http://scma.maragheh.ac.ir

## ANALYTICAL SOLUTIONS FOR THE FRACTIONAL

## FISHER’S EQUATION

H. KHEIRI1_{∗}

, A. MOJAVER2

, AND S. SHAHI3

Abstract. In this paper, we consider the inhomogeneous time-fractional nonlinear Fisher equation with three known boundary conditions. We first apply a modified Homotopy perturbation method for translating the proposed problem to a set of linear problems. Then we use the separation variables method to solve obtained problems. In examples, we illustrate that by right choice of source term in the modified Homotopy perturbation method, it is possible to get an exact solution.

## 1.

## Introduction

## Fractional partial differential equations (FPDEs) have recently aroused

## considerable interest in mathematics and its applications. Scientists

## used them to model many physical, biological and chemical processes

## [16, 18, 19]. Besides, they have applications in sampling, hold

## algo-rithms, and signal processing. There are various analytical method for

## solving nonlinear FPDEs including the Adomian decomposition [2, 4, 9],

## Homotopy perturbation method [5, 6, 13], Variation iteration method

## [14, 15], and Homotopy analysis method [10, 11]. Finding exact solutions

## for FPDEs are often too complicated and required so much calculations.

## Nowadays, biological models have been the focus of many mathematical

## scientists. Fisher’s equation

## (1.1)

## u

t## =

## u

xx## +

## u

## (1

## −

## u

## )

2010Mathematics Subject Classification. 34B24, 34B27.

Key words and phrases. Fractional Fisher’s equation, Mittag-Leffler, Method of separating variables.

Received: 5 November 2014, Accepted: 7 December 2014.

∗_{Corresponding author.}

## was first proposed by Fisher as a model for the propagation of a mutant

## gene [10]. In this model,

## u

## (

## x, t

## ) is the population density and

## u

## (

## u

## −

## 1)

## denotes the logistic form.

## In this paper, we consider the inhomogeneous time-fractional Fisher’s

## equation which is examined in [22, 21]:

## (1.2)

## D

α_{t}

## u

## (

## x, t

## ) =

## u

xx## (

## x, t

## )+

## u

## (

## x, t

## )(1

## −

## u

## (

## x, t

## ))+

## f

## (

## x, t

## )

## ,

## 0

## < x < L,

## 0

## < α

## ≤

## 1

## ,

## where the fractional derivative in (1.2) is the Caputo derivative.

## We introduce a scheme to solve (1.2) which is a combination of

## mod-ified homotopy perturbation [5], an especial case of homotopy analysis

## method, Laplace transform, and separation of variables. This kind of

## modification of homotopy perturbation method has the capability of

## transforming nonlinear terms into linear ones while homotopy analysis

## method and homotopy perturbation method don’t possess this

## charac-teristic.

## 2.

## Preliminaries

## In this section, we give some necessary definitions and lemmas about

## fractional calculus. For some details, you can refer to [18, 20].

## Definition 2.1

## ([3])

## .

## A function

## f

## :

## R

_{→}

## R

+_{is said to be in the space}

## C

ν## , with

## ν

## ∈

## R

## , if it can be written as

## f

## (

## x

## ) =

## x

p## f

1## (

## x

## ) with

## p > ν

## ,

## f

1## (

## x

## )

## ∈

## C

## [0

## ,

## ∞

## ) and it is said to be in the space

## C

νm## if

## f

(m)## ∈

## C

ν## for

## m

## ∈

## N

## ∪

_{{}

_{0}

_{}}

_{.}

## Definition 2.2

## ([12])

## .

## The Riemann-Liouville fractional integral of

## f

## ∈

## C

ν## with order

## α >

## 0 and

## ν

## ≥ −

## 1 is defined as:

## J

α## f

## (

## t

## ) =

## 1

## Γ(

## α

## )

## ∫

t0

## (

## t

## −

## τ

## )

α−_{1}

## f

## (

## τ

## )

## dτ,

## α >

## 0

## ,

## t >

## 0

## ,

## (2.1)

## J

0## f

## (

## t

## ) =

## f

## (

## t

## )

## .

## Definition 2.3

## ([20])

## .

## The Riemann-Liouville fractional derivative of

## f

## ∈

## C

m−_{1}

## with order

## α >

## 0 and

## m

## ∈

## N

## ∪

## {

## 0

## }

## , is defined as:

## D

αt

## f

## (

## t

## ) =

dm

dtm

## J

m−α

_{f}

_{(}

_{t}

_{)}

_{,}

_{m}

_{−}

_{1}

_{< α}

_{≤}

_{m,}

_{m}

_{∈}

_{N}

_{.}

## (2.2)

## Definition 2.4

## ([20])

## .

## The Caputo fractional derivative of

## f

## ∈

## C

m−1

## with order

## α >

## 0 and

## m

## ∈

## N

## ∪

_{{}

_{0}

_{}}

_{, is defined as:}

## (2.3)

C## D

_{t}α

## f

## (

## t

## ) =

## {

## J

m−α_{f}

(m)_{(}

_{t}

_{)}

_{,}

dm
f(t) dtm

## ,

## Definition 2.5

## ([20])

## .

## A two-parameter Mittag-Leffler function is

## de-fined by the following series

## E

α,β## (

## t

## ) =

∞

## ∑

k=0

## t

k## Γ(

## αk

## +

## β

## )

## .

## (2.4)

## Definition 2.6

## ([12])

## .

## A multivariate Mittag-Leffler function is defined

## as

## E

_{(}a1,a2,···,an),b

## (

## z

1## , z

2## ,

## · · ·

## , z

n## )

## (2.5)

## =

∞

## ∑

k=0

## ∑

l1+l2+···_{+}_{l}n=k

## k

## !

## l

1## !

## ×

## l

2## !

## × · · · ×

## l

n## !

n

## ∏

i=1

## z

lii

## Γ

## (

## b

## +

n

## ∑

i=1

## a

i## l

i## )

## ,

## where

## b >

## 0,

## l

1## , l

2## , . . . , l

n## ≥

## 0,

## |

## z

i## |

## <

## ∞

## , i

## = 1

## ,

## 2

## , . . . , n.

## Definition 2.7.

## Let us define the Laplace-transform (LT) operator

## ϕ

## on a function

## u

## (

## x, t

## )

## ,

## (

## t

## ≥

## 0) by

## ϕ

## {

## u

## (

## x, t

## );

## t

## 7→

## s

## }

## =

## ∫

∞0

## e

−_{st}

## u

## (

## x, t

## )

## dt,

## (2.6)

## and denote it by

## ϕ

## {

## u

## (

## x, t

## );

## t

## 7→

## s

## }

## =

## L

## (

## u

## (

## x, t

## )), where

## s

## is the LT

## parameter. For our purpose here, we shall take

## s

## to be real and positive.

## As a consequence, the LT of Mittag-Leffler function takes the following

## form

## L

## (

## E

α,β## (

## t

## )) =

## ∫

∞0

## e

−st_{E}

α,β

## (

## t

## )

## dt

## (2.7)

## =

∞

## ∑

k=0

## 1

## s

k+1_{Γ(}

_{αk}

_{+}

_{β}

_{)}

## .

## Lemma 2.8

## (see [12])

## .

## Let

## µ > µ

1## > µ

2## > . . . > µ

n## ≥

## 0

## , m

i## −

## 1

## <

## µ

i## ≤

## m

i## , m

i## ∈

## N

0## =

## N

## ∪

## {

## 0

## }

## ,

## d

i## ∈

## R

## , i

## = 1

## ,

## 2

## , . . . , n.

## Consider the initial

## value problem

##

##

##

##

##

## (

## D

µ_{y}

_{)(}

_{x}

_{)}

_{−}

n
## ∑

i=1

## λ

i## (

## D

µi## y

## )(

## x

## ) =

## g

## (

## x

## )

## ,

## y

(k)_{(0) =}

_{c}

k

## ∈

## R

## ,

## k

## = 0

## ,

## 1

## , . . . , m

## −

## 1

## , m

## −

## 1

## < µ

## ≤

## m,

## C

m−1

## .

## This has solution

## y

## (

## x

## ) =

## y

g## (

## x

## ) +

m−1## ∑

k=0

## c

k## u

k## (

## x

## )

## ,

## x

## ≥

## 0

## ,

## where

## y

g## (

## x

## ) =

## ∫

x0

## t

µ−_{1}

## E

_{(}·

_{)},µ

## (

## t

## )

## g

## (

## x

## −

## t

## )

## dt

## and

## u

k## (

## x

## ) =

## x

k## k

## !

## +

n## ∑

i=lk+1

## d

i## x

k+µ−µi

## E

_{(}·

_{)},k+1+µ−µi

## (

## x

## )

## ,

## k

## = 0

## ,

## 1

## , . . . , m

## −

## 1

## ,

## fulfills the initial conditions

## u

(_{k}l)

## (0) =

## δ

kl## ,

## k, l

## = 0

## ,

## 1

## , . . . , m

## −

## 1

## .

## The function

## E

_{(}·

_{)},σ

## (

## x

## ) =

## E

(µ−µ1,...,µ−µn),σ## (

## d

1## x

µ−µ1

## , . . . , d

n## x

µ−µn## )

## ,

## is a particular case of the multivariate Mittag-Leffler function (see

## [12]

## )

## and the natural numbers

## l

k## , k

## = 0

## ,

## 1

## , . . . , m

## −

## 1

## ,

## are determined from the

## condition

_{{}

## m

lk## ≥

## k

## + 1

## ,

## m

lk+1## ≤

## k.

## In the case

## m

i## ≤

## k, i

## = 1

## ,

## 2

## , . . . , n,

## we set

## l

k## := 0

## ,

## and if

## m

i## ≥

## k

## + 1

## , i

## =

## 1

## ,

## 2

## , . . . , n,

## then

## l

k## :=

## n.

## 3.

## Modified homotopy perturbation method (MHPM)

## The homotopy perturbation method is one of the most effective

## meth-ods for solving nonlinear problems. Several modifications of this method

## is presented. In this paper, we used a modified one for solving the

## frac-tional Fisher’s equation as follows:

## D

α_{t}

## u

## (

## x, t

## ) =

## u

xx## (

## x, t

## ) +

## u

## (

## x, t

## ) +

## ph

## (

## u

## (

## x, t

## ))

## (3.1)

## +

## f

1## (

## x, t

## ) +

## pf

2## (

## x, t

## )

## ,

## where

## f

1## (

## x, t

## ) +

## f

2## (

## x, t

## ) =

## f

## (

## x, t

## ),

## f

## (

## x, t

## ) is the source term of Eq. (1.2),

## and

## p

## is an embedding parameter that varies from zero to one. For more

## details see [5]. By choosing proper functions

## f

1## and

## f

2## , we can improve

## the success of our method.

## converting the nonlinear problems to linear ones, we can also use a

## special case of modified homotopy analysis method.

## 4.

## Inhomogeneous fractional Fisher’s equation

## 4.1.

## Dirichlet boundary condition.

## In this subsection, we determine

## the solution of the fractional nonlinear Fisher’s equation

## (4.1)

## D

tα## u

## (

## x, t

## ) =

## u

xx## (

## x, t

## ) +

## u

## (

## x, t

## )

## −

## u

2## (

## x, t

## ) +

## f

## (

## x, t

## )

## ,

## with the initial and Dirichlet boundary conditions

## u

## (

## x,

## 0) =

## φ

## (

## x

## )

## ,

## u

## (0

## , t

## ) =

## µ

1## (

## t

## )

## ,

## 0

## ≤

## x

## ≤

## L,

## u

## (

## L, t

## ) =

## µ

2## (

## t

## )

## ,

## t

## ≥

## 0

## .

## (4.2)

## In order to solve the problem with inhomogeneous boundary conditions,

## first transform it into a problem with homogeneous boundary conditions.

## For this purpose let

## u

## (

## x, t

## ) =

## W

## (

## x, t

## ) +

## V

## (

## x, t

## )

## ,

## where

## W

## (

## x, t

## ) is a new unknown function and

## (4.3)

## V

## (

## x, t

## ) =

## µ

2## (

## t

## )

## −

## µ

1## (

## t

## )

## L

## x

## +

## µ

1## (

## t

## )

## ,

## satisfies the boundary conditions as

## (4.4)

## V

## (0

## , t

## ) =

## µ

1## (

## t

## )

## ,

## V

## (

## L, t

## ) =

## µ

2## (

## t

## )

## .

## Furthermore, the function

## W

## (

## x, t

## ) satisfies in problem with

## homoge-neous boundary conditions as follows:

## (4.5)

##

##

##

## D

αt## W

## (

## x, t

## ) =

## W

xx## (

## x, t

## ) +

## W

## (

## x, t

## ) +

## h

## (

## W

## +

## V

## ) + ˜

## f

## (

## x, t

## )

## ,

## W

## (

## x,

## 0) =

## g

## (

## x

## )

## ,

## W

## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## W

## (

## L, t

## ) = 0

## ,

## where

## ˜

## f

## (

## x, t

## ) =

## f

## (

## x, t

## ) +

## x

## L

## [

## D

α

t

## µ

1## (

## t

## )

## −

## D

tα## µ

2## (

## t

## )]

## −

## D

tα## µ

1## (

## t

## )

## (4.6)

## +

## µ

2## (

## t

## )

## −

## µ

1## (

## t

## )

## L

## x

## +

## µ

1## (

## t

## )

## ,

## and

## g

## (

## x

## ) =

## φ

## (

## x

## )

## −

## x

## L

## [

## µ

2## (0)

## −

## µ

1## (0)]

## −

## µ

1## (0)

## .

## (4.7)

## For solving (4.5) we use MHPM

## By assuming

## W

## (

## x, t

## ) =

∞

## ∑

i=0

## W

i## p

i## , and substituting it in (4.8), we obtain

## p

0## :

##

##

##

## D

αt

## W

0## (

## x, t

## ) =

∂ 2W0(x,t)

∂x2

## +

## W

0## (

## x, t

## ) + ˜

## f

1## (

## x, t

## )

## ,

## W

0## (

## x,

## 0) =

## g

## (

## x

## )

## W

0## (0

## ,

## t) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## W

0## (L

## ,

## t) = 0

## ,

## (4.9)

## p

1## :

##

##

##

## D

tα## W

1## (

## x, t

## ) =

∂ 2_{W}

1(x,t)

∂x2

## +

## W

1## (

## x, t

## ) + ˜

## f

2## (

## x, t

## ) +

## A

0## ,

## W

1## (

## x,

## 0) = 0

## ,

## W

1## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## W

1## (

## L, t

## ) = 0

## ,

## (4.10)

## ..

## .

## p

k## :

##

##

##

## D

2αt

## W

k## (

## x, t

## ) =

∂ 2_{W}

k(x,t)

∂x2

## +

## W

_{k}

## (

## x, t

## ) +

## A

_{k}−1

## ,

## W

k## (

## x,

## 0) = 0

## ,

## W

k## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## W

k## (

## L, t

## ) = 0

## ,

## (4.11)

## ..

## .

## where ˜

## f

1## (

## x, t

## ) + ˜

## f

2## (

## x, t

## ) = ˜

## f

## (

## x, t

## ) and ˜

## f

1## (

## x, t

## ) must be satisfied in

## ini-tial and boundary conditions (4.9) [5] and

## A

k## , k

## = 0

## ,

## 1

## , . . .

## are Adomian

## polynomials [1] and are obtained

## A

k## =

## d

k## dp

k## h

## (

∞## ∑

i=0

## W

i## p

i## +

## V

## )

_{}

p=0

## ,

## k

## = 0

## ,

## 1

## , . . . .

## (4.12)

## We solve the corresponding homogeneous equation in (4.9) by the method

## of separation of variables. By assuming

## W

0## (

## x, t

## ) =

## X

0## (

## x

## )

## T

0## (

## t

## ) and

## sub-stituting it in (4.9), we obtain an ordinary linear differential equation

## for

## X

0## (

## x

## ) as

## (4.13)

## X

′′0

## (

## x

## ) +

## λ

2## X

0## (

## x

## ) = 0

## ,

## X

0## (0) =

## X

0## (

## L

## ) = 0

## ,

## and a fractional ordinary linear differential equation for

## T

0## (

## t

## ) as follows:

## (4.14)

## D

α_{t}

## T

0## −

## T

0## +

## λ

2## T

0## = 0

## .

## The Sturm-Liouville problem given by (4.13) has eigenvalues

## (4.15)

## λ

n## =

## n

2_{π}

2
## L

2## ,

## n

## = 1

## ,

## 2

## , . . . ,

## and corresponding eigenfunctions are

## (4.16)

## (

## X

0## )

n## (

## x

## ) = sin

## (

_{nπx}

## L

## )

## Now we seek a solution of the inhomogeneous problem in (4.9) of the

## form

## (4.17)

## W

0## (

## x, t

## ) =

∞

## ∑

n=1

## (

## B

0## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## .

## We assumed that the series can be differentiated term by term. In order

## to determine (

## B

0## )

n## (

## t

## ), we expanded ˜

## f

1## (

## x, t

## ) as a Fourier series by the

## eigenfunctions sin(

nπx_{L}

## ) as follows:

## (4.18)

## f

## ˜

1## (

## x, t

## ) =

∞

## ∑

n=1

## ( ˜

## f

1## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## ,

## then

## (4.19)

## ( ˜

## f

1## )

n## (

## t

## ) =

## 2

## L

## ∫

L0

## ˜

## f

1## (

## x, t

## ) sin

## (

_{nπx}

## L

## )

## dx.

## Substituting (4.17) and (4.18) into (4.9) yields

∞

## ∑

n=1

## D

α## (

## B

0## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## +

## (

## n

2_{π}

2
## L

2## −

## 1

## )

∞## ∑

n=1

## D

α## (

## B

0## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## (4.20)

## =

∞

## ∑

n=1

## ( ˜

## f

1## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## .

## By orthogonality properties of sin(

nπxL

## ), we get

## D

_{t}α

## (

## B

0## )

n## (

## t

## ) +

## (

## n

2## π

2## L

## −

## 1

## )

## (

## B

0## )

n## (

## t

## ) = ( ˜

## f

1## )

n## (

## t

## )

## .

## (4.21)

## Since

## W

0## (

## x, t

## ) satisfies the initial conditions in (4.9), we have

## (4.22)

∞

## ∑

n=1

## (

## B

0## )

n## (0) sin

## (

_{nπx}

## L

## )

## =

## g

## (

## x

## )

## ,

## which yields

## (4.23)

## (

## B

0## )

n## (0) =

## 2

## L

## ∫

X0

## g

1## (

## x

## ) sin

## (

_{nπx}

## L

## )

## dx.

## According to Lemma 2.8, the fractional initial value problem with

## µ

## =

## α, µ

1## = 0

## , m

1## = 0

## , λ

1## = 1

## −

n 2_{π}2

L

## , m

## = 1

## ,

## has the solution

## (

## B

0## )

n## (

## t

## ) =

## ∫

t0

## τ

α## E

(α,α)## (

## λ

1## τ

α## )( ˜

## f

1## )

n## (

## t

## −

## τ

## )

## dτ

## (4.24)

## + (

## B

0## )

n## (0)

## [

## 1 +

## λ

1## E

(α,α+1)## (

## λ

1## t

α## )

## ]

## .

## Hence we get the solution of the initial boundary value problem (4.9) in

## the form

## (4.25)

## W

0## (

## x, t

## ) =

∞

## ∑

n=1

## (

## B

0## )

n## (

## t

## ) sin

## (

_{nπx}

## L

## )

## .

## In a similar way, we can get

## W

k## , k

## = 1

## , . . .

## from (4.10) and (4.11). Since

## in calculating

## W

k+1## the value of

## A

k## is known from pervious stages, then

## all of problems in (4.9)-(4.11) are linear and hence solving them is

## sim-pler than main problems. Note that the success of these methods relies

## mainly on the proper choice of the functions ˜

## f

1## and ˜

## f

2## . Furthermore,

## this proper choice of ˜

## f

1## and ˜

## f

2## may provide the solution only in one

## iteration of MHPM.

## 4.2.

## Neumann boundary condition.

## Now, we obtain the solution of

## the inhomogeneous fractional Fisher’s equation (1.2) with the initial and

## Neumann boundary conditions as follows:

## u

## (

## x,

## 0) =

## φ

## (

## x

## )

## ,

## 0

## ≤

## x

## ≤

## L,

## (4.26)

## u

x## (0

## , t

## ) =

## µ

1## (

## t

## )

## ,

## u

x## (

## L, t

## ) =

## µ

2## (

## t

## )

## ,

## t

## ≥

## 0

## ,

## in which

## φ

## (

## x

## )

## , µ

1## (

## t

## )

## , µ

2## (

## t

## ) are as defined in subsection 4.1.

## For solving the problem with inhomogeneous boundary conditions,

## as before, we transform it into a problem with homogeneous boundary

## conditions. Thus we suppose that

## u

## (

## x, t

## ) = ˜

## W

## (

## x, t

## ) + ˜

## V

## (

## x, t

## )

## ,

## where ˜

## W

## (

## x, t

## ) is an unknown function and

## (4.27)

## V

## ˜

## (

## x, t

## ) =

## µ

2## (

## t

## )

## −

## µ

1## (

## t

## )

## 2

## L

## x

2

_{+}

_{µ}

1## (

## t

## )

## x,

## which satisfies the following boundary conditions:

## (4.28)

## V

## ˜

x## (0

## , t

## ) =

## µ

1## (

## t

## )

## ,

## V

## ˜

x## (

## L, t

## ) =

## µ

2## (

## t

## )

## .

## (4.29)

_{}

##

##

##

##

## D

αt

## W

## ˜

## (

## x, t

## ) =

∂2_{W}

_{˜}

(x,t)

∂x2

## + ˜

## W

## (

## x, t

## ) +

## h

## ( ˜

## W

## (

## x, t

## ) + ˜

## V

## (

## x, t

## )) = ˜

## f

## (

## x, t

## )

## ,

## ˜

## W

## (

## x,

## 0) =

## g

## (

## x

## )

## ,

## ˜

## W

x## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## ˜

## W

x## (

## L, t

## ) = 0

## ,

## t

## ≥

## 0

## ,

## in which ˜

## f

## (

## x, t

## ) and

## g

## (

## x

## ) are as the form as below:

## ˜

## f

## (

## x, t

## ) =

## f

## (

## x, t

## ) +

## x

2

## 2

## L

## (

## D

α

t

## µ

1## (

## t

## )

## −

## D

αt## µ

2## (

## t

## ))

## −

## D

αt## µ

1## (

## t

## ) +

## µ

2## (

## t

## )

## −

## µ

1## (

## t

## )

## L

## (4.30)

## −

## µ

2## (

## t

## )

## −

## µ

1## (

## t

## )

## 2

## L

## x

2

_{+}

_{µ}

1## (

## t

## )

## x,

## g

## (

## x

## ) =

## φ

## (

## x

## )

## −

## x

2## 2

## L

## [

## µ

2## (0)

## −

## µ

1## (0)]

## −

## µ

1## (0)

## x.

## Now, for solving (4.29) by MHPM, we have

## D

α_{t}

## W

## ˜

## (

## x, t

## ) = ˜

## W

xx## (

## x, t

## ) + ˜

## W

## (

## x, t

## ) +

## ph

## ( ˜

## W

## (

## x, t

## ) + ˜

## V

## (

## x, t

## ))

## (4.31)

## + ˜

## f

1## (

## x, t

## ) +

## p

## f

## ˜

2## (

## x, t

## )

## .

## If we assume ˜

## W

## (

## x, t

## ) =

∞

## ∑

i=0

## ˜

## W

i## p

i## , and substitute it in (4.29), we obtain

## p

0## :

##

##

##

## D

αt

## W

## f

0## (

## x, t

## ) = (

## W

## f

0## )

xx## (

## x, t

## ) +

## W

## f

0## (

## x, t

## ) +

## f

## e

1## (

## x, t

## )

## ,

## ˜

## W

0## (

## x,

## 0) =

## g

## (

## x

## )

## ,

## ( ˜

## W

0## )

x## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## ( ˜

## W

0## )

x## (

## L, t

## ) = 0

## ,

## (4.32)

## p

1## :

##

##

##

## D

αt

## W

## f

1## (

## x, t

## ) = (

## W

## f

1xx## )(

## x, t

## ) +

## W

## f

1## (

## x, t

## ) +

## f

## e

2## (

## x, t

## ) +

## A

0## ˜

## W

1## (

## x,

## 0) = 0

## ,

## ( ˜

## W

1## )

x## (0

## , t

## ) = 0

## ,

## 0

## ≤

## x

## ≤

## L,

## ( ˜

## W

1## )

x## (

## L, t

## ) = 0

## ,

## (4.33)

## ..

## .

## p

k## :

##

##

##

## D

αt

## W

## f

k## (

## x, t

## ) = (

## W

## f

k## )

xx## (

## x, t

## ) +

## W

## f

k## (

## x, t

## ) +

## A

k−1## ,

## ˜

## W

k## (

## x,

## 0) = 0

## ,

## ( ˜

## W

k## )

x## (0

## , t

## ) = 0

## ,

## ( ˜

## W

k## )

x## (

## L, t

## ) = 0

## ,

## (4.34)

## and so on, in which ˜

## f

1## (

## x, t

## ) + ˜

## f

2## (

## x, t

## ) = ˜

## f

## (

## x, t

## ) and

## A

k## , k

## = 0

## ,

## 1

## , . . .

## are

## Adomian polynomials defined in (4.12).

## For solving the corresponding homogeneous equation in (4.32) by the

## method of separation of variables, we assume that ˜

## W

0## (

## x, t

## ) =

## X

0## (

## x

## )

## T

0## (

## t

## )

## FDE for

## T

0## (

## t

## ) as

## X

′′0

## (

## x

## ) +

## λX

## (

## x

## ) = 0

## ,

## X

## (0) =

## X

## (

## L

## ) = 0

## ,

## D

_{t}α

## T

0## (

## t

## ) + (

## λ

## −

## 1)

## T

0## (

## t

## ) = 0

## .

## (4.35)

## The Sturm-Liouville problem, which is given by (4.35), has eigenvalues

## and corresponding eigenfunctions as:

## λ

n## =

## n

2## π

2## L

2## ,

## n

## = 1

## ,

## 2

## , . . . ,

## (

## X

0## )

n## (

## x

## ) = cos

## (

_{nπx}

## L

## )

## ,

## n

## = 1

## ,

## 2

## , . . . .

## (4.36)

## Now we are going to seek a solution of the inhomogeneous problem in

## (4.29) which takes the form

## (4.37)

## W

## ˜

0## (

## x, t

## ) =

∞

## ∑

n=1

## (

## B

0## )

n## (

## t

## ) cos

## (

_{nπ}

## L

## x

## )

## .

## For determining (

## B

0## )

n## (

## t

## ), by expanding ˜

## f

1## (

## x, t

## ) as a Fourier series by

## the eigenfunctions cos(

nπ_{L}

## x

## ) we have:

## (4.38)

## f

## ˜

1## (

## x, t

## ) =

∞

## ∑

n=1

## ( ˜

## f

1## )

n## (

## t

## ) cos

## (

_{nπ}

## L

## x

## )

## ,

## in which the Fourier coefficients are as the following form:

## (4.39)

## ( ˜

## f

1## )

n## (

## t

## ) =

## 2

## L

## ∫

L0

## ˜

## f

1## (

## x, t

## ) cos

## (

_{nπ}

## L

## x

## )

## dx.

## Then substituting (4.37), (4.38) into (4.29) implies

## D

α_{t}

## (

## B

0## )

n## (

## t

## ) +

## (

## −

## 1 +

## n

2

_{π}

2
## L

2## )

## (

## B

0## )

n## (

## t

## ) = ( ˜

## f

1## )

n## (

## t

## )

## .

## (4.40)

## Since ˜

## W

## (

## x, t

## ) fulfills the initial conditions in (4.29), we have

∞

## ∑

n=1

## (

## B

0## )

n## (0) cos

## (

_{nπ}

## L

## x

## )

## =

## g

## (

## x

## )

## ,

## (4.41)

## which yields

## (

## B

0## )

n## (0) =

## 2

## L

## ∫

L0

## g

## (

## x

## ) cos

## (

## nπ

## L

## x

## )

## dx.

## Therefore, Lemma 2.8 implies that the fractional initial value problem

## has the solution as follows:

u(x, t) = ∞

∑

n=1

(B0)n(t) cos

(_{nπx}

L

)

(4.43)

=µ1(t)x+µ2(t)−µ1(t) 2L x

2

+ ∞

∑

n=1

cos(nπ

L x

) [∫ t

0

τα−1_{E(α,α)}

((

1−n

2_{π}2
L2

)

τα_{(}_{f1}_{˜}_{)}

n(t−τ)

)

dτ

]

.

4.3. Robin boundary condition. In this subsection, we try to solve (4.1) with the initial and Robin boundary conditions as

u(x,0) =φ(x),

u(0, t) +α1ux(0, t) =µ1(t), u(L, t) +β1ux(L, t) =µ2(t),

0≤x≤L, t≥0, t≥0,

where α1, β1 are nonzero constants. To solve this problem, we translate the inhomogeneous bondary conditions to the homogenouse ones. So, suppose that

u(x, t) = ¯W(x, t) + ¯V(x, t),

where ¯W(x, t) is a new unknown function and

(4.44) V¯(x, t) = µ1(t)−µ2(t)

α1−β1−Lx−

(L+β1)µ1(t)−α1µ2(t)

α1−β1−L .

Therefore, we will have

{ _{¯}

V(0, t) +α1Vx¯ (0, t) =µ1(t),

¯

V(L, t) +β1V¯(L, t) =µ2(t).

The function ¯W(x, t) is the solution of nonlinear problem with homogeneous boundary conditions:

(4.45)

Dα

tW¯(x, t) = ¯Wxx(x, t) + ¯W(x, t) +h( ¯W + ¯V) + ˜f(x, t),

¯

W(x,0) =g(x),

¯

W(0, t) +α1Wx¯ (0, t) = 0,

¯

W(L, t) +β1W¯(L, t) = 0,

0≤x≤L,

where

˜

f(x, t) =f(x, t)−Dα tV¯(x, t).

(4.46)

For solving (4.45), again we use MHPM. By assuming

¯

W(x, t) = ∞

∑

i=0

¯

and substituting it in (4.45), we obtain

p0_{:}

Dα

tW0¯ (x, t) =
∂2_{¯}

W0(x,t)

∂x2 + ¯W0(x, t) + ˜f1(x, t), ¯

W0(x,0) =g(x),

¯

W0(0, t) +α1( ¯W0)x(0, t) =µ1(t),

¯

W0(L, t) +β1( ¯W0)x(L, t) =µ2(t),

0≤x≤L,

t≥0,

(4.47)

p1:

Dα

tW1¯ (x, t) =
∂2_{¯}

W1(x,t)

∂x2 + ¯W1(x, t) + ˜f2(x, t) +A0, ¯

W1(x,0) = 0,

¯

W1(0, t) +α1( ¯W1)x(0, t) =µ1(t),

¯

W1(L, t) +β1( ¯W1)x(L, t) =µ2(t),

0≤x≤L,

t≥0,

(4.48)

.. .

pk :

D2α

t Wk¯ (x, t) =
∂2_{¯}

Wk(x,t)

∂x2 + ¯Wk(x, t) +Ak−1, ¯

Wk(x,0) = 0,

¯

Wk(0, t) +α1( ¯Wk)x(0, t) =µ1(t),

¯

Wk(L, t) +β1( ¯Wk)x(L, t) =µ2(t),

0≤x≤L,

t≥0,

(4.49)

.. .

where ˜f1(x, t) + ˜f2(x, t) = ˜f(x, t) and Ak, k = 0, 1, . . . are Adomian polyno-mials and are obtained from (4.12). We solve the corresponding homogeneous equation in (4.47) by the method of separation of variables. By assuming

¯

W0(x, t) =X0(x)T0(t) and substituting it in (4.9), we obtain an ordinary lin-ear differential equation forX0(x):

X′′

0(x) +λ2X0(x) = 0, X0(0) +α1X0′(0) = 0, X0(L) +β1X0′(L) = 0.

(4.50)

and a fractional ordinary linear differential equation for T0(t) as (4.14). The Sturm-Liouville problem given by (4.50) has eigenvaluesλ2

n and corresponding

eigenfunctions are

(4.51) (X0)n(x) =−α1λncos(λnx) + sin(λnx), n= 1,2, . . . .

Now we seek a solution for the nonhomogeneous problem in (4.47) of the form

(4.52) W0¯ (x, t) = ∞

∑

n=1

(B0)n(t)(X0)n(x).

Like previous section, in order to determine (B0)n(t), we expand ˜f1(x, t) as a

Fourier series by the eigenfunctions (X0)n(x) as follows

(4.53) f1˜(x, t) = ∞

∑

n=1

( ˜f1)n(t)(X0)n(x).

We know that

(4.54) ( ˜f1)n(t) =

2

L

∫ L

0

˜

By substituting (4.52) and (4.53) into (4.47) we have ∞

∑

n=1

Dα(B0)n(t)(X0)n(x) + (λ2n+ 1)

∞

∑

n=1

(B0)n(t)(X0)n(x)

= ∞

∑

n=1

( ˜f1)n(t)(X0)n(x).

(4.55)

We know that eigenfunctions of Stumr-Lioville equations are orthogonal, so

Dα

t(B0)n(t) + (λ2n−1)(B0)n(t) = ( ˜f1)n(t).

(4.56)

Since ¯W0(x, t) satisfies the initial conditions in (4.47), we have

(4.57)

∞

∑

n=1

(B0)n(0)(X0)n(x) =g(x),

which yields

(4.58) (B0)n(0) =

2

L

∫ L

0

g(x)(X0)n(x)dx.

According to Lemma 2.8, the fractional initial value problem with

µ=α, µ1= 0, m1= 0, λ1=−(1 +λ2

n), m= 1,has the solution

(B0)n(t) =

∫ t

0

τα−1_{E(α,α)}_{(}_{λ1τ}α_{)( ˜}_{f1}_{)}

n(t−τ)dτ

(4.59)

+ (B0)n(0)

[

1 +λ1tαE(α,α+1)(λ1tα)].

Hence we get the solution of the initial boundary value problem (4.47) in the form

(4.60) W0¯ (x, t) = ∞

∑

n=1

(B0)n(t) (−α1λncos(λnx) + sin(λnx)).

By applying the same way we can get ¯Wk(x, t).

5. Examples

In this section, we consider three examples with different initial and bound-ary conditions and source term. We show that the solution obtained above agree with those established in these examples.

Example 5.1. Consider the fractional nonlinear Fisher’s equation (4.1) with the initial and Dirichlet boundary conditions

u(x,0) = 2,

u(0,t) = 2,

0≤x≤1,

u(1, t) =t2_{+ 2}_{,} _{t}_{≥}_{0}_{,}

(5.1)

where

f(x, t) = sin(3πx)

(

t6sin(3πx) + 2t5x+ (9π2+ 3)t3+ Γ(4) Γ(4−α)t

3−_{α}

)

+x

(

t4x+ 3t2− Γ(3) Γ(3−α)t

2−α

)

In order to solve this problem, we first transform it into a problems homoge-neous boundary conditions as

u(x, t) =W(x, t) +V(x, t) (5.2)

=W(x, t) +t2x+ 2,

(5.3)

Dα

tW(x, t) = ∂2

W(x,t)

∂x2 +W(x, t) + (W +V)2+ ˜f(x, t),

W(x,0) =, W(0, t) = 0,

0≤x≤1, W(1, t) = 0,

where

˜

f(x, t) = sin(3πx)

(

t6sin(3πx) + 2t5x+ (9π2+ 3)t3+ Γ(4) Γ(4−α)t

3−α

)

+t4x2+ 4t2+ 4.

˜

f(x, t) = sin(3πx)

(

t6sin(3πx) + 2t5x+ (9π2+ 3)t3+ Γ(4) Γ(4−α)t

3−α

)

+t4x2+ 4t2+ 4.

For solving (5.3) we apply MHPM

(5.4) DαtW(x, t) =

∂2_{W}_{(}_{x, t}_{)}

∂x2 +W(x, t)−p(W+V)

2_{+ ˜}_{f1}_{(}_{x, t}_{) +}_{p}_{f2}_{˜}_{(}_{x, t}_{)}_{.}

By assuming W(x, t) = ∞

∑

i=0

Wipi, and substituting it in (5.4), we obtain

p0:

Dα

tW0(x, t) =W0(x, t) + ∂2

W0(x,t)

∂x2 + ˜f1(x, t),

W0(x,0) = 2, W0(0, t) = 0,

0≤x≤1, W0(L, t) = 0,

(5.5)

p1:

Dα

tW1(x, t) =W1(x, t) +∂

2

W1(x,t)

∂x2 + ˜f2(x, t) +A0,

W1(x,0) = 0, W1(0, t) = 0,

0≤x≤1, W1(L, t) = 0,

(5.6)

.. .

pk:

Dα

tWk(x, t) =Wk(x, t) + ∂2

Wk(x,t)

∂x2 +Ak−1,

Wk(x,0) = 0, Wk(0, t) = 0,

0≤x≤1, Wk(L, t) = 0,

(5.7)

.. .

where ˜f1(x, t) + ˜f2(x, t) = ˜f(x, t) and

(5.8) f1˜(x, t) = sin(3πx)

(

Γ(4) Γ(4−α)t

3−_{α}

+t3(9π2−1)

)

and

With similar calculation, in subsection 4.1, we obtain a Sturm-Liouville prob-lem and an ordinary linear differential equation respect toxandtrespectively. The eigenvalues and eigenfunctions of the Sturm-Liouville problem are

λn=n2π2, (X0)n(x) = sin(nπx), n= 1,2, . . . .

(5.10)

Furthermore, we have

(f1)n(t) =

2 1

∫ 1 0

˜

f1(x, t) sin(nπx)dx=

H(t), n= 3

0, n̸= 3

with

(5.11) H(t) = Γ(4) Γ(4−α)t

3−α_{+}_{t}3_{(9}_{π}2_{−}_{1)}_{.}

So

(B0)n(t) =

∫t 0Eα,α

(

(1−n2_{π}2_{)}_{τ}α)_{H}_{(}_{t}_{−}_{τ}_{)}_{dτ,} _{n}_{= 3}

0, n̸= 3.

(5.12)

To evaluate (B0)3(t), we take laplace transform from both side of (5.12): L[(B0)3(t)] =L

[∫ t

0

τα−1_{E(α,α)}_{((1}_{−}_{9}_{π}2_{)}_{τ}α_{)}_{H}_{(}_{t}_{−}_{τ}_{)}_{dτ}

]

(5.13)

=L[τα−1_{E(α,α)}(_{(1}_{−}_{9}_{π}2_{)}_{τ}α)]_{L}_{[}_{H}_{(}_{t}_{)]}

=L

[∞

∑

k=0

1−9π2

Γ(αk+α)t

α(k+1)−1

]

L[H(t)]

=

(∞

∑

k=0

(1−9π2_{)}k
sα_{(}_{k}_{+ 1)}

)

L[H(t)]

=

(

1

sα

∞

∑

k=0

(1−9π

2 sα )

k

)

L[H(t)]

= 1

sα

1 1−1−9π2

sα

(

6

s4−α +

9π2_{−}_{1}
s4

)

= 1

sα_{−}_{(1}_{−}_{9}_{π}2_{)}

sα_{+ 9}_{π}2_{−}_{1}
s4

= 6

s4.

From (5.12) and (5.13), we get

(B0)n(t) =

{

t3_{,} _{n}_{= 3}_{,}

0, n̸= 3.

Therefore, the solution for (5.18) is ˜

Again by arguments in section 4.1, we have

(W)i(x, t) = 0, i= 1, 2, . . . .

Then the exact solution for the fractional Fisher’s equation given in Exam-ple 5.1 is

u(x, t) =t3sin(3πx) +t2x+ 2.

Example 5.2. Once again, we consider the fractional Fisher’s equation

(5.14) Dα

tu(x, t) =uxx(x, t) +u(x, t) (1−u(x, t)) +f(x, t),

with the initial and Neumann boundary conditions as

u(x,0) = cos(5πx),

(5.15)

ux(0, t) =t3_{,} _{ux}_{(1}_{, t}_{) = 2}_{t}4_{+}_{t}3_{,} _{t}_{≥}_{0}_{,}

and

f(x, t) = cos(5πx)[ Γ(γ+ 1) Γ(γ+ 1−α)t

γ−α_{+ 25}_{π}2_{(}_{t}γ_{+ 1)}_{−}_{(}_{t}γ_{+ 1)}

+ (tγ+1)2cos(5πx) + 2t3(tγ+1)x+ 2t4(tγ+1)x2]

+ Γ(5) Γ(5−α)t

4−α_{x}2_{+} Γ(4)

Γ(4−α)t

3−α_{x}_{−}_{2}_{t}4_{−}_{x}2_{t}4_{−}_{t}3_{x}

+t6x2+t8x4+ 2t7x3.

Now, if we assume that

u(x, t) = ˜W(x, t) + ˜V(x, t) = ˜W(x, t) +t4_{x}2_{+}_{t}3_{x,}

we get
(5.16)_{}

Dα

tW˜(x, t) =
∂2_{˜}

W(x,t)

∂x2 + ˜W(x, t)−( ˜W(x, t) +t4x2+t3x)2+ ˜f(x, t), ˜

W(x,0) = cos(5πx),

˜

Wx(0, t) = 0, Wx˜ (1, t) = 0, t≥0,

in which

˜

f(x, t) = cos(5πx)

[

Γ(γ+ 1) Γ(γ+ 1−α)t

γ−_{α}

+ (25π2−1)(tγ_{+ 1)}

]

+[(tγ_{+ 1) cos(5}_{πx}_{) +}_{t}4_{x}2_{+}_{t}3_{x}]2_{.}

To solve (5.16) we use MHPM

(5.17) DtαW˜(x, t) =

∂2_{W}_{˜}_{(}_{x, t}_{)}

Therefore if we assume ˜W(x, t) = ∞

∑

i=0

˜

Wipi, and substitute it in (5.17), we

derive

p0:

Dα

tW0˜ (x, t) =
∂2_{˜}

W0(x,t)

∂x2 + ˜W0(x, t) + ˜f1(x, t), ˜

W0(x,0) = cos(3πx),

( ˜W0)x(0, t) = 0,

0≤x≤L,

( ˜W0)x(L, t) = 0,

(5.18)

p1:

Dα

tW1˜ (x, t) =
∂2_{˜}

W1(x,t)

∂x2 + ˜W1(x, t) + ˜f2(x, t) +A0, ˜

W1(x,0) = 0,

( ˜W1)x(0, t) = 0,

0≤x≤L,

( ˜W1)x(L, t) = 0,

(5.19)

.. .

pk_{:}

Dα

tWk˜ (x, t) =
∂2_{˜}

Wk(x,t)

∂x2 + ˜Wk(x, t) +Ak−1, ˜

Wk(x,0) = 0,

( ˜Wk)x(0, t) = 0,

0≤x≤L,

( ˜Wk)x(L, t) = 0,

(5.20)

in which we have ˜f1(x, t) + ˜f2(x, t) = ˜f(x, t) and ˜f1(x, t) must be satisfied in initial and boundary conditions (5.18) [8]. We choose

˜

f1(x, t) = cos(5πx)

[

Γ(γ+ 1) Γ(γ+ 1−α)t

γ−_{α}

+ (25π2−1)(tγ_{+ 1)}

]

,

˜

f2(x, t) =[(tγ_{+ 1) cos(5}_{πx}_{) +}_{t}4_{x}2_{+}_{t}3_{x}]2_{.}

With a similar manner as Example 5.1, we first apply separation method for the corresponding homogeneous equation in (5.18). We obtain the eigenvalue and eigenfunctions of the Sturm-Liouville problem as

λn=n2π2, (X0)n(x) = cos(nπx), n= 1,2, . . . .

(5.21)

By supposing that

(5.22) W0˜ (x, t) = ∞

∑

n=1

(B0)n(t) cos(nπx),

and substituting in (5.18) we derive

Dα

t(B0)n(t) +

[

(nπ)2_{−}_{1}]_{(}_{B0}_{)}

n(t) = ( ˜f1)n(t),

(5.23)

so, same as Example 5.1, we have

H(t) = Γ(γ+ 1) Γ(γ+ 1−α)t

γ−_{α}

+ (25π2−1)(tγ_{+ 1)}_{.}

Since ˜W0(x, t) satisfies the initial conditions in (5.18), we have

∞

∑

n=1

(B0)n(0) cos(nπx) = cos(5πx),

which gives

(B0)n(0) =

2 1

∫ 1 0

cos(5πx) cos(nπx)dx

(5.25)

=

{

1,

0,

n= 5, n̸= 5.

where

(f1)n(t) =

2 1

∫ 1 0

˜

f1(x, t) cos(nπx)dx

=

{

H(t),

0,

n= 5, n̸= 5.

Furthermore Lemma 2.8 implies that

(B0)n(t) = t

∫

0

τα−1_{Eα,α}(_{(1}_{−}_{25}_{π}2_{)}_{τ}α)_{( ˜}_{f1}_{)}

n(t−τ)dτ

(5.26)

+ cos(5πx)((B0)_{n})_{0}(t)

=

t

∫

0

τα−1_{Eα,α}(_{(1}_{−}_{25}_{π}2_{)}_{τ}α)

{

H(t−τ),

0,

n= 5, n̸= 5, dτ

+ 1 + (1−25π2)tα_{Eα,α+1}(_{(1}_{−}_{25}_{π}2_{)}_{t}α)_{.}

Now, if we take the Laplace transform from both side of (5.26) we obtain

L[(B0)n(t)] = 0, n̸= 5

and

L[(B0)5(t)] =L

t

∫

0

τα−1_{Eα,α}(_{(1}_{−}_{25}_{π}2_{)}_{τ}α)_{H}_{(}_{t}_{−}_{τ}_{)}_{dτ}

(5.27)

+L[1 + (1−25π2)tαEα,α+1((1−25π2)tα)]

= 1

sα_{−}_{1 + 25}_{π}2 ×L[H(t)] +

1

s

+ (1−25π2)L

[∞

∑

k=0

tα(1−25π2) k

tαk

Γ(αk+α+ 1)

]

= 1

sα_{−}_{1 + 25}_{π}2 ×

(

Γ(γ+ 1)

sγ+1−α_{+ (25}_{π}2_{−}_{1)(} 1
sγ+1 +1_{s})

)

+1

s+

1−25π2
s(sα_{−}_{(1}_{−}_{25}_{π}2_{))}

= 1

s+

Γ(γ+ 1)

Hence from (5.26) and (5.27), we obtain

(B0)5(t) =

{

tγ_{+ 1}_{,} _{n}_{= 5}_{,}

0, n̸= 5.

Thus, the solution for (5.18) with above Neumann boundary conditions takes the form as

˜

W0(x, t) = (tγ+ 1) cos(5πx).

Hence like as Example 5.1, and by some computational algebra we derive

( ˜W)i(x, t)≡0, i= 1,2, . . . .

Then the analytical solution for the fractional Fisher’s equation with given conditions is as follows:

u(x, t) = (tγ_{+ 1) cos(5}_{πx}_{) +}_{t}4_{x}2_{+}_{t}3_{x.}

Example 5.3. One more time we consider the fractional Fisher’s equation as follows

(5.28) Dαtu(x, t) =uxx(x, t) +u(x, t) (1−u(x, t)) +f(x, t),

with the initial and Robin boundary conditions as

u(x,0) = 0,

u(0, t)−_{π}1ux(0, t) =t3_{−} 1
πt

5_{,}
u(1, t)−_{π}1ux(1, t) = (1−_{π}1)t5_{+}_{t}3_{,}

0≤x≤1, t≥0, t≥0,

(5.29)

and

f(x, t) = (cos(πx) + sin(πx))

[

Γ(5) Γ(5−α)t

4−α_{+}_{π}2_{t}4_{−}_{t}4_{+ 2}_{t}9_{x}_{+ 2}_{t}7

]

(5.30)

+ Γ(6) Γ(6−α)t

5−_{α}

x+ Γ(4) Γ(4−α)t

3−_{α}

−t5x−t3

+t8_{(cos(}_{πx}_{) + sin}_{πx}_{)}2

+t10_{x}2_{+}_{t}6_{+ 2}_{t}8_{x.}

Next, by assuming

u(x, t) = ¯W(x, t) + ¯V(x, t)

= ¯W(x, t) +t5x+t3,

we get
(5.31)_{}

Dα

tW¯(x, t) =
∂2_{¯}

W(x,t)

∂x2 + ¯W(x, t)−( ˜W(x, t) +t5x+t3)2+ ˜f(x, t), ¯

W(x,0) = 0,

¯

W(0, t)−_{π}1Wx¯ (0, t) = 0,

¯

W(1, t)−_{π}1Wx¯ (1, t) = 0,

0≤x≤1, t≥0, t≥0,

in which

˜

f(x, t) = (cos(πx) + sin(πx))

[

Γ(5) Γ(5−α)t

4−α_{+}_{π}2_{t}4_{−}_{t}4_{+ 2}_{t}9_{x}_{+ 2}_{t}7

]

To solve (5.31) we use MHPM as

(5.32) Dα

tW¯(x, t) =

∂2_{W}_{¯}_{(}_{x, t}_{)}

∂x2 + ¯W(x, t) +ph( ¯W+ ¯V) + ˜f1(x, t) +pf2˜(x, t).

Therefore if we assume ˜W(x, t) = ∞

∑

i=0

˜

Wipi_{, and substitute it in (5.32), we}

obtain

p0:

Dα

tW0¯ (x, t) =
∂2_{¯}

W0(x,t)

∂x2 + ¯W0(x, t) + ˜f1(x, t), ¯

W0(x,0) = 0,

¯

W0(0, t)−1

π( ¯W0)x(0, t) = 0,

¯

W0(1, t)−1

π( ¯W0)x(1, t) = 0,

0≤x≤L, t≥0, t≥0.

(5.33)

p1_{:}

Dα

tW1¯ (x, t) =
∂2_{¯}

W1(x,t)

∂x2 + ¯W1(x, t) + ˜f2(x, t) +A0, ¯

W1(x,0) = 0,

¯

W1(0, t)−1

π( ¯W1)x(0, t) = 0,

¯

W1(1, t)−1

π( ¯W1)x(1, t) = 0,

0≤x≤L, t≥0, t≥0.

(5.34)

.. .

pk:

Dα

tWk¯ (x, t) =
∂2_{¯}

Wk(x,t)

∂x2 + ¯Wk(x, t) +Ak−1, ¯

Wk(x,0) = 0,

¯

Wk(0, t)−1_{π}( ¯Wk)x(0, t) = 0,

¯

Wk(1, t)−1_{π}( ¯Wk)x(1, t) = 0,

0≤x≤L, t≥0, t≥0,

(5.35)

in which we have ˜f1(x, t) + ˜f2(x, t) = ˜f(x, t) and ˜f1(x, t) must be satisfied in initial and boundary conditions (5.33) [8]. Here,

˜

f1(x, t) = (cos(πx) + sin(πx))

[

Γ(5) Γ(5−α)t

4−_{α}

+π2t4−t4

]

.

With a similar manner as Examples 5.1 and 5.2, we use separation method for the corresponding homogeneous equation in (5.33). We obtain the eigen-value and eigenfunction of the Sturm-Liouville problem as follows

λn =n2π2, (X0)n(x) = cos(nπx) + sin(nπx), n= 1,2, . . . .

(5.36)

By assuming that

(5.37) W0¯ (x, t) = ∞

∑

n=1

(B0)n(t) (cos(nπx) + sin(nπx)),

and substituting in (5.33) we derive

Dtα(B0)n(t) +

[

(nπ)2−1](B0)n(t) = ( ˜f1)n(t),

(5.38)

so, like as Examples 5.1 and 5.2, we have

H(t) = Γ(5) Γ(5−α)t

4−_{α}

Since ¯W0(x, t) satisfies the initial conditions in (5.33), we have ∞

∑

n=1

(B0)n(0)(cos(nπx) + sin(nπx)) = cos(πx) + sin(nπx),

(5.39)

which gives

(B0)n(0) =

2 1

∫ 1 0

(cos(πx) + sin(πx)) (cos(nπx) + sin(nπx))dx

(5.40)

=

{

1,

0,

n= 1, n̸= 1,

where

( ˜f1)n(t) =

2 1

∫ 1 0

˜

f1(x, t) (cos(nπx) + sin(nπx))dx

=

{

H(t),

0,

n= 1, n= 1.

Furthermore Lemma 2.8 implies that

(B0)n(t) = t

∫

0

τα−1_{Eα,α}(_{(1}_{−}_{π}2_{)}_{τ}α)_{( ˜}_{f1}_{)}

n(t−τ)dτ

(5.41)

=

t

∫

0

τα−1_{Eα,α}(_{(1}_{−}_{π}2_{)}_{τ}α)

{

H(t−τ),

0,

n= 1, n= 1, dτ.

Now, if we take the Laplace transform from both side of (5.41) we obtain

L[(B0)n(t)] = 0, n̸= 1

and

L[(B0)1(t)] =

24

sα_{−}_{1 +}_{π}2×

sα_{+}_{π}2_{−}_{1}
s5

(5.42)

= 24

s5.

Thus from (5.41) and (5.42), we obtain

(B0)1(t) =

{

t4_{,} _{n}_{= 1}_{,}

0, n̸= 1.

Hence, the solution for (5.33) with above Robin boundary conditions takes the form as

¯

W0(x, t) =t4(cos(πx) + sin(πx)).

Therefore, we derive

( ¯W2)i(x, t)≡0, i= 1,2, . . . .

Then the analytical solution for the fractional Fisher’s equation with given conditions is as follows:

6. Conclusion

In this article, we obtained analytical solutions for the time-fractional Fisher’s nonlinear differential equation. We showed that by choosing proper functions

˜

f1and ˜f2, the solution can be obtained only in one iteration of MHPM. Finally, we illustrated the effectiveness of this method by some examples.

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1

Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.

E-mail address: h-kheiri@tabrizu.ac.ir

2

Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.

E-mail address: aida mojaver1987@yahoo.com

3

Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.