Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE

Prince Singh; Dinkar Sharma

doi:10.1515/nleng-2018-0136

Article Open Access

Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE

Prince Singh and Dinkar Sharma

Published/Copyright: September 25, 2019

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From the journal Nonlinear Engineering Volume 9 Issue 1

Abstract

We apply homotopy perturbation transformation method (combination of homotopy perturbation method and Laplace transformation) and homotopy perturbation Elzaki transformation method on nonlinear fractional partial differential equation (fpde) to obtain a series solution of the equation. In this case, the fractional derivative is described in Caputo sense. To avow the adequacy and authenticity of the technique, we have applied both the techniques to Fractional Fisher’s equation, time-fractional Fornberg-Whitham equation and time fractional Inviscid Burgers’ equation. Finally, we compare the results obtained from homotopy perturbation transformation technique with homotopy perturbation Elzaki transformation.

Keywords: nonlinear fractional partial differential equation; HPTM; Caputo sense; He’s polynomial; Elzaki transform; HPETM

1 Introduction

Most of the real world problems arising in the field of biology, fluid mechanics, ecology and thermodynamics etc. are modelled as nonlinear partial differential equation(PDE). The fractional calculus is important tool to refine the description of most of the natural phenomenon. Fractional differential equations have attracted considerable interest of many researcher because of their successive appearance in diverse fields of science and engineering. Many numerical and semi-analytical techniques are used to obtain the solution of linear and nonlinear partial differential equations.

Dr. Ji Huan He in 1999, proposed homotopy perturbation method (HPM) [11, 13] which is coupling of homotpoy method and classical perturbation technique, has been successfully implemented on linear and nonlinear problems like nonlinear wave equation[15, 16], fractional diffusion equation [3], fractional convection–diffusion equation [19], space–time fractional advection–dispersion equation [35], fractional Zakharov–Kuznetsov equations [34], fractional partial differential equations in fluid mechanics [33], fractional Schrödinger equation [32]. The significance of HPM is that it doesn’t require a small parameter in the equation, so it overcome the impediments of classical perturbation technique. The other semi-analytical techniques such as HAM (Homotopy analysis method) [18], Laplace homotopy analysis method [20] Adomian decomposition method [4], HPTM(Homotopy perturbation transformation method) [10, 17, 24, 26, 28] and HPSTM(Homotopy perturbation Sumudu transform method) [23, 27], we can always obtain better result than the numerical one for partial differential equation. In recent years, many researchers have used numerical and analytical technique [1, 29, 30, 31] for the solution of fractional partial differential equation. In[9], author suggested a new form of fractional differentiation to model comlex physical problems. In this work, we apply HPTM [17, 24, 25] technique (which is combination of homotopy perturbation method and Laplace transformation) and HPETM[5, 6, 7, 8, 21] (Homotopy perturbation method with Elzaki transform) to find the solution of Fractional Fisher’s equation, time-fractional Fornberg-Whitham equation and time fractional Inviscid Burgers' equation and we get a power series solution in the form of a rapidly convergent series and only a few iterations lead to high accurate solutions. In these techniques, there is no need of algorithm like discritizing the problem, no linearization is required for nonlinear problem, only few iterations will lead to the solution which can be easily calculated. There are many symbolic computation software like Maple, Mathematica etc. with which we can easily calculate more terms very easily, hence it reduces the computational cost for solving such complex problem. Finally, we compare the result obtained by these methods.

2 Basic definitions and properties

Definition 2.1

A real function g(t) ∈ C_μ, t > 0, μ ∈ ℝ if ∃ q ∈ ℝ; (q > μ), such that g(t) = t^qk(t), where k(t) ∈ C[0, ∞) and g(t) ∈ Cμm if g^(m) ∈ C_μ, m ∈ ℕ.

Definition 2.2

The Caputo fractional derivative of g(τ) [2, 22] is defined as

∂α∂ταg(τ)=Jm−α∂m∂τmg(τ)=1Γ(m−α)∫0τ(τ−η)m−α−1gm(η)dη,

g ∈ C−1m, m – 1 < α ≤ m, m ∈ ℕ, τ > 0. Here ∂α∂τα and Γ denotes Caputo derivative operator and the Gamma function respectively.

Definition 2.3

The Elzaki transformation [5, 7] of g(τ) is defined as

E[g(τ)]=v∫0∞g(τ)e−τvdτ=F(v),τ>0.(2.1)

2.1 Properties

Elzaki transform of the Caputo fractional derivative is

E{∂α∂ταg(τ)}=E{g(τ)}vα−∑k=0n−1vk−α+2g(k)(0),n−1<α≤n.(2.2)

Definition 2.4

The Laplace transformation of f(τ) is defined as

L[f(τ)]=∫0∞f(τ)e−sτdτ=F(s),τ>0.(2.3)

2.2 Properties

Laplace transform of the Caputo fractional derivative is given by [2]

L{∂α∂ταf(τ)}=sαL{f(τ)}−∑k=0n−1sα−k−1f(k)(0),n−1<α≤n.(2.4)

Definition 2.5

The Mittag-Leffler function of two parameter α and β is given by [12]

Eα,β(τ)=∑n=0∞τnΓ(αn+β),α,β>0(2.5)

3 Homotopy perturbation method (HPM)

Consider the nonlinear partial differential equation

L(w)+N(w)=f(r),r∈Ω(3.1)

with boundary condition

B(w,∂w∂n)=0,r∈Γ(3.2)

where L and N are linear and nonlinear differential operator and f(r) is an analytic function. Ji-huan He [13], [14] construct a homotopy of eq. (3.1) as H : Ω × [0, 1] → ℝ which satisfies

H(v,p)=(1−p)(L(v)−L(w0))+p(L(v)+N(v)−f(r))=0(3.3)

H(v,p)=L(v)−L(w0)+pL(w0)+p(N(v)−f(r))(3.4)

where p ∈ [0, 1] is an embedding parameter and w₀ is an initial approximation which satisfies the boundary conditions. Clearly, from (3.3), we have

H(v,0)=L(v)−L(w0)=0(3.5)

H(v,1)=(L(v)+N(v)−f(r))=0(3.6)

The process of changing of p from zero to unity is that v varies from w₀ to w(x, t). The basic assumption for this method is that the solution of (3.1) can be expressed as

v=v0+v1p+v2p2+v3p3+…

The solution of (3.1) is given by

w(x,t) = limp→1(v0+v1p+v2p2+v3p3+…)(3.7)

=v0+v1+v2+…(3.8)

4 Homotopy perturbation Elzaki transform method (HPETM)

Consider the following general fractional nonlinear partial differential equation

∂α∂tαw(x,t)+Lw(x,t)+Nw(x,t)=f(x,t),t>0,x∈R,n−1<α≤n,(4.1)

here, ∂α∂tα, is the fractional Caputo derivative with respect to t, L and N are linear and non linear differential operators respectively which satisfy Lipschitz condition, f(x, t) is the source term. Now applying Elzaki transform, we get

E{∂α∂tαw+Lw+Nw}=E{f(x,t)}.

Using (2.2), we have

E{w}=∑k=0n−1vk+2w(k)(x,0)+vα(E{f(x,t)−Lw−Nw}).

Applying the inverse Elzaki transform, we have

w=∑k=0n−1tkk!w(k)(x,0)+E−1{vα(E{f(x,t)−Lw−Nw})}.(4.2)

By applying HPM, we get

0=(1−p)(w(x,t)−w(x,0))+p(w(x,t)−∑k=0n−1tkk!w(k)(x,0)−E−1{vαE{f(x,t)−Lw−Nw}}),

Let

w(x,t) = ∑n=0∞pnwn(x,t),Nw(x,t) = ∑n=0∞pnHn(w(x,t))(4.3)

where

Hn(w(x,t))=1n!∂n∂pn(∑i=0∞piwi)(p=0),n=0,1,2,3,…(4.4)

So, (4.2) becomes

∑n=0∞pnwn=w(x,0)+p∑k=1n−1tkk!w(k)(x,0)+E−1vαEf(x,t)−L∑n=0∞pnwn−∑n=0∞pnHn(w).

Comparing the coefficients of like powers of p, we have

p0:w0=w(x,0);p1:w1=∑k=1n−1tkk!w(k)(x,0)+E−1vαEf(x,t)−Lw0(x,t)−H0;p2:w2=−E−1vαELw1(x,t)+H1;p3:w3=−E−1vαELw2(x,t)+H2,⋮

therefore, the HPETM series solution is obtained as p → 1

w(x,t)=w0+w1+w2+w3+….

5 Convergence analysis

In this section, we emphasis on the condition of convergence of the proposed method for the series solution of eq. (4.1).

Theorem 5.1

Letwandw_n(x, t) be defined in Banach space, then the condition that series solution defined by eq. (4.3) converges to the solution of eq.(4.1) if ∃η ∈ (0, 1) such that ||w_n+1|| ≤ η||w_n||.The condition of convergence has been proved in [27, 28].

Theorem 5.2

The maximum absolute truncation error of the series solution eq. (4.3) of eq. (4.1) is given by

|w(x,t)−∑k=0nwk(x,t)|≤ηn+11−η||w0||

6 Application

Example 6.1

Consider the time fractional nonlinear Fisher’s equation [21]

∂αw∂tα=∂2w∂x2+6w(1−w),t>0,x∈R,0<α≤1,(6.1)

with initial condition w(x,0)=1(1+ex)2.

By applying HPETM on (6.1), we have

∑n=0∞pnwn=1(1+ex)2+p(E−1{vα(E{∑n=0∞(pnwn)xx+6{∑n=0∞pnwn−6∑n=0∞pnHn(w)}})}),(6.2)

where H_n(w) represents He’s polynomial. The first few components of He’s polynomial are given by

H0(w)=w02;H1(w)=2w0w1;H2(w)=w12+2w0w2,⋮

Comparing the like powers of p on both sides of (6.2), we have

p0:w0=1(1+ex)2,p1:w1=10extα(1+ex)3Γ(α+1),p2:w2=50ex(2ex−1)t2α(1+ex)4Γ(2α+1),p3:w3=50ex(−16e3x−15e2x+30ex+5)(1+ex)6+600e2xΓ(2α+1)(1+ex)6Γ(α+1)2t3αΓ(3α+1).⋮

Hence the solution is

w(x,t)=1(1+ex)2+10extα(1+ex)3Γ(α+1)+50ex(2ex−1)t2α(1+ex)4Γ(2α+1)+50ex(−16e3x−15e2x+30ex+5)(1+ex)6+600e2xΓ(2α+1)(1+ex)6Γ(α+1)2t3αΓ(3α+1)+….(6.3)

The above solution obtained is equivalent to the closed form solution when α = 1 i.e w(x,t)=1(1+ex−5t)2 up to fourth order term approximation.

Example 6.2

Consider the time-fractional Fornberg-Whitham equation [26]

∂α∂tαw(x,t)=∂3w∂x2∂t−∂w∂x+w∂3w∂x3−w∂w∂x+3∂w∂x∂2w∂x2,t>0,x∈R,0<α≤1,(6.4)

with initial condition w(x,0)=ex2.

By applying HPETM on (6.4), we have

∑n=0∞pnwn=ex2+p(E−1{vα(E{∑n=0∞pn(wn)xxt+∑n=0∞pn(−wn)x+{∑n=0∞pnHn(w)}})}),(6.5)

where H_n(w) are He’s polynomial represents nonlinear terms. The first few components of He’s polynomial are given by

H0(w) = w0w0xxx−w0w0x+3w0xw0xx;H1(w) = w0w1xxx+w1w0xxx−w0w1x−w1w0x+3w0xw1xx+3w1xw0xx;

H2(w)=w0w2xxx+w1w1xxx+w2w0xxx−w0w2x−w1w1x−w2w0x+3w2xw0xx+3w1xw1xx+3w0xw2xx.

Comparing the like powers of p on both sides of (6.5), we have

p0:w0=ex2;p1:w1=−ex22tαΓ(α+1);p2:w2=−ex28t2α−1Γ2α+ex24t2αΓ(2α+1);p3:w3=ex2−132t3α−2Γ(3α−1)+18t3α−1Γ(3α)−18t3αΓ(3α+1),⋮

Hence, the solution is

w(x,t)=ex2−ex22tαΓ(α+1)−ex28t2α−1Γ2α+ex24t2αΓ(2α+1)+ex2−132t3α−2Γ(3α−1)+18t3α−1Γ(3α)−18t3αΓ(3α+1)+….(6.6)

From the above solution, it is clear that the approximate solution(up to fourth order approximation) obtained from above said technique is very closed to the exact solution i.e. w(x,t)=e12(x−4t3) for α = 1.

Example 6.3

Consider the nonlinear nonhomogeneous time fractional Inviscid Burgers’ equation [36]

Dtαw+wwx=1+x+t,w(x,0)=x,0<α≤1(6.7)

By applying HPETM on eq.(6.7), we have

∑n=0∞pnwn=x+p(E−1{vαE{1+x+t}−1sαE{∑n=0∞pnHn(w)}})(6.8)

where

wwx=∑n=0∞pnHn(w)

i.e.

(∑n=0∞pnwn)(∑n=0∞pnwn)x=∑n=0∞pnHn(w)

The first few components of He’s polynomial i.e. H_n(w) are given as

H0=w0w0x;H1=w0w1x+w1w0x;H2=w0w2x+w1w1x+w2w0x, ⋮

On comparing the like powers of p on both sides of (6.8), we have

p0:wo=x;p1:w1=(1+x)tαΓ(1+α)+tα+1Γ(α+2)−E−1{vαE{H0}} =tαΓ(1+α)+tα+1Γ(α+2);p2:w2=−E−1{vαE{H1}} =−(t2αΓ(2α+1)+t2α+1Γ(2α+2));p3:w3=−E−1{vαE{H2}} =(t3αΓ(3α+1)+t3α+1Γ(3α+2)), ⋮

Hence the solution of (6.7) is

w(x,t)=x+tαΓ(1+α)+tα+1Γ(α+2)−(t2αΓ(2α+1)+t2α+1Γ(2α+2))+(t3αΓ(3α+1)+t3α+1Γ(3α+2))+…

w(x,t)=x+(tαΓ(1+α)−t2αΓ(2α+1)+t3αΓ(3α+1)+…)+(tα+1Γ(α+2)−t2α+1Γ(2α+2)+t3α+1Γ(3α+2)+…)

w(x,t)=x−∑n=1∞(−1)ntnαΓ(nα+1)−t∑n=1∞(−1)ntnαΓ(nα+2)

w(x,t)=x+1+t−Eα,1(−tα)−tEα,2(−tα)(6.9)

where E_α,2(–t^α) in eq. (6.9) is Mittag-Leffler function defined in (2.5). When α = 1, the exact solution of (6.7) is w(x, t) = x + t.

7 Homotopy perturbation transformation method (HPTM)

Now we present the solution of (4.1) using Laplace transformation,

L{∂α∂tαw+Lw+Nw}=L{f(x,t)}.

Using (2.4), we have

L{w}=1sα∑k=0n−1sα−k−1w(k)(x,0)+1sαL{f(x,t)−Lw−Nw}.L{w}=∑k=0n−1s−k−1w(k)(x,0)+1sαL{f(x,t)−Lw−Nw}.

Operating inverse Laplace transform, we get

w(x,t)=∑k=0n−1tkk!w(k)(x,0)+L−1{1sαL{f(x,t)−Lw−Nw}},

By applying HPM, we get

0=(1−p)(w(x,t)−w(x,0))+p(w(x,t)−∑k=0n−1tkk!w(k)(x,0)−L−1{1sαL{f(x,t)−Lw−Nw}}),

w(x,t)=w(x,0)+p(∑k=1n−1tkk!w(k)(x,0)+L−1{1sαL{f(x,t)−Lw−Nw}})(7.1)

Let

w(x,t)=∑n=0∞pnwn(x,t),Nw(x,t)=∑n=0∞pnHn(w(x,t)) and w(x,0)=w0(x,t)(7.2)

where

Hn(w(x,t))=1n!∂n∂pn(∑i=0∞piwi)(7.3)

Substituting (7.2) and (7.3) in (7.1), we get

∑n=0∞pnwn(x,t)=w0(x,t)+p(∑k=1n−1tkk!w(k)(x,0)+L−1{1sαL{f(x,t)−L(∑n=0∞pnwn(x,t))−∑n=0∞pnHn(w(x,t))}})(7.4)

On comparing the coefficients of the like powers of p, we get

p0:w0=w(x,0);p1:w1=∑k=1n−1tkk!w(k)(x,0)+L−11sαL{f(x,t)−Lw0−H0(w)};p2:w2=−L−11sαL{Lw1+H1(w)};p3:w3=−L−11sαL{Lw2+H2(w)},⋮

hence, the approximate solution is obtained as p → 1

w(x,t)=w0+w1+w2+….

8 Application

Example 8.1

Consider the time fractional nonlinear Fisher’s equation[21]

∂αw∂tα=∂2w∂x2+6w(1−w),t>0,x∈R,0<α≤1,(8.1)

with initial condition w(x,0)=1(1+ex)2.

By applying HPTM on (8.1), we have

∑n=0∞pnwn=1(1+ex)2+p(L−1{1sα(L{∑n=0∞(pnwn)xx+6{∑n=0∞pnwn−∑n=0∞pnHn(w)}})})(8.2)

where H_n(w) are He’s polynomial represents nonlinear terms. The first few components of He’s polynomial are given by

H0(w)=w02;H1(w)=2w0w1;H2(w)=w12+2w0w2,⋮

Comparing the like powers of p on both sides of (8.2), we have

p0:w0=1(1+ex)2;p1:w1=10extα(1+ex)3Γ(α+1);p2:w2=50ex(2ex−1)t2α(1+ex)4Γ(2α+1);p3:w3=50ex(−16e3x−15e2x+30ex+5)(1+ex)6+600e2xΓ(2α+1)(1+ex)6Γ(α+1)2t3αΓ(3α+1),⋮

Hence, the solution is given as

w(x,t)=1(1+ex)2+10extα(1+ex)3Γ(α+1)+50ex(2ex−1)t2α(1+ex)4Γ(2α+1)+50ex(−16e3x−15e2x+30ex+5)(1+ex)6+600e2xΓ(2α+1)(1+ex)6Γ(α+1)2t3αΓ(3α+1)+….(8.3)

Example 8.2

Consider the time-fractional Fornberg-Whitham equation [26]

∂α∂tαw(x,t)=∂3w∂x2∂t−∂w∂x+w∂3w∂x3−w∂w∂x+3∂w∂x∂2w∂x2,t>0,x∈R,0<α≤1,(8.4)

with initial condition w(x,0)=ex2.

By applying HPTM on (8.4), we have

∑n=0∞pnwn=ex2+p(L−1{1sα(L{∑n=0∞pn(wn)xxt+∑n=0∞pn(−wn)x+{∑n=0∞pnHn(w)}})})(8.5)

where H_n(w) are He’s polynomial represents nonlinear terms. The first few components of He’s polynomial are given by

H0(w)=w0w0xxx−w0w0x+3w0xw0xx;H1(w)=w0w1xxx+w1w0xxx−w0w1x−w1w0x+3w0xw1xx+3w1xw0xx;

H2(w)=w0w2xxx+w1w1xxx+w2w0xxx−w0w2x−w1w1x−w2w0x+3w2xw0xx+3w1xw1xx+3w0xw2xx.

Comparing the like powers of p on both sides of (8.5), we have

p0:w0=ex2;p1:w1=−ex22tαΓ(α+1);p2:w2=−ex28t2α−1Γ2α+ex24t2αΓ(2α+1);p3:w3=ex2−132t3α−2Γ(3α−1)+18t3α−1Γ(3α)−18t3αΓ(3α+1),⋮

Hence, the solution is given as

w(x,t)=ex2−ex22tαΓ(α+1)−ex28t2α−1Γ2α+ex24t2αΓ(2α+1)+ex2−132t3α−2Γ(3α−1)+18t3α−1Γ(3α)−18t3αΓ(3α+1)+….(8.6)

Example 8.3

Consider the nonlinear nonhomogeneous time fractional Inviscid Burgers’ equation[36]

Dtαw+wwx=1+x+t,w(x,0)=x,0<α≤1(8.7)

By applying HPTM on eq.(8.7), we have

∑n=0∞pnwn=x+p(L−1{1sαL{1+x+t}−1sαL{∑n=0∞pnHn(w)}})(8.8)

where

wwx=∑n=0∞pnHn(w)

i.e.

(∑n=0∞pnwn)(∑n=0∞pnwn)x=∑n=0∞pnHn(w)

The first few components of He’s polynomial i.e. H_n(w) are given as

H0=w0w0x;H1=w0w1x+w1w0x;H2=w0w2x+w1w1x+w2w0x, ⋮

On comparing the like powers of p on both sides of (8.8), we have

p0:wo=x;p1:w1=(1+x)tαΓ(1+α)+tα+1Γ(α+2)−L−1{1sαL{H0}}=tαΓ(1+α)+tα+1Γ(α+2);p2:w2=−L−1{1sαL{H1}}=−(t2αΓ(2α+1)+t2α+1Γ(2α+2));p3:w3=L−1{1sαL{H2}}=(t3αΓ(3α+1)+t3α+1Γ(3α+2)), ⋮

Hence the solution of (8.7) is

w(x,t)=x+tαΓ(1+α)+tα+1Γ(α+2)−(t2αΓ(2α+1)+t2α+1Γ(2α+2))+(t3αΓ(3α+1)+t3α+1Γ(3α+2))+…

w(x,t)=x+(tαΓ(1+α)−t2αΓ(2α+1)+t3αΓ(3α+1)+…)+(tα+1Γ(α+2)−t2α+1Γ(2α+2)+t3α+1Γ(3α+2)+…)

w(x,t)=x−∑n=1∞(−1)ntnαΓ(nα+1)−t∑n=1∞(−1)ntnαΓ(nα+2).

w(x,t)=x+1+t−Eα,1(−tα)−tEα,2(−tα).

When α = 1, the exact solution of (8.7) is w(x, t) = x + t.

9 Analysis and conclusion

We Know that

L{f(t)}=∫0∞e−stf(t)=f¯(s)(9.1)

Also from (2.1) and (9.1), we have

E{f(t)}=F(v)=vf¯(1v)(9.2)

f¯(1v)=1vF(v)⇒f¯(s)=1vF(v), where v=1s

Hence

L{tn}=Γ(n+1)sn+1.⇒E{tn}=vn+2Γ(n+1), using(9.2).(9.3)

Also using (9.2) in (2.4), we have

L{∂α∂tαf(t)}=sαL{f(t)}−∑k=0n−1sα−k−1f(k)(0),n−1<α≤n,⇒1vE{fα(t)}=1vαF(v)v−∑k=0n−1(1v)α−k−1f(k)(0),n−1<α≤n,⇒E{fα(t)}=F(v)vα−∑k=0n−1vk−α+2f(k)(0),n−1<α≤n.

In this work, we intend to study two semi-analytical techniques to solve nonlinear fractional partial differential equations: homotopy perturbation with Elzaki transform and homotopy perturbation transformation method.So, from above analysis, we conclude that the Elzaki transformation and its properties could be derive from Laplace transformation. This is the reason that either we use HPTM or HPETM, we come out with same series solution of nonlinear PDE or fractional PDE. Fig. 1-4, represent the surface graph of approximate solution of (8.1) for various estimations of α and the exact solution for α = 1 and we find that approximate solution up to order 4 converges to exact solution for α = 1, in Table 1, the condition of convergence is verified i.e. we analyse that ||w₁|| < ||w₂|| < ||w₃||. Moreover, from Fig.5, we conclude that with the decrease in the value of α, the value of w(x, t) increases. On the other hand, Fig. 6-9 and Fig. 11-14 represents the surface graph of (8.4) and (8.7) for various estimations of α and the exact solution for α = 1, the approximate solution of w(x, t) converges to exact solution when α = 1, but by slightly decreasing the value of α, the value of w(x, t) also decreases which is shown in the Fig. 10 and Fig.15. We have applied both the techniques (i.e HPTM and HPETM) on nonlinear homogeneous and non homogenous fractional PDE and the outcome exhibit the efficiency, simplicity and high rate of accuracy of the suggested methodologies to solve this type of complex equation.

Fig. 1

Surface graph of w(x, t) of eq. (8.1), when α = 0.6

Fig. 2

Surface graph of w(x, t) of eq. (8.1), when α = 0.8

Fig. 3

Surface graph of w(x, t) of eq. (8.1), when α = 1

Fig. 4

Surface graph of w(x, t) of eq. (8.1), when α = 1 (exact solution)

Fig. 5

Plot of of w(x, t) of eq. (8.1) when x = 0.4 and α = 0.6, 0.8 and 1

Fig. 6

Surface graph of w(x, t) of eq. (8.4), when α = 0.6

Fig. 7

Surface graph of w(x, t) of eq. (8.4), when α = 0.8

Fig. 8

Surface graph of w(x, t) of eq. (8.4), when α = 1

Fig. 9

Surface graph of w(x, t) of eq. (8.4), when α = 1 (exact solution)

Fig. 10

Plot of of w(x, t) of eq. (8.4) when x = 2 and α = 0.6, 0.8 and 1

Fig. 11

Surface graph of w(x, t) of eq. (8.7), when α = 0.6

Fig. 12

Surface graph of w(x, t) of eq. (8.7), when α = 0.8

Fig. 13

Surface graph of w(x, t) of eq. (8.7), when α = 1

Fig. 14

Surface graph of w(x, t) of eq. (8.7), when α = 1 (exact solution)

Fig. 15

Plot of of w(x, t) of eq. (8.7) when x = 0.5 and α = 0.6, 0.8 and 1

Table 1

Approximate solution of Fisher’s equation (6.1) and (8.1) up to fourth order(when α = 1)

x	t	w_HPETM (approx.)	w_HPTM (approx.)	w (exact sol.)	abs.error	∥w₁∥	∥w₂∥	∥w₃∥
0.3	0.1	0.304691131	0.304691131	0.302317425	0.002373706	0.104031064	1.88E-02	7.49E-04
	0.11	0.319292625	0.319292625	0.316042418	0.003250207	0.11443417	2.28E-02	9.97E-04
	0.12	0.334319781	0.334319781	0.329984205	0.004335576	0.124837277	2.71E-02	1.29E-03
	0.13	0.34977709	0.34977709	0.344120184	0.005656906	0.135240383	3.18E-02	1.65E-03
	0.14	0.365669045	0.365669045	0.358426914	0.007242131	0.145643489	3.69E-02	2.05E-03

0.4	0.1	0.276611064	0.276611064	0.275603147	0.001007917	9.64E-02	1.92E-02	4.91507E-05
	0.11	0.29026645	0.29026645	0.288830839	0.001435611	0.106061562	2.32E-02	6.54196E-05
	0.12	0.304302372	0.304302372	0.302317425	0.001984947	0.115703523	2.76E-02	8.49324E-05
	0.13	0.318718535	0.318718535	0.316042418	0.002676117	0.125345483	3.24E-02	0.000107984
	0.14	0.333514645	0.333514645	0.329984205	0.00353044	0.134987443	3.76E-02	0.00013487

0.5	0.1	0.249765515	0.249765515	0.25	0.000234485	8.87E-02	1.92E-02	0.000734094
	0.11	0.262435106	0.262435106	0.262653581	0.000218475	9.76E-02	2.33E-02	0.00097708
	0.12	0.275441031	0.275441031	0.275603147	0.000162116	0.10646815	2.77E-02	0.001268515
	0.13	0.288778885	0.288778885	0.288830839	5.19537E-05	0.115340496	3.25E-02	0.001612806
	0.14	0.302444264	0.302444264	0.302317425	0.000126839	0.124212842	3.77E-02	0.002014355

Table 2

Approximate solution of Fornberg-Whitham equation (6.4) and (8.4) up to fourth order (when α = 1)

t	x	w_HPETM (approx.)	w_HPTM (approx.)	w (exact sol.)	abs.error	∥w₁∥	∥w₂∥	∥w₃∥
0.1	1	1.543580941	1.543580941	1.542390265	1.19E-03	0.082436064	0.018548114	0.004156152
	2	2.544934731	2.544934731	2.542971638	1.96E-03	0.135914091	0.030580671	0.006852335
	3	4.195888024	4.195888024	4.19265143	3.24E-03	0.224084454	0.050419002	0.011297591
	4	6.917849834	6.917849834	6.912513593	5.34E-03	0.369452805	0.083126881	0.018626579
	5	11.40560617	11.40560617	11.3968082	8.80E-03	0.609124698	0.137053057	0.030710037

0.3	1	1.351024036	1.351024036	1.349858808	1.17E-03	0.247308191	0.043278933	0.00711011
	2	2.227462066	2.227462066	2.225540928	1.92E-03	0.407742274	0.071354898	0.01172259
	3	3.672464088	3.672464088	3.669296668	3.17E-03	0.672253361	0.117644338	0.019327284
	4	6.054869657	6.054869657	6.049647464	5.22E-03	1.108358415	0.193962723	0.031865304
	5	9.982792395	9.982792395	9.974182455	8.61E-03	1.827374094	0.319790467	0.052537005

0.5	1	1.180724868	1.180724868	1.181360413	6.36E-04	0.412180318	0.05152254	0.004293545
	2	1.946686205	1.946686205	1.947734041	1.05E-03	0.679570457	0.084946307	0.007078859
	3	3.209542954	3.209542954	3.211270543	1.73E-03	1.120422268	0.140052783	0.011671065
	4	5.291641738	5.291641738	5.29449005	2.85E-03	1.847264025	0.230908003	0.019242334
	5	8.72444229	8.72444229	8.729138364	4.70E-03	3.04562349	0.380702936	0.031725245

Table 3

Approximate solution of Inviscid Burger’s Equation (6.7) and (8.7) up to fourth order(α = 1)

x	t	w_HPETM (approx.)	w_HPTM (approx.)	w (exact sol.)	abs.error	∥w₁∥	∥w₂∥	∥w₃∥
0.25	0.25	0.50016276	0.50016276	0.5	0.00016276	0.28125	0.033854167	0.002766927
	0.5	0.752604167	0.752604167	0.75	0.002604167	0.625	0.145833333	0.0234375
	0.75	1.013183594	1.013183594	1	0.013183594	1.03125	0.3515625	0.083496094
	1	1.291666667	1.291666667	1.25	0.041666667	1.5	0.666666667	0.208333333

0.5	0.25	0.75016276	0.75016276	0.75	0.00016276	0.28125	0.033854167	0.002766927
	0.5	1.002604167	1.002604167	1	0.002604167	0.625	0.145833333	0.0234375
	0.75	1.263183594	1.263183594	1.25	0.013183594	1.03125	0.3515625	0.083496094
	1	1.541666667	1.541666667	1.5	0.041666667	1.5	0.666666667	0.208333333

0.75	0.25	1.00016276	1.00016276	1	0.00016276	0.28125	0.033854167	0.002766927
	0.5	1.252604167	1.252604167	1.25	0.002604167	0.625	0.145833333	0.0234375
	0.75	1.513183594	1.513183594	1.5	0.013183594	1.03125	0.3515625	0.083496094
	1	1.791666667	1.791666667	1.75	0.041666667	1.5	0.666666667	0.208333333

References

[1] Atangana A., Gómez-Aguilar J.F., Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Meth. Part. Diff. Equat., 2018, 34(5), 1502-1523.10.1002/num.22195Search in Google Scholar

[2] Caputo M., Elasticità e dissipazione. Zanichelli, 1969.Search in Google Scholar

[3] Das S., Gupta P.K., An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method, Zeitschrift Naturforsch. A, 2010, 65(3), 182-190.10.1515/zna-2010-0305Search in Google Scholar

[4] Dhaigude D.B., Birajdar G.A., Nikam V.R., Adomain decomposition method for fractional Benjamin-Bona-Mahony-Burger’s equations, Int. J. Appl. Math. Mech., 2012, 8(12), 42-51.Search in Google Scholar

[5] Elzaki T.M., Application of new transform “Elzaki transform” to partial differential equations, Glob. J. Pure Appl. Math., 2011, 7(1), 65-70.Search in Google Scholar

[6] Elzaki T.M., On the connections between Laplace and Elzaki Transforms, Adv. Theoret. Appl. Math., 2011, 6(1), 1-11.Search in Google Scholar

[7] Elzaki T.M., The new integral transform “Elzaki transform”. Glob. J. Pure Appl. Math., 2011, 7(1), 57-64.Search in Google Scholar

[8] Elzaki T.M., Hilal E.M.A., Arabia J.S., Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations, Math. Theory Model., 2012, 2(3), 33-42.Search in Google Scholar

[9] Gómez-Aguilar J.F., Atangana A., New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications, Europ. Phys. J. Plus, 2017, 132(1), 13.10.1140/epjp/i2017-11293-3Search in Google Scholar

[10] Gómez-Aguilar J.F., Yépez-Martínez H., Torres-Jiménez J., Córdova-Fraga T., Escobar-Jiménez R.F., Olivares-Peregrino V.H., Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv. Differ. Equat., 2017, (1)68, 2017.10.1186/s13662-017-1120-7Search in Google Scholar

[11] Gupta P.K., Singh M., Homotopy perturbation method for fractional Fornberg-Whitham equation, Comput. Math. Appl., 2011, 61(2), 250-254.10.1016/j.camwa.2010.10.045Search in Google Scholar

[12] Haubold H.J., Mathai A.M., Saxena R.K., Mittag-Leffler functions and their applications, J. Appl. Math, 2011, 2011, 298628.10.1155/2011/298628Search in Google Scholar

[13] He J.H., Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 1999, 178(3), 257-262.10.1016/S0045-7825(99)00018-3Search in Google Scholar

[14] He J.H., Homotopy perturbation method: a new nonlinear analytical Technique, Appl. Math. Comput., 2003, 135(1), 73-79.10.1016/S0096-3003(01)00312-5Search in Google Scholar

[15] He J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solit. Fract., 2005, 26(3), 695-700.10.1016/j.chaos.2005.03.006Search in Google Scholar

[16] He J.H., Limit cycle and bifurcation of nonlinear problems, Chaos, Solit. Fract., 2005, 26(3), 827-833.10.1016/j.chaos.2005.03.007Search in Google Scholar

[17] Khan Y., Wu Q., Homotopy perturbation transform method for nonlinear equations using He0s polynomials, Comput. Math. Appl., 2011, 61(8), 1963-1967.10.1016/j.camwa.2010.08.022Search in Google Scholar

[18] Liao S., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput., 2005, 169(2), 1186-1194.10.1016/j.amc.2004.10.058Search in Google Scholar

[19] Momani S., Yıldırım A., Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He0s homotopy perturbation method, Int. J. Comp. Math., 2010, 87(5), 1057-1065.10.1080/00207160903023581Search in Google Scholar

[20] Morales-Delgado V.F., Gómez-Aguilar J.F., Yépez-Martínez H., Baleanu D., Escobar-Jimenez R.F., Olivares-Peregrino V.H., Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equat., 2016, 2016(1), 164.10.1186/s13662-016-0891-6Search in Google Scholar

[21] Neamaty A., Agheli B., Darzi R., Applications of homotopy perturbation method and Elzaki transform for solving nonlinear partial differential equations of fractional order, J. Nonlin. Evolut. Equat. Appl., 2016, 2015(6), 91-104.Search in Google Scholar

[22] Podlubny I., Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Math. Sci. Eng, 1999, 198.Search in Google Scholar

[23] Sharma D., Singh P., Chauhan S., Homotopy perturbation Sumudu transform method with He0s polynomial for solutions of some fractional nonlinear partial differential equations, Int. J. Nonlin. Sci., 2016, 21(2), 91-97.Search in Google Scholar

[24] Sharma D., Singh P., Chauhan S., Homotopy perturbation transform method with He0s polynomial for solution of coupled nonlinear partial differential equations, Nonlin. Eng., 2016, 5(1), 17-23.10.1515/nleng-2015-0029Search in Google Scholar

[25] Sharma D., Singh P., Chauhan S., Solution of fifth-order Korteweg and de Vries equation by homotopy perturbation transform method using He0s polynomial, Nonlin. Eng., 2017, 6(2), 89-93.10.1515/nleng-2016-0011Search in Google Scholar

[26] Singh J., Kumar D, Kumar S., New treatment of fractional Fornberg-Whitham equation via Laplace transform, Ain Shams Eng. J., 2013, 4(3), 557-562.10.1016/j.asej.2012.11.009Search in Google Scholar

[27] Singh P., Sharma D., Convergence and error analysis of series solution of nonlinear partial differential equation, Nonlin. Eng., 2018, 7(4), 303-308.10.1515/nleng-2017-0113Search in Google Scholar

[28] Singh P., Sharma D., On the problem of convergence of series solution of non-linear fractional partial differential equation, In: AIP Conf. Proc., 2017, 1860, 020027.10.1063/1.4990326Search in Google Scholar

[29] Yépez-Martínez H., Gómez-Aguilar J.F., Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Math. Model. Nat. Phenom., 2018, 13(1), 13.10.1051/mmnp/2018002Search in Google Scholar

[30] Yépez-Martínez H., Gómez-Aguilar J.F., Sosa I.O., Reyes J.M., Torres-Jiménez J., The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mex. de Física, 2016, 62(4), 310.10.1155/2016/7047126Search in Google Scholar

[31] Yépez-Martínez H., Gómez-Aguilar J.F., Atangana A., First integral method for non-linear differential equations with conformable derivative, Math. Model. Nat. Phenom., 2018, 13(1), 14.10.1051/mmnp/2018012Search in Google Scholar

[32] Yıldırım A., An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, Int. J. Nonlin. Sci. Numer. Simul., 2009, 10(4), 445-450.10.1515/IJNSNS.2009.10.4.445Search in Google Scholar

[33] Yıldırım A., Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method, Int. J. Numer. Meth. Heat Fluid Flow, 2010, 20(2), 186-200.10.1108/09615531011016957Search in Google Scholar

[34] Yıldırım A., Gülkanat Y., Analytical approach to fractional Zakharov-Kuznetsov equations by He’s homotopy perturbation method, Comm. Theor. Phys., 2010, 53(6), 1005.10.1088/0253-6102/53/6/02Search in Google Scholar

[35] Yıldırım A., Koçak H., Homotopy perturbation method for solving the space-time fractional advection-dispersion equation, Adv. Water Res., 2009, 32(12), 1711-1716.10.1016/j.advwatres.2009.09.003Search in Google Scholar

[36] Yousif E.A., Hamed S.H.M., Solution of nonlinear fractional differential equations using the homotopy perturbation Sumudu transform method, Appl. Math. Sci, 2014, 8(44), 2195-2210.10.12988/ams.2014.4285Search in Google Scholar

Received: 2018-09-14

Revised: 2018-12-04

Accepted: 2019-01-28

Published Online: 2019-09-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2018-0136

Keywords for this article

nonlinear fractional partial differential equation; HPTM; Caputo sense; He’s polynomial; Elzaki transform; HPETM

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