In this section, five test examples of fractional Newell- Whitehead-Segel equation demonstrate the efficiency of proposed ADM.
Ex. 5.1. We study the linear time-fractional Newell- Whitehead-Segel equation
utα
=uxx−2u, t>0, 0< α≤1, (5.1) with initial condition
u(x, 0)=ex. (5.2)
Applying the operatorJtαon both side of above defined problem, we have
u(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk
Ŵ k+1+Jtα{uxx−2u}.
This gives the following recursive relation:
u0(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk Ŵ k+1, un+1(x,t)=Jαt{(un)xx−2un}, n≥0.
u0=ex, u1= −ex tα
Ŵ(α+1), u2=ex t2α
Ŵ(2α+1), u3= −ex t3α
Ŵ (3α+1),
∞
X
n=0
un(x,t)=ex−ex tα
Ŵ (α+1)+ex t2α Ŵ (2α+1)
−ex t3α
Ŵ (3α+1)+. . . ,
Now, for the standard case when α = 1, we get u(x, t) = ex−t, which is the exact solution of the classical Newell- Whitehead-Segel equation as obtained by HPM [25] and VIM [26]. Here the numerical results obtained by ADM upto eight terms of approximation and exact solution as shown inFigures 1, 2are almost identical. It can be observed that as the value oft increases,udecreases, and asxincreases,ualso increases. Hence, the accuracy of ADM can be enhanced by increasing the number of iterations.
Ex. 5.2. We study the non-linear time-fractional Newell- Whitehead-Segel equation
uαt =uxx+2u−3u2, t>0, 0< α≤1, (5.3) with initial condition
u(x, 0)=η. (5.4)
FIGURE 1 |Surface represents eight order approximate solution forα=1, for Ex. 5.1.
FIGURE 2 |Surface represents exact solution forα=1, for Ex. 5.1.
Applying the operatorJαt on both side of above defined problem, we have
u(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk
Ŵ k+1+Jtα{uxx+2u+An}. This gives the following recursive relation:
u0(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk Ŵ k+1,
un+1(x, t)=Jtα{(un)xx+2un+An}, n≥0.
u0=η
u1=η(2−3η) tα Ŵ(α+1), u2=2η (2−3η)(1−3η) t2α
Ŵ(2α+1), u3=2η(2−3η)(18η2−12η+2) t3α
Ŵ(3α+1)
−3η2(2−3η)2Ŵ (2α+1) Ŵ (α+1)2
t3α Ŵ (3α+1), u4= −12η2(2−3η) 18η2−12η+2 t4α
Ŵ (4α+1) +18η3(2−3η)2Ŵ (2α+1)
Ŵ (α+1)2 t4α Ŵ (4α+1)
−12η2(2−3η)2(1−3η) Ŵ (3α+1) Ŵ (α+1) Ŵ (2α+1)
t4α Ŵ (4α+1) +4η(2−3η)(18η2−12η+2) t4α
Ŵ(4α+1)
−6η2(2−3η)2Ŵ (2α+1) Ŵ (α+1)2
t4α
Ŵ (4α+1) +. . .
∞
X
n=0
un(x,t)=η+η(2−3η) tα Ŵ (α+1) +2η(2−3η) (1−3η) t2α
Ŵ (2α+1) +2η(2−3η) 18η2−12η+2 t3α
Ŵ (3α+1)
−3η2(2−3η)2Ŵ (2α+1) Ŵ(α+1)2
t3α Ŵ (3α+1)
−12η2(2−3η) 18η2−12η+2 t4α Ŵ (4α+1) +18η3(2−3η)2Ŵ (2α+1)
Ŵ(α+1)2 t4α Ŵ (4α+1)
−12η2(2−3η)2(1−3η) Ŵ (3α+1) Ŵ (α+1) Ŵ (2α+1)
t4α Ŵ (4α+1) +4η(2−3η)(18η2−12η+2) t4α
Ŵ(4α+1)
−6η2(2−3η)2Ŵ (2α+1) Ŵ(α+1)2
t4α
Ŵ (4α+1) +. . .
In particular whenα=1, we get the solution in the form
u(x,t)=η+η (2−3η)t+2η (2−3η) (1−3η) t2 Ŵ (3) +2η (2−3η) 27η2−18η+2 t3
Ŵ (4) +12η (2−3η)
−54η3+54η2−14η+2 3
t4
Ŵ (5). . . .,
Prakash and Verma Fractional Newell-Whitehead-Segal Equation
FIGURE 3 |Comparison of approx. sol. for different values ofαand exact sol.
atα=1, for Ex. 5.2.
Which converge to the exact solution of the classical Newell- Whitehead-Segel equation very fastly [25,26].
u(x, t)=
−2 3 ηe2t
−23+η−ηe2t.
Figure 3 shows the comparison of approximate solution for different value of fractional order α = 0.25, 0.50, 0.75, 1 and exact solution at α = 1, when η = 1. It is observed from the Figure 3 that there is a good agreement between exact solution and approximate solution at α = 1. It is also noticed that solution depends on the time-fractional derivative.
Accuracy and efficiency can be enhanced by increasing the number of iterations.
Ex. 5.3. We study the non-linear time-fractional Newell- Whitehead-Segel equation.
uαt =uxx+u−u2=0, t>0, 0< α≤1, (5.5) With initial condition,
u(x, 0)= 1 (1+e
√x 6)2
. (5.6)
Applying the operatorJαt on both side of above equation, we get u(x, t)=
X1−1 k=0
∂ku
∂tk
!
t=0
tk
Ŵ k+1+Jtα{uxx+u+An}.
This gives the following recursive relation:
u0(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk Ŵ k+1,
un+1(x, t)=Jtα{(un)xx+2un+An}, n≥0.
u0= 1 (1+e
√x 6)
2, u1= 5
3 e
√x 6
(1+e
√x 6)3
tα Ŵ(α+1), u2= 25
18( e
√x 6
−1+2e
√x 6
1+e
√x 6
4 ) t2α Ŵ(2α+1), u3= {25
18 1
1+e
√x 6
5[8 6(e
√x
6)2−4(e
√x 6)3
+ 8 6(e
√x 6)2−(e
√x 6) 6
!
1+e
√x 6
+4 6(e
√x 6)2−16
6(e
√x 6)3+
2(e
√x 6)2−e
√x 6
1+e
√x 6
+
−20 6 (e
√x
6)3+406(e
√x 6)4
1+e
√x 6
−2((−e
√x
6)1+2(e
√x 6)2
1+e
√x 6
)]} t3α
Ŵ (3α+1) −25 9
(e
√x 6)2
1+e
√x 6
6
Ŵ (2α+1)t3α Ŵ (3α+1) Ŵ (α+1)2.
∞
X
n=0
un(x,t)= 1
1+e
√x 6
2 +5 3
e
√x 6
1+e
√x 6
3
tα Ŵ (α+1)
+25 18
e
√x 6
−1+2e
√x 6
1+e
√x 6
4
t2α Ŵ (2α+1) +{25
18 1
1+e
√x 6
5[8 6(e
√x
6)2−4(e
√x 6)3
+ 8 6(e
√x 6)2−(e
√x 6) 6
!
1+e
√x 6
+ 4
6(e√x6)2− 16
6(e√x6)3+
2(e√x6)2−e√x6
1+e√x6
+
−20
6 (e√x6)3+406(e√x6)4
1+e
√x 6
−2((−e
√x 6)
1
+2(e
√x 6)
2
1+e
√x 6
)]} t3α
Ŵ (3α+1) −25 9
(e
√x 6)
2
1+e
√x 6
6
Ŵ (2α+1)t3α
Ŵ (3α+1) Ŵ (α+1)2 +. . .
In particular whenα=1, we get the solution in the form
FIGURE 4 |Comparison of approx. sol. for different values of fractional order αand exact sol. atα=1, for Ex. 5.3.
u(x,t)= 1
1+e
√x 6
2 +5 3
e
√x 6
1+e
√x 6
3
t 1
+25 18
e
√x 6
−1+2e
√x 6
1+e
√x 6
4
t2
2
+
125 216
(e
√x 6(4
e
√x 6
2
−7e
√x 6 +1)
1+e
√x 6
5
t3
3 +. . .. Which converge to the exact solution of the classical Newell- Whitehead-Segel equation very fastly [25].
u(x, t)= 1
1+e
√x 6−56t2.
Figure 4 shows the comparison of third order approximate solution for different value of fractional order α= 0.25, 0.50, 0.75, 1 and exact solution atα=1, forx = 1. It is observed from the Figure 4 that there is a good agreement between exact solution and approximate solution at α = 1. It is also noticed that solution depends on the time-fractional derivative.
Accuracy and efficiency can be enhanced by increasing the number of iterations.
Ex. 5.4. We study the non-linear time-fractional Newell- Whitehead-Segel equation
uαt =uxx+u−u4=0, t>0, 0< α≤1, (5.7)
with initial condition u(x, 0)=( 1
1+e
√3x 10
)
2
3. (5.8)
Applying the operatorJtαon both side of above equation, we have u(x, t)=
X1−1 k=0
∂ku
∂tk
!
t=0
tk
Ŵ k+1+Jαt{uxx+u+An}.
This gives the following recursive relation:
u0(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk Ŵ k+1, un+1(x, t)=Jtα{(un)xx+un+An}, n≥0.
u0=( 1 1+e
√3x 10
)
2 3,
u1= 7
5( e√3x10 (1+e
√3x 10)
5 3
) tα Ŵ(α+1),
u2= 49 50{
e
√3x 10
2e
√3x 10−3
1+e
√3x 10
83 } t2α Ŵ(2α+1),
u3= 1
1+e
√3x 10
113 {3528 500
e
√3x 10
2 1+e
√3x 10
−4704 500 (e
√3x 10)
3
−1323 500 (e
√3x 10)
1+e
√3x 10
+3528
500
e
√3x 10
2
−7056 500 (e
√3x 10)
3
+7056 500
e
√3x 10
2
+8624 500
e√3x10
4
1+e
√3x 10
−12936 500
(e
√3x 10)
3
1+e
√3x 10
+49 50 2
e
√3x 10
2
−3e
√3x 10
! 1+e
√3x 10
−
196
50 2
e
√3x 10
2
−3e
√3x 10
!
1+e
√3x 10
} t3α Ŵ (3α+1)
−294 25
(e
√3x 10)
2
(1+e
√3x 10)
14 3
Ŵ (2α+1)t3α Ŵ (3α+1) Ŵ (α+1)2,
Prakash and Verma Fractional Newell-Whitehead-Segal Equation
∞
X
n=0
un(x,t)= 1 1+e
√3x 10
!23 +7
5( e
√3x 10
1+e
√3x 10
53) tα
Ŵ (α+1)
+49 50
e√3x10
2e√3x10 −3
1+e
√3x 10
83
t2α Ŵ (2α+1)
+ 1
1+e√3x10
113 {3528 500
e√3x10
2 1+e√3x10
−4704 500 (e
√3x 10)
3
−1323 500(e
√3x 10)
1+e
√3x 10
+3528 500
e
√3x 10
2
−7056 500 (e
√3x 10)
3
+7056 500
e
√3x 10
2
+8624 500
e
√3x 10
4
1+e
√3x 10
−12936 500
(e√3x10)
3
1+e√3x10
+49 50 2
e√3x10
2
−3e√3x10
! 1+e√3x10
−196 50
2
e√3x10 2
−3e√3x10
!
1+e
√3x 10
t3α Ŵ (3α+1)}
− 294
25
e
√3x 10
2
1+e
√3x 10
143
Ŵ (2α+1)t3α Ŵ (3α+1) Ŵ (α+1)2 +. . ..
Takingα=1, we get the solution in the form
u(x,t)= 1 1+e
√3x 10
!23 +7
5( e
√3x 10
1+e
√3x 10
53)t 1
+49 50
e
√3x 10
2e
√3x 10 −3
1+e
√3x 10
83
t2
2
FIGURE 5 |Comparison of approx. sol. for different values ofαand exact sol.
atα=1, for Ex. 5.4.
+343 1000
4
e
√3x 10
2
−27e
√3x 10 +9
! e
√3x 10
1+e
√3x 10
113
t3
3 +. . ..,
Which converge to the exact solution of the classical Newell- Whitehead-Segel equation very fastly [25,26].
u(x,t)=[1 2tanh
− 3 2√
10
x− 7
√10t
+1 2]
2 3
. Figure 5 shows the comparison of third order approximate solution for different value of fractional order α = 0.25, 0.50, 0.75, 1 and exact solution atα = 1 for x = 1.
It is observed from theFigure 5that there is a good agreement between exact solution and approximate solution at α = 1.
It is also noticed that solution depends on the time-fractional derivative. Accuracy and efficiency can be enhanced by increasing the number of iterations.
Ex. 5.5. We study the nonlinear time-fractional Newell- Whitehead-Segel equation of the form
uαt =uxx+3u−4u4=0, t>0, 0< α≤1, (5.9) with initial condition
u(x, 0)= r3
4 e
√6x
e√6x+e
√6
2x. (5.10)
Applying the operatorJαt on both side of above defined problem, we have
u(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk
Ŵ k+1+Jtα{uxx+2u+An}.
This gives the following recursive relation:
u0(x, t)= X1−1
k=0
∂ku
∂tk
!
t=0
tk Ŵ k+1,
un+1(x,t)=Jαt{(un)xx+3un+An}, n≥0.
u0= r3
4 e
√6x
e√6x+e
√6 2 x, u1=9
2 r3
4
e√6xe
√6 2x
(e√6x+e
√6 2 x)
2
tα Ŵ(α+1),
u2=81 4
r3 4(
e√6xe
√6 2x
−e√6x+e
√6 2x
(e√6x+e
√6 2x)
3 ) t2α
Ŵ(2α+1), u3=81
4 r3
4 1
1+e−
√6 2 x
4{−3 2
e−
√6 2x
1+e−
√6 2x
+9 2
e−
√6 2 x
2
+6
e−
√6 2x
2 1+e−
√6 2 x
−9
e−
√6 2x
3
+9
e−
√6 2 x
2
−18
e−
√6 2 x
3
1+e−
√6 2 x
−27 2
e−
√6 2x
3
+18
e−
√6 2x
4
1+e−
√6 2x
+3
−e−
√6 2 x
+
e−
√6 2x
2!
1+e−
√6 2 x
−9e−
√6
2 x(−1+e−
√6 2 x) (1+e−
√6 2 x)
} t3α
Ŵ(3α+1)−729 4 r3
4 (−e−
√6 2 x
)
2
(1+e−
√6 2x)
5
Ŵ (2α+1)t3α Ŵ (3α+1) Ŵ (α+1)2,
∞
X
n=0
un(x,t)= r3
4
e√6x e√6x+e
√6 2 x
+9 2
r3 4
e
√6xe
√6 2 x
(e√6x+e
√6 2 x)
2
tα Ŵ (α+1)
+ 81
4 r3
4 e√6xe
√6 2x
−e√6x+e
√6 2x
(e√6x+e
√6 2x)
3
t2α Ŵ (2α+1) +81
4 r3
4 1
1+e−
√6 2 x
4{−3 2
e−
√6 2 x
∗
1+e−
√6 2x
+9
2
e−
√6 2x
2
+ +6
e−
√6 2x
2 1+e−
√6 2 x
−9
e−
√6 2 x
3
+9
e−
√6 2x
2
−18
e−
√6 2 x
3
1+e−
√6 2 x
−27 2
e−
√6 2 x
3
+18
e−
√6 2 x
4
1+e−
√6 2 x
+3 −e−
√6 2 x
+
e−
√6 2 x
2!
∗
1+e−
√6 2x
−9 e−
√6 2 x
−1+e−
√6 2 x
1+e−
√6 2 x
} t3α Ŵ (3α+1)
−729 4
r3 4
−e−
√6 2 x
2
1+e−
√6 2x
5
Ŵ (2α+1)t3α
Ŵ (3α+1) Ŵ (α+1)2 +. . .
FIGURE 6 |Comparison of approx. sol. for different values of fractional order αand exact sol. atα=1, for Ex. 5.5.
Prakash and Verma Fractional Newell-Whitehead-Segal Equation
Takingα=1, we get the solution in the form
u(x,t)= r3
4
e√6x e√6x+e
√6
2 x +9
2 r3
4
e√6xe
√6 2x
(e√6x+e
√6 2 x)
2
t 1
+81 4
r3 4
e√6xe
√6 2x
−e√6x+e
√6 2 x
(e√6x+e
√6 2x)
3
t2 2
+243 16
r3 4 e
√6xe
√6 2 x
−4e
√6xe
√6 2 x
+(e
√6x)2+(e
√6 2 x
)
2
(e
√6x
+e
√6 2 x)
4
t3 3 +. . ..
which converge to the exact solution of the classical Newell- Whitehead-Segel equation very fastly [25,26].
u(x, t)= r3
4 e
√6x
e√6x+e(
√6 2 x−92t).
Figure 6 shows the comparison of third order approximate solution for different value of fractional order α = 0.25, 0.50, 0.75, 1 and exact solution at α = 1, for
x = 1. It is observed from the Figure 6 that there is a good agreement between exact solution and approximate solution at α = 1. It is also noticed that solution depends on the time- fractional derivative. Accuracy and efficiency can be enhanced by increasing the number of iterations.
CONCLUSION
In this article, we have successfully applied the ADM to obtain the approximate analytic solutions of fractional model of Newell-Whitehead-Segel equation. The plotted graph and numerical result shows the accuracy of proposed method.
We observed an excellent agreement between ADM and the exact solution. The results reveal that ADM is an efficient and computationally very attractive approach to investigate non-linear fractional model. Therefore, ADM can be further applied to solve various types of linear and non-linear fractional model arising in the field of science and engineering.
AUTHOR CONTRIBUTIONS
AP and VV designed the study, collected the data, performed the analysis, and wrote the manuscript.
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Conflict of Interest Statement: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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