# Analytical solutions for the fractional Fisher's equation

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http://scma.maragheh.ac.ir

## 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.

t

xx

## )

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.

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αt

xx

+

ν

p

1

1

νm

(m)

ν

ν

α

t

0

α−1

0

m

1

α

t

d

m

dtm

m

−α

m

−1

C

tα

m−α

(m)

dm

f(t) dtm

(3)

α,β

k=0

k

(a1,a2,···,an),b

1

2

n

k=0

l1+l2+···+ln=k

1

2

n

n

i=1

li

i

n

i=1

i

i

1

2

n

i

0

st

α,β

0

−st

α,β

k=0

k+1

1

2

n

i

i

i

i

0

i

µ

n

i=1

i

µi

(k)

k

(4)

m

−1

g

m−1

k=0

k

k

g

x

0

µ−1

(·)

k

k

n

i=lk+1

i

k+µ

−µi

(·),k+1+µ−µi

(kl)

kl

(·)

## E

(µ−µ1,...,µ−µn),σ

1

µ−µ1

n

µ−µn

k

lk

lk+1

i

k

i

k

αt

xx

1

2

1

2

1

2

(5)

xx

2

1

2

2

1

1

1

2

αt

xx

α

t

1

2

1

2

1

1

2

1

1

(6)

i=0

i

i

0

α

t

0

∂ 2

W0(x,t)

∂x2

0

1

0

0

0

1

1

∂ 2W

1(x,t)

∂x2

1

2

0

1

1

1

k

t

k

∂ 2W

k(x,t)

∂x2

k

k−1

k

k

k

1

2

1

k

k

k

k

i=0

i

i

p=0

0

0

0

0

′′

0

2

0

0

0

0

αt

0

0

2

0

n

2

2

2

0

n

(7)

0

n=1

0

n

0

n

1

nπxL

1

n=1

1

n

1

n

L

0

1

n=1

α

0

n

2

2

2

n=1

α

0

n

n=1

1

n

nπx

L

tα

0

n

2

2

0

n

1

n

0

n=1

0

n

0

n

X

0

1

(8)

1

1

1

n 2π2

L

0

n

t

0

α

(α,α)

1

α

1

n

0

n

1

(α,α+1)

1

α

0

n=1

0

n

k

k+1

k

1

2

1

2

x

1

x

2

1

2

2

1

2

1

x

1

x

2

(9)

α

t

∂2W˜

(x,t)

∂x2

x

x

2

α

t

1

αt

2

αt

1

2

1

2

1

2

1

2

2

1

1

αt

xx

1

2

i=0

i

i

0

α

t

0

0

xx

0

1

0

0

x

0

x

1

α

t

1

1xx

1

2

0

1

1

x

1

x

k

α

t

k

k

xx

k

k−1

k

k

x

k

x

1

2

k

0

0

0

(10)

0

′′

0

tα

0

0

n

2

2

2

0

n

0

n=1

0

n

0

n

1

L

1

n=1

1

n

1

n

L

0

1

αt

0

n

2

2

2

0

n

1

n

n=1

0

n

0

n

L

0

(11)

## 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−1E(α,α)

((

1−n

2π2 L2

)

τα(f1˜)

n(t−τ)

)

]

.

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)

      

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

¯

(12)

and substituting it in (4.45), we obtain

p0:

      

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:

      

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

˜

(13)

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

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−1E(α,α)(λ1τα)( ˜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, t0,

(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

)

(14)

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)

  

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) =

∂2W(x, t)

∂x2 +W(x, t)−p(W+V)

2+ ˜f1(x, t) +pf2˜(x, t).

By assuming W(x, t) = ∞

i=0

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

p0:

  

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:

  

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:

  

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

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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+t3(9π21).

So

(B0)n(t) =

  

∫t 0Eα,α

(

(1−n2π2)τα)H(tτ)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−1E(α,α)((19π2)τα)H(tτ)

]

(5.13)

=L[τα−1E(α,α)((19π2)τα)]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

k=0

(1−9π

2 sα )

k

)

L[H(t)]

= 1

1 1−1−9π2

(

6

s4−α +

9π21 s4

)

= 1

(19π2)

+ 9π21 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 ˜

(16)

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) = 2t4+t3, t0,

and

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

γ−α+ 25π2(tγ+ 1)(tγ+ 1)

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

+ Γ(5) Γ(5−α)t

4−αx2+ Γ(4)

Γ(4−α)t

3−αx2t4x2t4t3x

+t6x2+t8x4+ 2t7x3.

Now, if we assume that

u(x, t) = ˜W(x, t) + ˜V(x, t) = ˜W(x, t) +t4x2+t3x,

we get (5.16)

   

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(5πx) +t4x2+t3x]2.

To solve (5.16) we use MHPM

(5.17) DtαW˜(x, t) =

∂2W˜(x, t)

(17)

Therefore if we assume ˜W(x, t) = ∞

i=0

˜

Wipi, and substitute it in (5.17), we

derive

p0:

    

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:

    

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:

    

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(5πx) +t4x2+t3x]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

t(B0)n(t) +

[

(nπ)21](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),

(18)

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−1Eα,α((125π2)τα)( ˜f1)

n(t−τ)dτ

(5.26)

+ cos(5πx)((B0)n)0(t)

=

t

0

τα−1Eα,α((125π2)τα)

{

H(t−τ),

0,

n= 5, n̸= 5, dτ

+ 1 + (1−25π2)tαEα,α+1((125π2)tα).

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−1Eα,α((125π2)τα)H(tτ)

 

(5.27)

+L[1 + (1−25π2)tαEα,α+1((1−25π2)tα)]

= 1

1 + 25π2 ×L[H(t)] +

1

s

+ (1−25π2)L

[∞

k=0

tα(1−25π2) k

tαk

Γ(αk+α+ 1)

]

= 1

1 + 25π2 ×

(

Γ(γ+ 1)

sγ+1−α+ (25π21)( 1 sγ+1 +1s)

)

+1

s+

1−25π2 s(sα(125π2))

= 1

s+

Γ(γ+ 1)

(19)

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

(B0)5(t) =

{

+ 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(5πx) +t4x2+t3x.

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)−π1ux(0, t) =t3 1 πt

5, u(1, t)−π1ux(1, t) = (1−π1)t5+t3,

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+π2t4t4+ 2t9x+ 2t7

]

(5.30)

+ Γ(6) Γ(6−α)t

5−α

x+ Γ(4) Γ(4−α)t

3−α

−t5x−t3

+t8(cos(πx) + sinπx)2

+t10x2+t6+ 2t8x.

Next, by assuming

u(x, t) = ¯W(x, t) + ¯V(x, t)

= ¯W(x, t) +t5x+t3,

we get (5.31)

     

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)−π1Wx¯ (0, t) = 0,

¯

W(1, t)−π1Wx¯ (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+π2t4t4+ 2t9x+ 2t7

]

(20)

To solve (5.31) we use MHPM as

(5.32) Dα

tW¯(x, t) =

∂2W¯(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)−1π( ¯Wk)x(0, t) = 0,

¯

Wk(1, t)−1π( ¯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

(21)

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−1Eα,α((1π2)τα)( ˜f1)

n(t−τ)dτ

(5.41)

=

t

0

τα−1Eα,α((1π2)τα)

{

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

1 +π

+π21 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:

(22)

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.

References

2. G. Adomian,A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Model., 13 (7) (1992) 17.

3. I. Dimovski,Convolutional calculus, Publishing House of the Bulgarian Academy of Sciences, 1982.

4. J.-S. Duan and R. Rach, A new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differen-tial equations, Appl. Math. Comput., 218 (2011) 4090-4118.

5. J. H. He,homotipy perturbation technique, Comput. Methods Appl. Mech. En-grg., 178 (1999) 257-262.

6. J. H. He,A coupling method of homotopy tecknique and perturbation tecknique for nonlinear problems, Int. J. Nonlinear Mech., 35 (2000) 37-43.

7. R. Hilfer,Applications of Fractional Calculus in Physics, World Scientific, Sin-gapore, 2000.

8. S. Irandoust-Pakchin, H. Kheiri and S. Abdi-Mazraeh,Efficient computational algorithms for solving one class of fractional boundary value problems, Comp. Math. Math Phys., 53 (7) (2013) 920-932.

9. M. Kumar and N. Singh,Modified Adomian Decomposition Method and computer implementation for solving singular boundary value problems arising in various physical problems, Comput. and Chem. Eng., 34 (2010) 1750-1760.

10. S. J. Liao,on the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. Dissertation. Shanghai Jiao Tong University, Shang-hai, 1992.

11. S. J. Liao,On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004) 499-513.

12. Y. Luchko, and R. Gorenflo,An operational method for solving fractional differ-ential equations with the caputo derivatives, Acta Math. Vietnam, 24(2) (1999) 207-233.

13. Z. Odibat, and S. Momani,Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Soliton Frac., 36 (2008) 167-74.

14. Z. Odibat, and S. Momani,Application of variational iteration method to nonlin-ear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 1(7) (2006) 271-9.

15. Z. Odibat, and S. Momani,Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Soliton Frac., 31 (2007) 1248-55. 16. I. Podlubny,Fractional Differential Equations, Academic Press, New York, NY,

USA, 1999.

17. S. Reich,Constructive techniques for accretive and monotone operators in Ap-plied Nonlinear Analysis, Academic Press, New York, 1979, 335-345.

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19. J. Sabatier, O. P. Agrawal, and J. A. T. Machado,Advances in Fractional Cal-culus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.

20. S. Samko, A. Kilbas, and O. Marichev, Fractional integrals and derivatives: theory and applications,USA: Gordon and breach science publishers, 1993. 21. H. Yepez-Martinez, J. M. Reyes, and I. O. Sosa,Analytical solutions to the

frac-tional Fisher equation by applying the fracfrac-tional Sub-equation method, British Journal of Mathematics & Computer Science, 4 (11) (2014).

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1

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

2

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

3

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

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