minus119905] cos [119910] erf [119909]

+4 exp [minus119905 minus 1199092] sin [119910]

radic120587minus 2 exp [minus119905] sin [119910] erf [119909]

(22)

4 Mathematical Problems in Engineering

with initial condition119877 (119909 119910 0) = cos (119910) erf (119909) (23)

To derive the exact solution to (22) and (23) we apply onboth sides of (22) the double Fourier-Laplace operator aspresented earlier and then we obtain the desired Greenfunction

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (24)With the above Green function in hand and making useof the initial condition equation (23) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883 119889119884 119889119879

= cos (119910) exp (minus119905) erf [119909]

(25)

This is the exact solution of the Agaciro equation(22)

Example 5 Let us now consider the nonhomogeneous Aga-ciro equation given as follows

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= minus4 exp [minus119905 minus 1199092 + 119910]

radic120587119910+ 2

exp [119910 minus 119905] erf [119909]119910

+4 exp [minus119905 minus 1199092] expIntegralEi [119910]

radic120587

minus 2 exp [minus119905] erf [119909] expIntegralEi [119910]

(26)

with initial condition119877 (119909 119910 0) = expIntegralEi [119910] erf [119909] (27)

where expIntegralEi [119910] is the exponential integral defined as

expIntegralEi [119910] = minusintinfin

minus119910

119890minus119909

119909119889119909 (28)

and erf[119909] is the error function defined as

erf [119909] = 2

radic120587int119909

0

119890minus1199102

119889119910 (29)

To obtain the exact solution to (26) and (27) we apply on bothsides of (22) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (30)

With the above Green function in hand and making useof the initial condition equation (27) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= expIntegralEi [119910] erf [119909] exp [minus119905]

(31)

This is the exact solution of the nonhomogeneous Agaciroequation (26)

Example 6 Let us consider the following nonlinear Agaciroequation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=2 exp [minus119905 + 119910]Bessel119869 [1 119909]

119910

(32)

together with initial condition119877 (119909 119910 0) = Bessel119869 [1 119909] expIntegralEi [119910] (33)

where Bessel119869[1 119909] is the Bessel function of first kind and isdefined as

Bessel119869 [120572 119909] =infin

sum119898=0

(minus1)119898

119898Γ (119898 + 120572 + 1)(119909

2)2119898+120572

(34)

To obtain the exact solution to (32) and (33) we apply on bothsides of (33) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (35)With the above Green function in hand and making useof the initial condition equation (21) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= Bessel119869 [1 119909] expIntegralEi [119910] exp [minus119905]

(36)

This is the exact solution of the nonhomogeneous Agaciroequation (32)

Mathematical Problems in Engineering 5

252015105

10

5

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

24

68

10

Time

Agaciro

x-distance

Figure 1 Exact solution of Agaciro equation for 119910 = 0

252015105

105

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

2143

658 9

710

Time

Agaciro

y-distance

Figure 2 Exact solution of Agaciro equation (22) for 119909 = 10

33 Analytical Results of the Agaciro Equations We presentin this section the numerical results of the nonhomogeneousAgaciro equation as function of time and space The numer-ical results have been depicted in Figures 1 2 3 and 4

4 Green Function of Space-Time FractionalAgaciro Equation

Wedevote this section to the symposium supporting the con-struction of the Green function for the space-time fractionalAgaciro equation (1)

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 lt 1

(37)

The construction of this Green function will be achievedvia the application of the double Fourier-Laplace operator asfollows Therefore applying the Laplace on both sides of (1)we arrive at the following equation

119904120572120597120573+120574

119909119910119877119897(119909 119910 119904) minus 120597120573+120574

119909119910119877119897(119909 119910 0) + 119904120572120597120573

119909119877119897(119909 119910 119904)

minus 120597120573119909119877119897(119909 119910 0) minus 120597120573+120574

119909119910119877119897(119909 119910 119904) minus 119904120572120597120574

119910119877119897(119909 119910 119904)

2000

minus2000

10

50

10

5

0

0

minus5minus10

Agaciro

y-distance

x-distance

Figure 3 Exact solution of Agaciro equation (26) for 119905 = 10

1000

10

500

0

10

5

5

00

minus5

minus10

minus5

minus10

Agaciro

y-distance

x-distance

Figure 4 Exact solution of Agaciro equation (32) for 119905 = 10

+ 120597120574119910119877119897(119909 119910 0) minus 119904120572120597120573

119909119877119897(119909 119910 119904) + 120597120573

119909119877119897(119909 119910 0)

+ 119904120572119877119897(119909 119910 119904) minus 119877

119897(119909 119910 0) + 120597120572

119909119877119897(119909 119910 119904)

+ 120597120573119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(38)

Now if we affect the double Fourier transformon both sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904120572(119894119901)120573

(119894119902)120574119877119897119891119891

(119901 119902 119904) minus 119904120572minus1 (119894119901)120573

(119894119902)120574

119877119891119891

(119901 119902 0)

+ 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904) minus 119904120572minus1(119894119901)120573

119877119891119891

(119901 119902 0)

minus (119894119901)120573

(119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119904120572(119894119902)120574

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119902)120574

119877119891119891

(119901 119902 0) minus 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119901)120573

119877119891119891

(119901 119902 0) + 119904120572119877119897119891119891

(119901 119902 119904)

minus 119904120572minus1119877119891119891

(119901 119902 0) + (119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ (119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(39)

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

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Mathematical Problems in Engineering

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4 Mathematical Problems in Engineering

with initial condition119877 (119909 119910 0) = cos (119910) erf (119909) (23)

To derive the exact solution to (22) and (23) we apply onboth sides of (22) the double Fourier-Laplace operator aspresented earlier and then we obtain the desired Greenfunction

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (24)With the above Green function in hand and making useof the initial condition equation (23) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883 119889119884 119889119879

= cos (119910) exp (minus119905) erf [119909]

(25)

This is the exact solution of the Agaciro equation(22)

Example 5 Let us now consider the nonhomogeneous Aga-ciro equation given as follows

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= minus4 exp [minus119905 minus 1199092 + 119910]

radic120587119910+ 2

exp [119910 minus 119905] erf [119909]119910

+4 exp [minus119905 minus 1199092] expIntegralEi [119910]

radic120587

minus 2 exp [minus119905] erf [119909] expIntegralEi [119910]

(26)

with initial condition119877 (119909 119910 0) = expIntegralEi [119910] erf [119909] (27)

where expIntegralEi [119910] is the exponential integral defined as

expIntegralEi [119910] = minusintinfin

minus119910

119890minus119909

119909119889119909 (28)

and erf[119909] is the error function defined as

erf [119909] = 2

radic120587int119909

0

119890minus1199102

119889119910 (29)

To obtain the exact solution to (26) and (27) we apply on bothsides of (22) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (30)

With the above Green function in hand and making useof the initial condition equation (27) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= expIntegralEi [119910] erf [119909] exp [minus119905]

(31)

This is the exact solution of the nonhomogeneous Agaciroequation (26)

Example 6 Let us consider the following nonlinear Agaciroequation

1205973119909119910119905

119877 (119909 119910 119905) + 1205972119909119905119877 (119909 119910 119905) minus 1205972

119909119910119877 (119909 119910 119905)

minus 1205972119910119905119877 (119909 119910 119905) minus 1205972

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 1205971119909119877 (119909 119910 119905) + 1205971

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

=2 exp [minus119905 + 119910]Bessel119869 [1 119909]

119910

(32)

together with initial condition119877 (119909 119910 0) = Bessel119869 [1 119909] expIntegralEi [119910] (33)

where Bessel119869[1 119909] is the Bessel function of first kind and isdefined as

Bessel119869 [120572 119909] =infin

sum119898=0

(minus1)119898

119898Γ (119898 + 120572 + 1)(119909

2)2119898+120572

(34)

To obtain the exact solution to (32) and (33) we apply on bothsides of (33) the double Fourier-Laplace operator as offeredearlier and then we obtain the desired Green function

119866 (119909 119910 119905) = exp [119909 + 119910 + 119905] (35)With the above Green function in hand and making useof the initial condition equation (21) we can now obtainstraightforward the following solution

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866 (119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

= Bessel119869 [1 119909] expIntegralEi [119910] exp [minus119905]

(36)

This is the exact solution of the nonhomogeneous Agaciroequation (32)

Mathematical Problems in Engineering 5

252015105

10

5

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

24

68

10

Time

Agaciro

x-distance

Figure 1 Exact solution of Agaciro equation for 119910 = 0

252015105

105

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

2143

658 9

710

Time

Agaciro

y-distance

Figure 2 Exact solution of Agaciro equation (22) for 119909 = 10

33 Analytical Results of the Agaciro Equations We presentin this section the numerical results of the nonhomogeneousAgaciro equation as function of time and space The numer-ical results have been depicted in Figures 1 2 3 and 4

4 Green Function of Space-Time FractionalAgaciro Equation

Wedevote this section to the symposium supporting the con-struction of the Green function for the space-time fractionalAgaciro equation (1)

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 lt 1

(37)

The construction of this Green function will be achievedvia the application of the double Fourier-Laplace operator asfollows Therefore applying the Laplace on both sides of (1)we arrive at the following equation

119904120572120597120573+120574

119909119910119877119897(119909 119910 119904) minus 120597120573+120574

119909119910119877119897(119909 119910 0) + 119904120572120597120573

119909119877119897(119909 119910 119904)

minus 120597120573119909119877119897(119909 119910 0) minus 120597120573+120574

119909119910119877119897(119909 119910 119904) minus 119904120572120597120574

119910119877119897(119909 119910 119904)

2000

minus2000

10

50

10

5

0

0

minus5minus10

Agaciro

y-distance

x-distance

Figure 3 Exact solution of Agaciro equation (26) for 119905 = 10

1000

10

500

0

10

5

5

00

minus5

minus10

minus5

minus10

Agaciro

y-distance

x-distance

Figure 4 Exact solution of Agaciro equation (32) for 119905 = 10

+ 120597120574119910119877119897(119909 119910 0) minus 119904120572120597120573

119909119877119897(119909 119910 119904) + 120597120573

119909119877119897(119909 119910 0)

+ 119904120572119877119897(119909 119910 119904) minus 119877

119897(119909 119910 0) + 120597120572

119909119877119897(119909 119910 119904)

+ 120597120573119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(38)

Now if we affect the double Fourier transformon both sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904120572(119894119901)120573

(119894119902)120574119877119897119891119891

(119901 119902 119904) minus 119904120572minus1 (119894119901)120573

(119894119902)120574

119877119891119891

(119901 119902 0)

+ 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904) minus 119904120572minus1(119894119901)120573

119877119891119891

(119901 119902 0)

minus (119894119901)120573

(119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119904120572(119894119902)120574

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119902)120574

119877119891119891

(119901 119902 0) minus 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119901)120573

119877119891119891

(119901 119902 0) + 119904120572119877119897119891119891

(119901 119902 119904)

minus 119904120572minus1119877119891119891

(119901 119902 0) + (119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ (119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(39)

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

252015105

10

5

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

24

68

10

Time

Agaciro

x-distance

Figure 1 Exact solution of Agaciro equation for 119910 = 0

252015105

105

0minus5minus10

0

0

minus5minus10

minus15minus20minus25

2143

658 9

710

Time

Agaciro

y-distance

Figure 2 Exact solution of Agaciro equation (22) for 119909 = 10

33 Analytical Results of the Agaciro Equations We presentin this section the numerical results of the nonhomogeneousAgaciro equation as function of time and space The numer-ical results have been depicted in Figures 1 2 3 and 4

4 Green Function of Space-Time FractionalAgaciro Equation

Wedevote this section to the symposium supporting the con-struction of the Green function for the space-time fractionalAgaciro equation (1)

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= 119891 (119909 119910 119905) 0 lt 120572 120573 120574 lt 1

(37)

The construction of this Green function will be achievedvia the application of the double Fourier-Laplace operator asfollows Therefore applying the Laplace on both sides of (1)we arrive at the following equation

119904120572120597120573+120574

119909119910119877119897(119909 119910 119904) minus 120597120573+120574

119909119910119877119897(119909 119910 0) + 119904120572120597120573

119909119877119897(119909 119910 119904)

minus 120597120573119909119877119897(119909 119910 0) minus 120597120573+120574

119909119910119877119897(119909 119910 119904) minus 119904120572120597120574

119910119877119897(119909 119910 119904)

2000

minus2000

10

50

10

5

0

0

minus5minus10

Agaciro

y-distance

x-distance

Figure 3 Exact solution of Agaciro equation (26) for 119905 = 10

1000

10

500

0

10

5

5

00

minus5

minus10

minus5

minus10

Agaciro

y-distance

x-distance

Figure 4 Exact solution of Agaciro equation (32) for 119905 = 10

+ 120597120574119910119877119897(119909 119910 0) minus 119904120572120597120573

119909119877119897(119909 119910 119904) + 120597120573

119909119877119897(119909 119910 0)

+ 119904120572119877119897(119909 119910 119904) minus 119877

119897(119909 119910 0) + 120597120572

119909119877119897(119909 119910 119904)

+ 120597120573119910119877119897(119909 119910 119904) minus 119877

119897(119909 119910 119904) = 119891

119897(119909 119910 119904)

(38)

Now if we affect the double Fourier transformon both sides ofthe above equation meaning we apply the Fourier transformin 119909 and 119910 direction we obtain the following homogeneousequation

119904120572(119894119901)120573

(119894119902)120574119877119897119891119891

(119901 119902 119904) minus 119904120572minus1 (119894119901)120573

(119894119902)120574

119877119891119891

(119901 119902 0)

+ 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904) minus 119904120572minus1(119894119901)120573

119877119891119891

(119901 119902 0)

minus (119894119901)120573

(119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119904120572(119894119902)120574

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119902)120574

119877119891119891

(119901 119902 0) minus 119904120572(119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ 119904120572minus1 (119894119901)120573

119877119891119891

(119901 119902 0) + 119904120572119877119897119891119891

(119901 119902 119904)

minus 119904120572minus1119877119891119891

(119901 119902 0) + (119894119901)120573

119877119897119891119891

(119901 119902 119904)

+ (119894119902)120574

119877119897119891119891

(119901 119902 119904) minus 119877119897119891119891

(119901 119902 119904) = 119891119897119891119891

(119901 119902 119904)

(39)

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The above equation can be transformed as follows

119877119897119891119891

(119901 119902 119904)

= 1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1)minus1

times (119891119897119891119891

(119901 119902 sdot 119904) + (119894119901)120573

(119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

+ (119894119901)120573

119904120572minus1119877119891119891

(119901 119902 0) minus (119894119902)120574

119904120572minus1119877119891119891

(119901 119902 0)

minus 119904120572minus1119877119891119891

(119901 119902 0) )

(40)

The broad-spectrum fractional Green function that connectsto the space-time fractional Agaciro equation is provided as

119866120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

(1 (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+(119894119902)120574

minus 1)minus1

)))

(41)

It is perhaps important to point out that the below equation isthe fractional characteristic equation associate to the space-time fractional Agaciro equation

119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+ 119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120573

+ (119894119902)120574

minus 1(42)

To discover the Green function it will conceivably relapsethe above fractional polynomial equation to the appropriateform that will consent to the inversion of the two operatorsconsequently one can see that the denominator can befactorised as follows

1 times (119904120572(119894119901)120573

(119894119902)120574

minus 119904120572(119894119901)120573

minus 119904120572(119894119902)120574

+119904120572 + (119894119901)120573

minus (119894119902)120574

(119894119901)120572

+ (119894119902)120574

minus 1)minus1

= 1 times ((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

(43)

In order to find the inverse of the above equation we willfirst accommodate readers that are not used to relation-ships between some special functions and some propertiesof Fourier and Laplace transforms An important specialfunction in the field of fractional calculus is theMittag-Lefflerfunction which is regarded as the generalized exponentialfunction and is defined as

119864120572[119911] =

infin

sum119899=0

119911119899

Γ [120572119899 + 1] (119911 isin CRe [120572] gt 0) (44)

The Laplace transform of the above function is given as

L (119864120572[120591119911]) (119904) =

119904120572minus1

119904120572 minus 120591 119877 (119904) gt 0 120591 isin C

1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(45)

Another useful relation here is the following

L (119911120573minus1119864120572[120591119911]) (119904) =

119904120572minus120573

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(46)

If we assume from the above equation that 120572 = 120573 then weobtain the following useful relationship

L (119911120572minus1119864120572[120591119911]) (119904) =

1

119904120572 minus 120591

119877 (119904) gt 0 120591 isin C1003816100381610038161003816120591119904minus1205721003816100381610038161003816 lt 1

(47)

With this new version in hand together with the above usefulproperties it will be possible for us to conclude that theGreenfunction for the space-time fractional Agaciro equation isgiven as

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (48)

Now to have a clear relationship between the inverse of thefractional Green function and the remaining terms we let

1198661120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

(119894119902)120574

119904120572minus1

times((119904120572minus1) ((119894119901)120573

minus1) ((119894119902)120574

minus 1))minus1

)))

1198662120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

( minus (119894119902)120574

119904120572minus1

times ((119904120572minus1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198663120572120573120574 (119909 119910 119905)

= Lminus1

119904(Fminus1119901

times (Fminus1119902

((119894119901)120573

119904120572minus1

times((119904120572 minus 1) ((119894119901)120573

minus 1) ((119894119902)120574

minus 1))minus1

)))

1198664120572120573120574

(119909 119910 119905)

= Lminus1

119904(Fminus1119901

(Fminus1119902

(minus1

(119904 minus 1) (119894119901 minus 1) (119894119902 minus 1))))

= minus119866120572120573120574

(119909 119910 119905)

(49)

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

With these functions together with the fractional Greenfunction we can further derive the exact solution of the classof fractional partial differential equation as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

+5

sum119895=1

int119905

0

∬infin

minusinfin

119866119895120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119877 (119883 119884 0) 119889119883119889119884119889119879

(50)

Example 7 Consider the following time-space fractionalAgaciro equation

120597120572+120573+120574

119909119910119905119877 (119909 119910 119905) + 120597

120572+120573

119909119905119877 (119909 119910 119905) minus 120597120573+120574

119909119910119877 (119909 119910 119905)

minus 120597120574+120572

119910119905119877 (119909 119910 119905) minus 120597

120573+120572

119909119905119877 (119909 119910 119905) + 120597120572

119905119877 (119909 119910 119905)

+ 120597120573119909119877 (119909 119910 119905) + 120597120574

119910119877 (119909 119910 119905) minus 119877 (119909 119910 119905)

= exp [119909 + 119910 + 119905] 0 lt 120572 120573 120574 lt 1

(51)

with initial condition

119877 (119909 119910 0) = 0 (52)

Now to solve the above equation we follow the discussionpresented earlier to obtain the desired Green function

119866120572120573120574

(119909 119910 119905) = 119905120572minus1119864120572(119905) 119909120573minus1119864120573(119909) 119910120574minus1119864120574(119910) (53)

Now using the convolution theorem together with the initialcondition we obtain the exact solution of (51) as

119877 (119909 119910 119905) = int119905

0

∬infin

minusinfin

119866120572120573120574

(119909 minus 119883 119910 minus 119884 119905 minus 119879)

times 119891 (119883 119884 119879) 119889119883119889119884119889119879

(54)

since the convolution is commutative we can reformulate theabove equation as

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119866120572120573120574

(119883 119884 119879)

times 119891 (119909 minus 119883 119910 minus 119884 119905 minus 119879) 119889119883119889119884119889119879

119877 (119909 119910 119905) = int119905

0

int119909

0

int119910

0

119879120572minus1119864120572(119879)119883

120573minus1119864120573(119883) 119884

120574minus1119864120574(119884)

times exp (119909 minus 119883) exp (119910 minus 119884)

times exp (119905 minus 119879) 119889119883119889119884119889119879

(55)

Now using the notation of the Mittag-Leffler function givenin (44) we can reformulate the above equation as follows

119877 (119909 119910 119905) = exp (119909 + 119910 + 119905)

times int119905

0

infin

sum119899=0

(minus119879)119899+120572minus1

Γ (120572119899 + 1)exp (minus119879) 119889119879

times int119909

0

infin

sum119896=0

(minus119883)119896+120573minus1

Γ (120573119896 + 1)exp (minus119883) 119889119883

times int119909

0

infin

sum119898=0

(minus119884)119898+120574minus1

Γ (120574119898 + 1)exp (minus119884) 119889119884

(56)

Integrating the above equation we obtain the following exactsolution of the space-time Agaciro equation (51) as

119877 (119909 119910 119905)

= exp (119909 + 119910 + 119905)infin

sum119899=0

infin

sum119898=0

infin

sum119896=0

(minus1)119899+120572

times (minus1)119896+1

(minus1)119898+1

times [minusΓ [119899 + 120572] + Γ [119899 + 120572 119905]]

times [minusΓ [119896 + 120573] + Γ [119896 + 120573 119909]]

times [minusΓ [119898 + 120574] + Γ [119898 + 120574 119910]]

(57)

5 Conclusion

The aim of this work was to investigate a class of partialdifferential equations within the concept of integer andfractional order derivative This class of equations is referredto as Agaciro equations We presented the general solutiontogether with some examples of this equation within thescope of ordinary and fractional derivation To achieve thiswemade use of the so-calledGreen functionmethod togetherwith some well-known integral transform operators

Conflict of Interests

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

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley-InterscienceJohn Wiley amp Sons New York NY USA 1993

[3] S S Bayin Mathematical Methods in Science and Engineeringchapter 18-19 John Wiley amp Sons Hoboken NJ USA 2006

[4] A D Polyanin andV F ZaitsevHandbook of Exact Solutions forOrdinary Differential Equations Chapman amp HallCRC PressBoca Raton Fla USA 2nd edition 2003

[5] S G Samko A A Kilbas and O I Marichev Fractional Inte-grals andDerivativesTheory andApplications Gordon and Bre-ach Yverdon Switzerland 1993

[6] I Podlubny ldquoGeometric and physical interpretation of fraction-al integration and fractional differentiationrdquo Fractional Calculusamp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[7] A A Kilbas HM Srivastava and J J TrujilloTheory and App-lications of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science AmsterdamTheNetherlands 2006

[8] A Atangana and A Secer ldquoA note on fractional order deriva-tives and table of fractional derivatives of some special func-tionsrdquo Abstract and Applied Analysis vol 2013 Article ID279681 8 pages 2013

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[9] A Anatoly J Juan and M S Hari Theory and Application ofFractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Amsterdam The Netherlands2006

[10] Y Luchko and R Groneflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativePreprint Series A08-98 Fachbereich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[11] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency independentmdashIIrdquo Geophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of