# Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations

Authors: Rashid Nawaz; Laiq Zada; Abraiz Khattak; Muhammad Jibran; Adam Khan
Journal: Complexity
 (2019)

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## Abstract

In this paper, the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial differential equations. The fractional order Zakharov–Kuznetsov equation is solved as a test example, while the time fractional derivatives are described in the Caputo sense. The solutions of the problem are computed in the form of rapidly convergent series with easily calculable components using Mathematica. Reliability of the proposed method is given by comparison with other methods in the literature. The obtained results showed that the method is powerful and efficient for determination of solution of higher-dimensional fractional order partial differential equations.


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## Body

## 1. Introduction

Fractional calculus is simply an extension of integer order calculus. For many years, it was assumed that fractional calculus is a pure subject of mathematics and having no such applications in real-world phenomena, but this concept is now wrong because of the recent applications of fractional calculus in modeling of the sound waves propagation in rigid porous materials [1], ultrasonic wave propagation in human cancellous bone [2], viscoelastic properties of soft biological tissues [3], the path tracking problem in an autonomous electric vehicles [4], etc. Differential equations of fractional order are the center of attention of many studies due to their frequent applications in the areas of electromagnetic, electrochemistry, acoustics, material science, physics, viscoelasticity, and engineering [5–9]. These kinds of problems are more complex as compared to integer order differential equations. Due to the complexities of fractional calculus, most of the fractional order differential equations do not have the exact solutions, and as an alternative, the approximate methods are extensively used for solution of these types of equations [10–14]. Some of the recent methods for approximate solutions of fractional order differential equations are the Adomian Decomposition Method (ADM), the Homotopy Perturbation Method (HPM), the Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), etc. [15–26].Marinca and Herisanu introduced the Optimal Homotopy Asymptotic Method (OHAM) for solving nonlinear differential equations which made the perturbation methods independent of the assumption of small parameters and huge computational work [27–31]. The method was recently extended by Sarwar et al. for solution of fractional order differential equations [32–35].In this paper, OHAM formulation is extended to two-dimensional fractional order partial differential equations. Particularly, the extended formulation is demonstrated by illustrative examples of the following fractional version of the Zakharov–Kuznetsov equations shortly called FZK (p,q,r):(1)Dtαζ+aζpx+bζqxxx+cζryyx=0.In above equation,α is a parameter describing theory of the fractional derivative 0<α≤1,a,b and c are arbitrary constants, and p,q,r≠0 are integers which govern the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. The FZK equation has been solved by many researchers using different techniques. Some recent well-known techniques are [36–40].The present paper is divided into six sections. In Section2, some basic definitions and properties from fractional calculus are given. Section 3 is devoted to analysis of the OHAM for two-dimensional partial differential equations of fractional order. In Section 4, the 1st order approximate solutions of FZK (2, 2, 2) and FZK (3, 3, 3) equations are given, in which the time fractional derivatives are described in the Caputo sense. In Section 5, comparisons of the results of 1st order approximate solution by the proposed method are made with 3rd order variational iteration method (VIM), Perturbation-Iteration Algorithm (PIA), and residual power series method (RPS) solutions [36, 37]. In all cases, the proposed method yields better results.


## 2. Basic Definitions

In this section, some definitions and results from the literature are stated which are relevant to the current work. Riemann–Liouville, Welyl, Reize, Compos, and Caputo proposed many definitions.Definition 1.
A real functionfx, x>0, is said to be in space Cη, η∈ℜ, if there is a real number p>η, such that fx = xpf1x, where f1x∈C0,∞ and it is said to be in the space Cηm if only if fm∈Cη,m∈N.Definition 2.
The Riemann–Liouville fractional integral operator of orderα≥0 of a function f∈Cη,η≥−1 is defined as(2)Iaαfx=1Γα∫axx−ηα−1fηdη,α>0,x>0,Ia0fx=fx.
When we formulate the model of real-world problems with fractional calculus, the Riemann–Liouville operator have certain disadvantages. Caputo proposed a modified fractional differential operatorD∗α in his work on the theory of viscoelasticity.Definition 3.
The fractional derivative offx in Caputo sense is defined as(3)Daαfx=Iam−αDmfx=1Γm−a∫axx−ζm−α−1fmζdζ,form−1<α≤m,m∈N,x>0,f∈C−1m.Definition 4.
Ifm−1<α≤m,m∈N, and f∈Cηm,η≥−1, then(4)DaαIaαfx=fx,DaαIaαfx=fx−∑k=0m−1fkx−ak!,x>0.
One can found the properties of the operatorIα in the literature. We mention the following:
Forf∈Cηm,α,β>0,η≥−1andγ≥−1.Iaαfx exists for almost every x∈a,b.IaαIaβfx=Iaα+βfx.IaαIaβfx=IaβIaαfx.Iaαx−aγ=Γγ+1/Γα+γ+1x−aα+γ.


## 3. OHAM Analysis for Fractional Order PDEs

In this section, the OHAM for fractional order partial differential equation is introduced. The proposed method is presented in the following steps.Step 1: write the general fractional order partial differential equation as(5)∂αζr,t∂tα=Aζr,t+fr,t,α>0.Subject to the initial conditions,(6)D0α−kζr,0=hkr,k=0,1,2,…,n−1,D0α−nζr,0=0,n=α,D0kζr,0=gkr,k=0,1,2,…,n−1,D0nζr,0=0,n=α.In above equations,∂α/∂tα is the Caputo or Riemann–Liouville fraction derivative operator, A is the differential operator, ζr,t is the unknown function and fr,t is known analytic function, r∈Ω is an n-tuple which denotes spatial independent variables, and t represents the temporal independent variable, respectively.Step 2: construct an optimal homotopy for fractional order partial differential equation,ϕr,t;p:Ω×0,1⟶R which is(7)1−p∂αϕr,t∂tα−fr,t−Hr,p∂αϕr,t∂tα−Aϕr,t+fr,t=0.In equation (7) p∈0,1 is the embedding parameter and Hr,p is auxiliary function which satisfies the following relation:Hr,p≠0 for p≠0 and Hr,0=0.The solutionϕr,t converges rapidly to the exact solution as the value of p increases in the interval 0,1. The efficiency of OHAM depends upon the construction and determination of the auxiliary function which controls the convergence of the solution.An auxiliary functionHr,p can be written in the form(8)Hr,p=pk1r,Ci+p2k2r,Ci+p3k3r,Ci+…+pmkmr,Ci.In the above equation,Ci, i=1,2,3… are the convergence control parameters and kir,i=1,2,3,… is a function of r.Step 3: expandingϕr,t;p,Ci in Taylor’s series about p, we have(9)ϕr,t;Ci=ζ0r,t+∑k=1mζkr,t;Cipk,i=1,2,3,….Remarks: it is clear from equation (9), the convergence of the series depends upon the auxiliary convergence control parameter Ci, i=1,2,3,4,…,mIf it converges atp=1, one has(10)ζr,t;Ci=ζ0r,t+∑k=1∞ζkr,t;Ci,i=1,2,3,….Step 4: equating the coefficients of like powers ofp after substituting equation (10) in equation (7), we get zero order, 1st order, 2nd order, and high-order problems:(11)p0:∂αζ0r,t∂tα−f=0,p1:∂αζ1r,t,C1∂tα−1+C1∂αζ0r,t∂tα+1+C1f+C1Aζ0r,t=0,p2:∂αζ2r,t,C1,C2∂tα−1+C1∂αζ1r,t,C1∂tα−C2∂αζ0r,t∂tα+C1Aζ1r,t,C1+C2f+Aζ0r,t=0.Step 5: these problems contain the time fractional derivatives. Therefore, we apply theIα operator on the above problems and obtain a series of solutions as follows:(12)ζ0r,t=Iαf,ζ1r,t;C1=Iα1+C1∂αζ0r,t∂tα−1+C1f−C1Aζ0r,t,ζ2r,t;C1,C2=Iα1+C1∂αζ1r,t;C1∂tα+C2∂αζ0r,t∂tα−C1Aζ1r,t;C1−C2f+Aζ0r,t….By putting the above solutions in equation (12), one can get the approximate solution ζr,t;Ci. The residual Rr,t;Ci is obtained by substituting approximate solution ζr,t;Ci in equation (5).Step 6: the convergence control parametersC1,C2,… can be found either by the Ritz method, Collocation method, Galerkin’s method, or least square method. In this presentation, least square method is used to calculate the convergence control parameters, in which we first construct the functional:(13)χCi=∫0t∫0xR2r,t;Cidrdt.And then the convergence control parameters are calculated by solving the following system:(14)∂χ∂C1=∂χ∂C2=⋯=∂χ∂Cm=0.The approximate solution is obtained by putting the optimum values of the convergence control parameters in equation (10). The method of least squares is a powerful technique and has been used in many other methods such as Optimal Homotopy Perturbation Method (OHPM) and Optimal Auxiliary Functions Method (OAFM) for calculating the optimum values of arbitrary constants [41, 42].


## 4. OHAM Convergence

If the series (10) converges to ζx,t, where ζkx,t∈LR+ is produced by zero order problem and the K-order deformation, then ζx,t is the exact solution of (5).Proof.
since the series(15)∑k=1∞ζi,kx,t;C1,C2,…,Ck,converges, it can be written as(16)ψix,t=∑k=1∞ζi,kx,t;C1,C2,…,Ck,and its holds that(17)limk⟶∞ζi,kx,t;C1,C2,…,Ck=0.
In fact, the following equation is satisfied:(18)ζi,1x,t;C1+∑k=2nζi,kx,t;C⟶k−∑k=2nζi,k−1x,t;C⟶k−1=ζi,2x,t;C⟶2−ζi,1x,t;C1+⋯+ζi,nx,t;C⟶n−ζi,n−1x,t;C⟶n−1=ζi,nx,t;C⟶n.
Now, we have(19)Li,1ζi,1x,t;C1+∑k=2∞L1ζi,kx,t;C⟶k−∑k=2∞Liζi,k−1x,t;C⟶k−1=Liζi,1x,t;C1+∑k=2∞Liζi,kx,t;C⟶k−∑k=2∞Liζi,k−1x,t;C⟶k−1=0,which satisfies(20)Li,1ζi,1x,t;C1+Li∑k=2∞ζi,kx,t;C⟶k−Li∑k=2∞ζi,k−1x,t;C⟶k−1=∑k=2∞CmLiζi,k−mx,t;C⟶k−m+Ni,k−mζi,k−1x,t;Ck−1+gix,t=0.
Now ifCm,m=1,2,3…, is properly chosen, then the equation leads to(21)Liζix,t+A=0,which is the exact solution.


## 5. Application of OHAM

### 5.1. Time Fractional FZK (2, 2, 2)

Consider the following Time Fractional FZK (2, 2, 2) equation with initial condition as(22)∂αw∂tα+∂w2∂x2+18∂3w2∂x3+18∂3w2∂x∂y2=0,0<α≤1,wx,y,0=43λsinh2x+y.The exact solution of equation (22) for α=1,(23)wx,y,t=43λsinh2x+y−λt,where λ is an arbitrary constant.Using the OHAM formulation discussed in Section3, we haveZero-order problem:(24)∂αw∂tα=0,w0x,y,0=43λsinh2x+y.First-order problem:(25)∂αw1x,y,t∂tα=∂αw0x,y,t∂tα+C1∂αw0x,y,t∂tα+2C1∂αw0x,y,t∂tαw0x,y,t+14C1∂2w0x,y,t∂y2,∂w0x,y,t∂x+12C1∂w0x,y,t∂y∂2w0x,y,t∂x∂y+14C1w0x,y,t∂3w0x,y,t∂x∂y2+34C1∂w0x,y,t∂x,∂2w0x,y,t∂x2+14C1w0x,y,t∂3w0x,y,t∂x3,w1x,y,0=0.The solutions of above problems are as follows:(26)w0x,y,t=43λsinh2x+y,w1x,y,t,C1=−8C1tαλ24sinh2x+y−5sinh4x+y9Γ1+α.The 1st order approximate solution by the OHAM is given by the following expression:(27)w˜x,y,t;Ci=w0x,y,t+w1x,y,t;C1,w˜x,y,t;C1=43λsinh2x+y+19Γ1+a−8C1tαλ24sinh2x+y−5sinh4x+y.


### 5.2. Time Fractional FZK (3, 3, 3)

Consider the following Time Fractional FZK (2, 2, 2) equation with initial condition as(28)∂αw∂tα+∂w3∂x+2∂3w3∂x3+2∂3w3∂xy2=0,0<α≤1,wx,y,0=32λsinh16x+y.The exact solution of equation (22) for α=1,(29)wx,y,t=32λsinh16x+y−λt,where λ is an arbitrary constant.Using the OHAM formulation discussed in Section3, we haveZero-order problem:(30)∂αwx,y,t∂tα=0,w0x,y,0=32λsinh16x+y.First-order problem:(31)∂αw1x,y,t∂tα=∂αw0x,y,t∂tα+C1∂αw0x,y,t∂tα+3C1∂w0x,y,t∂xw02x,y,t+12C1∂2w02x,y,t∂y2∂w0x,y,t∂x+12C1w0x,y,t∂2w0x,y,t∂y2∂w0x,y,t∂x+12C1∂w03x,y,t∂x3+24C1w0x,y,t∂w0x,y,t∂x∂2w0x,y,t∂x∂y+6C1w02x,y,t∂3w0x,y,t∂x∂y2+36C1w02x,y,t∂w0x,y,t∂x∂2w0x,y,t∂x2+6C1w02x,y,t∂3w0x,y,t∂y3,w1x,y,0=0.The solutions of above problems are as follows:(32)w0x,y,t=32λsinh16x+y,w1x,y,t;C1=132Γ1+α3C1tαλ3−5cosh16x+y+9cosh12x+y.The 1st order approximate solution by the OHAM is given by the following expression:(33)w˜x,y,t,Ci=w0x,y,t+w1x,y,t,C1.


## 5.1. Time Fractional FZK (2, 2, 2)

Consider the following Time Fractional FZK (2, 2, 2) equation with initial condition as(22)∂αw∂tα+∂w2∂x2+18∂3w2∂x3+18∂3w2∂x∂y2=0,0<α≤1,wx,y,0=43λsinh2x+y.The exact solution of equation (22) for α=1,(23)wx,y,t=43λsinh2x+y−λt,where λ is an arbitrary constant.Using the OHAM formulation discussed in Section3, we haveZero-order problem:(24)∂αw∂tα=0,w0x,y,0=43λsinh2x+y.First-order problem:(25)∂αw1x,y,t∂tα=∂αw0x,y,t∂tα+C1∂αw0x,y,t∂tα+2C1∂αw0x,y,t∂tαw0x,y,t+14C1∂2w0x,y,t∂y2,∂w0x,y,t∂x+12C1∂w0x,y,t∂y∂2w0x,y,t∂x∂y+14C1w0x,y,t∂3w0x,y,t∂x∂y2+34C1∂w0x,y,t∂x,∂2w0x,y,t∂x2+14C1w0x,y,t∂3w0x,y,t∂x3,w1x,y,0=0.The solutions of above problems are as follows:(26)w0x,y,t=43λsinh2x+y,w1x,y,t,C1=−8C1tαλ24sinh2x+y−5sinh4x+y9Γ1+α.The 1st order approximate solution by the OHAM is given by the following expression:(27)w˜x,y,t;Ci=w0x,y,t+w1x,y,t;C1,w˜x,y,t;C1=43λsinh2x+y+19Γ1+a−8C1tαλ24sinh2x+y−5sinh4x+y.


## 5.2. Time Fractional FZK (3, 3, 3)

Consider the following Time Fractional FZK (2, 2, 2) equation with initial condition as(28)∂αw∂tα+∂w3∂x+2∂3w3∂x3+2∂3w3∂xy2=0,0<α≤1,wx,y,0=32λsinh16x+y.The exact solution of equation (22) for α=1,(29)wx,y,t=32λsinh16x+y−λt,where λ is an arbitrary constant.Using the OHAM formulation discussed in Section3, we haveZero-order problem:(30)∂αwx,y,t∂tα=0,w0x,y,0=32λsinh16x+y.First-order problem:(31)∂αw1x,y,t∂tα=∂αw0x,y,t∂tα+C1∂αw0x,y,t∂tα+3C1∂w0x,y,t∂xw02x,y,t+12C1∂2w02x,y,t∂y2∂w0x,y,t∂x+12C1w0x,y,t∂2w0x,y,t∂y2∂w0x,y,t∂x+12C1∂w03x,y,t∂x3+24C1w0x,y,t∂w0x,y,t∂x∂2w0x,y,t∂x∂y+6C1w02x,y,t∂3w0x,y,t∂x∂y2+36C1w02x,y,t∂w0x,y,t∂x∂2w0x,y,t∂x2+6C1w02x,y,t∂3w0x,y,t∂y3,w1x,y,0=0.The solutions of above problems are as follows:(32)w0x,y,t=32λsinh16x+y,w1x,y,t;C1=132Γ1+α3C1tαλ3−5cosh16x+y+9cosh12x+y.The 1st order approximate solution by the OHAM is given by the following expression:(33)w˜x,y,t,Ci=w0x,y,t+w1x,y,t,C1.


## 6. Results and Discussion

OHAM formulation is tested upon the FZKequation. Mathematica 7 is used for most of computational work.Table1 shows the optimum values of the convergence control parameters for FZK (2, 2, 2) and FZK (3, 3, 3) equations at different values of α. In Tables 2 and 3, the results obtained by 1st order approximation of proposed method for the FZK (2, 2, 2) equation are compared with 3rd order approximation of Perturbation-Iteration Algorithm (PIA) and Residual power Series (RPS) method at different values of α. In Tables 4 and 5, the results obtained by 1st order approximation of the proposed method are compared with 3rd order approximation of VIM for FZK (3, 3, 3) equation. Figures 1–4 show the 3D plots of exact versus approximate solution by the proposed method for FZK (2, 2, 2) equation. Figures 1 and 2 show the 3D plots of exact versus approximate solution by the proposed method for FZK (3, 3, 3) equation. Figure 5 shows the 2D plots of approximate solution by the proposed method for FZK (2, 2, 2) equation at different values of α. Figure 6 shows the 2D plots of approximate solution by the proposed method for FZK (3, 3, 3) equation at different values of α.Table 1
Convergence-control parameters for FZK (2, 2, 2) and FZK (3, 3, 3).
FZK (2, 2, 2)
FZK (3, 3, 3)
α
C
1
α
C
1
1
.0
−
0.1654570202126229
1
.0
−
1.0008903783207066
0
.
75
−
0.11770797863128038
0
.
75
−
0.999990856107855
0
.
67
−
0.11303202695535328
0
.
67
−
0.9999963529361133Table 2
1st order approximate solution obtained by the OHAM in comparison with 3 terms approximate solution obtained by PIA and RPS for FZK (2, 2, 2) atλ=0.001.
Solution forα=1.0
Absolute errors forα=1.0
x
y
t
OHAM solution
Exact solution
PIA [34] error
RPS [34] error
OHAM error
0.1
0.1
0.2
5.3966 × 10−5
5.39388 × 10−5
3.85217 × 10−7
3.85217 × 10−7
2.71884 × 10−8
0.3
5.39248 × 10−5
5.38841 × 10−5
5.75911 × 10−7
5.75912 × 10−7
4.07394 × 10−8
0.4
5.38837 × 10−5
5.38294 × 10−5
7.65359 × 10−7
7.65352 × 10−7
5.42615 × 10−8
0.6
0.6
0.2
3.02967 × 10−3
3.03651 × 10−3
4.66337 × 10−5
4.66389 × 10−5
6.83433 × 10−6
0.3
3.02553 × 10−3
3.03578 × 10−3
6.86056 × 10−5
6.86314 × 10−5
1.02517 × 10−5
0.4
3.02138 × 10−3
3.03505 × 10−3
8.98263 × 10−5
8.99046 × 10−5
1.36692 × 10−5
0.9
0.9
0.2
1.14455 × 10−2
1.1537 × 10−2
5.12131 × 10−4
5.14241 × 10−4
9.14704 × 10−5
0.3
1.13973 × 10−2
1.15345 × 10−2
7.38186 × 10−4
7.48450 × 10−4
1.37206 × 10−4
0.4
1.13492 × 10−2
1.15321 × 10−2
9.57942 × 10−4
9.89139 × 10−4
1.82943 × 10−4Table 3
1st order approximate solution obtained by the OHAM in comparison with 3 terms approximate solution obtained by PIA and RPS for FZK (2, 2, 2) atλ=0.001.
α
=
0.67
α
=
0.75
x
y
t
PIA [36] solution
RPS [36] solution
OHAM solution
PIA [36] solution
RPS [36] solution
OHAM solution
0.1
0.1
0.2
5.31854 × 10−5
5.31244 × 10−5
5.39424 × 10−5
5.32747 × 10−5
5.32479 × 10−5
5.3953 × 10−5
0.3
5.28631 × 10−5
5.28410 × 10−5
5.39094 × 10−5
5.29757 × 10−5
5.29675 × 10−5
5.39191 × 10−5
0.4
5.25777 × 10−5
5.25897 × 10−5
5.38798 × 10−5
5.27039 × 10−5
5.27119 × 10−5
5.38881 × 10−5
0.6
0.6
0.2
2.95493 × 10−3
2.95185 × 10−3
3.0273 × 10−3
2.96356 × 10−3
2.96251 × 10−3
3.02837 × 10−3
0.3
2.92662 × 10−3
2.92709 × 10−3
3.02397 × 10−3
2.93717 × 10−3
2.93780 × 10−3
3.02496 × 10−3
0.4
2.90307 × 10−3
2.90522 × 10−3
3.02099 × 10−3
2.91448 × 10−3
2.91561 × 10−3
3.02182 × 10−3
0.9
0.9
0.2
1.06822 × 10−2
1.05522 × 10−2
1.14179 × 10−2
1.07716 × 10−2
1.07143 × 10−2
1.14303 × 10−2
0.3
1.04487 × 10−2
1.01199 × 10−2
1.13792 × 10−2
1.05488 × 10−2
1.03695 × 10−2
1.13907 × 10−2
0.4
9.02777 × 10−2
9.60606 × 10−2
1.13447 × 10−2
1.03736 × 10−2
9.96743 × 10−2
1.13543 × 10−2Table 4
1st order solution compared with three terms of VIM atλ=0.001, for FZK (3, 3, 3).
Solutionα=1
Absolute errors
x
y
t
VIM [37] solution
OHAM solution
Exact solution
OHAM error
0.1
0.1
0.2
5.00091 × 10−5
5.00092 × 10−5
4.99592 × 10−5
4.99519×10–8
0.3
5.00091 × 10−5
5.00091 × 10−5
4.99342 × 10−5
7.49278×10–8
0.4
5.00091 × 10−5
5.00091 × 10−5
4.99092 × 10−5
9.99037×10–8
0.6
0.6
0.2
3.02003 × 10−4
3.02004 × 10−4
3.01953 × 10−4
5.08987×10–8
0.3
3.02003 × 10−4
3.02004 × 10−4
3.01927 × 10−4
7.63479×10–8
0.4
3.02003 × 10−4
3.02004 × 10−4
3.01902 × 10−4
1.01797×10–7
0.9
0.9
0.2
4.56780 × 10−4
4.5678 × 10−4
4.56728 × 10−4
5.21227×10–8
0.3
4.56780 × 10−4
4.5678 × 10−4
4.56702 × 10−4
7.81839×10–8
0.4
4.56780 × 10−4
4.5678 × 10−4
4.56676 × 10−4
1.04245×10–7Table 5
1st order solution compared with 3 terms of VIM for,λ=0.001, for FZK (3, 3, 3).
α
=
0.67
α
=
0.75
x
y
t
VIM [37] solution
OHAM solution
VIM [37] solution
OHAM solution
0.1
0.1
0.2
5.00091 × 10−5
500091 × 10−5
5.00091 × 10−5
5.00091 × 10−5
0.3
5.00090 × 10−5
500091 × 10−5
5.00090 × 10−5
5.00091 × 10−5
0.4
5.00090 × 10−5
50009 × 10−5
5.00090 × 10−5
5.00091 × 10−5
0.6
0.6
0.2
3.02003 × 10−4
302004 × 10−4
3.02003 × 10−4
302004 × 10−4
0.3
3.02003 × 10−4
302004 × 10−4
3.02003 × 10−4
302004 × 10−4
0.4
3.02003 × 10−4
302004 × 10−4
3.02003 × 10−4
302004 × 10−4
0.9
0.9
0.2
4.56780 × 10−4
45678 × 10−4
4.5678 × 10−4
45678 × 10−4
0.3
4.56780 × 10−4
45678 × 10−4
4.56780 × 10−4
45678 × 10−4
0.4
4.56780 × 10−4
45678 × 10−4
4.56780 × 10−4
45678 × 10−4Figure 1
Approximate solution for equation (22) for α=1, (y=0.9).Figure 2
Exact solution for equation (22) for α=1, (y=0.9).Figure 3
Approximate solution for equation (28) for α=1, (y=0.9).Figure 4
Exact solution for equation (28) for α=1, (y=0.9).Figure 5
Convergence at different value of alpha for equation (22) at t=0.1,y=0.2.Figure 6
Convergence at different value of alpha for equation (28) at t=0.1,y=0.2.It is clear from 2D figures that as value ofα increases to 1, the approximate solutions tend close to the exact solutions.


## 7. Conclusion

The 1st order OHAM solution gives more encouraging results in comparison to 3rd order approximations of PIA, RPS, and VIM. From obtained results, it is concluded that the proposed method is very effective and convenient for solving higher-dimensional partial differential equations of fractional order. The accuracy of the method can be further improved by taking higher-order approximations.



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*Source: 1741958-2019-12-10.xml*