# New Modified Variational Iteration Laplace Transform Method Compares Laplace Adomian Decomposition Method for Solution Time-Partial Fractional Differential Equations

Authors: Mohamed Z. Mohamed; Tarig M. Elzaki; Mohamed S. Algolam; Eltaib M. Abd Elmohmoud; Amjad E. Hamza
Journal: Journal of Applied Mathematics
 (2021)

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## Abstract

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.


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## Body

## 1. Introduction

Fractional calculus has tremendous applications in applied science such as mechanical engineering, heat conduction, viscoelasticity, electrode-electrolyte polarization, nanotechnology, diffusion equations, and nearly every part of science and technology. One of the most important uses of fractional calculus is in fractional partial differential equations (PDEs) as most natural phenomena could be modeled by PDEs. Significant perception has been given to exact and convergent solutions of partial differential equations that embrace fractional system derivative because of its new framework and applications in assorted fields.Contemporary improvements of fractional differential equations are supported by the earliest examples of applications in scientific fields, such Liouville’s example. Due to the usual occurrence of FPDEs in different developments of engineering and science, many leading researchers in the field developed the study in this direction by both theoretical and application means [1–4]. The nonlinear oscillation could be represented by fractional derivatives and the fluid dynamic traffic pattern with fractional derivatives [5, 6]; this leads to the use of FPDEs. Several of the main tools used in solving this type of equations are LADM, VIM, and homotopy. It should be noted that LADM effects have shown higher accuracy as compared to other methods in the literature [7]. LADM and the VIM are comparatively new approaches to fit an analytical approximation to linear and nonlinear problems [8–12]. ADM is also compared with LADM to construe the solution of FPDEs given in [13]. The investigation shows that the emergence of VIM and MVIM is strong and simple in calculation when solving the heat equation in the Dirichlet boundary [14]. The MVIM is an approximation numerical algorithm to resolve nonlinear and nonhomogeneous differential equations [15]. In [16], they proposed a combined form for solving nonlinear interval Volterra-Fredholm integral equations of the second kind based on the modifying Laplace Adomian decomposition method. They have been concerned with the solution of Caputo fractional integrodifferential equations by the modified variational iteration technique via the Laplace approach [17]. From the computational viewpoint, the MVIM is more efficient, convenient, and easy to use [18].In recent times, there has been a wide concentration in the VIM for solving a large range of equations such as algebraic, differential, partial-differential, functional-delay, and integral-differential equations. The main feature of this technique is to build a correction functional using a general Lagrange multiplier selected in a suitable way that its adjustment solution is better concerning the initial assessment function. The target of this work is to extend the application of MLVIM to outfit convergent solutions for initial value problems of nonlinear partial fractional differential equations and to create a rapprochement with that obtained by LADM.


## 2. Definitions and Preliminaries

Definition 1.
Let a real functionfω,ω>0, which is said to be the space Cυ,υ∈ℝ, if the there exists a real number η>υ, such that fω=ωηf1ω, where f1ω∈C0,∞ [1, 2].Definition 2.
The Riemann-Liouville fractional integral operator of orderσ≥0, of a function fω∈Cυ,υ≥−1, is known as [1–4]
(1)Jσfω=1Γσ∫0ωω−ςσ−1fςdς,σ,ω>0,fω,σ=0.Definition 3.
The fractional derivative off,f∈C−1n,n∈ℕ∪0 is known as [1–3]
(2)Dσfω=∂σfω∂ωσ=Jn−σ∂nfω∂ωn,n−1<σ<n,n∈ℕ,∂nfω∂ωn,σ=n.
Hence, we have the following properties:(3)1JσJαfω=Jσ+αfω,σ,α≥0,2Jσtγ=Γγ+1Γσ+γ+1tσ+γ,σ>0,γ>−1,ω>0,3JσD∗σfω=fω−∑ρ=0n−1fρ0+ωρρ!,ρ>0,n−1<σ≤n.Definition 4.
The Laplace transform of the fractional derivativeD∗vft is known as
(4)LD∗σfω=sσFs−∑ρ=0n−1fρ0+sσ−ρ−1,ω>0,n−1<σ≤n,n∈ℕ,where Fs is the Laplace transform of ft [8].


## 3. Laplace Adomian Decomposition Method (LADM)

In this section, we will consider a class of nonlinear fractional partial differential equation of the form(5)Dωσuψ,ω+Ruψ,ω+Nuψ,ω=fψ,ω,ψ,ω≥0,n−1<σ≤n,where Dωσ=∂σ/∂ωσ, meaning the fractional derivative is of the order σ, where R is a linear operator, N is a nonlinear term function, and f is the source function. The initial and boundary associated with equation (5) are of the from
(6)uψ,0=hψ,0<σ≤1,uψ,ω⟶∞,ω>0,uψ,0=hψ,∂uψ,0∂ω=kψ,1<σ≤2,uψ,ω⟶∞,ω>0.Applying the Laplace transform to both sides of equation (5) by using linearity of Laplace transform, the result is
(7)LDσuψ,ω+LRuψ,ω+Nuψ,ω=Lfψ,ω.Now, using property of Laplace transform, we get(8)sσuψ,s−sσ−1uψ,0−sσ−2uωψ,0=Lfψ,ω−LRuψ,ω+Nuψ,ω,(9)uψ,s=hψs+kψs2+Lsσfψ,ω−LsσRuψ,ω+Nuψ,ω.The standard Laplace decomposition method defines the solutionux,t known by the series
(10)uψ,ω=∑n=0∞unψ,ω.The nonlinear operator is decomposed as(11)Nuψ,ω=∑n=0∞An,An is the mean Adomian polynomial [3, 4] that is given by
(12)An=1n!dndλnN∑i=0∞λiuiλ=0,n=1,2,⋯.Substituting equations (10), (11), and (12) into equation (9), we have
(13)∑n=0∞unψ,s=hψs+kψs2+Lsσfψ,ω−LsσR∑n=0∞unψ,ω+∑n=0∞An.Both sides of equation (13) yield the following iterative algorithm:
(14)u0ψ,s=hψs+kψs2+Lsσfψ,ω=gψ,s,(15)u1ψ,s=−LsσRu0ψ,ω+A0,u2ψ,s=−LsσRu1ψ,ω+A1,⋮⋮un+1ψ,s=−LsσRunψ,ω+An,n≥1.Applying inverse Laplace transform equations (14) and (15), we obtain
(16)u0ψ,ω=gψ,ω,un+1ψ,ω=−L−1LsσRunψ,ω+An,n≥1.


## 4. Modified Laplace Variational Iteration Method (MLVIM)

The new tactic of the modified Laplace variation iteration technique is instituted on the following steps.Step 1.
Removing the fractional derivative of orderσ with respect to unknown function uψ,ω by applying Laplace and inverse Laplace transforms of equation (5), we obtain
(17)uψ,s=hψs+kψs2+Lsσfψ,ω−LsvNuψ,ω,uψ,ω=gψ,ω−L−1LsσNuψ,ω.Step 2.
Differentiating the results obtained in Step 1 with respect toσ, we then get the value of the general Lagrange multiplier, for the correction functional iterative formula to equal one
(18)∂uψ,ω∂ω=∂gψ,ω∂ω−∂∂ωL−1LsσNuψ,ω.Therefore, equation (18) can be put above in the following formula:
(19)un+1ψ,ω=unψ,ω+∫0ωλ∂unψ,ξ∂ξ−∂Gψ,ξ∂ξ+∂∂ξL−1LsσNuψ,ξdξ.The general Lagrange multiplier for equation (19) can be identified optimally via variation theory to get
(20)1+λξ=ω=0,λξ=ω=−1.Substitutingλ=−1 into equation (19), we get the iterative formula n=1,2,⋯ as follows:
(21)un+1ψ,ω=unψ,ω−∫0ω∂unψ,ξ∂ξ−∂gψ,ξ∂ξ+∂∂ξL−1LsσNunψ,ξdξ.Equation (21) is the call new modified function of Laplace transform and variational iteration method. Start with the initial iteration u0ψ,ω=uψ,0. The exact solution is present as the sequent approximation un+1ψ,ω,n=0,1,2,⋯; in other words, uψ,ω=limn⟶∞unψ,ω.


## 5. Illustrative Examples

In this section, we first implement the Laplace decomposition method and apply the fractional modified Laplace variational iteration method (MLVIM) for solving time-fractional partial differential equations in a tube.Example 5.
Assume the nonlinear time-fractional advection partial differential equation [19].
(22)Dσuσψ,ω+uψ,ωuωψ,ω=ψ+ψω2,ω>0,0<σ≤1.
Follow the first condition(23)uψ,0=0.Now, by enforcement of the Laplace transform to the sides of equation (22), we get
(24)sσuψ,ω=sσ−1uψ,0+Lψ+ψω2+Luψ,ωuψψ,ω.By using the initial condition, we obtain
(25)uψ,s=ψsσ+1+2ψsσ+3−1sσLuψ,ωuψψ,ω,Taking the inverse Laplace transform to the sides of equation (25), we obtain
(26)uψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2−L−11sσLuψ,ωuωψ,ω.


### 5.1. Laplace Adomian Decomposition

By applying the Laplace’s standard decomposition method to determine the solution known by the series equations (11) and (12) into equation (26), we obtain
(27)∑n=0∞unψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2−L−11sσL∑n=0∞An.The few ingredients of Adomian polynomials(28)A0=u0u0ψ,A1=u0ψu1+u0u1ψ,A2=u0ψu2+u1ψu1+u2ψu0,A3=u0ψu3+u1ψu2+u2ψu1+u3ψu0.⋮⋮⋮We have the following recurrent relations:
(29)u0ψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2,u1ψω=−ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4,u2ψ,ω=2ψΓ2σ+1Γ4σ+1Γ3σ+1Γ3σ+1Γ5σ+1ω5σ+8ψΓ2σ+3Γ2σ+1Γσ+3Γ3σ+3+4ψΓ2σ+3Γσ+1Γ2σ+3Γ3σ+3Γ4σ+3Γ5σ+3ω5σ+2+8ψΓ2σ+5Γσ+1Γ2σ+3Γ3σ+5+16ψΓ2σ+3Γσ+1Γ2σ+3Γ3σ+3Γ4σ+5Γ5σ+5ω5σ+4+16ψΓ2σ+5Γ4σ+7Γ3σ+3Γ3σ+5Γ5σ+7ω5σ+6.The third term approximate solution for equation (22) is given by
(30)uψ,ω=ψΓσ+1ωσ+2xΓσ+3ωσ+2+ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4+⋯.


### 5.2. Modified Laplace Variational Iteration

Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (26), and we get
(31)∂uψ,ω∂ω=∂∂ωψωσΓσ+1+∂∂ω2ψωσ+2Γσ+3−∂∂ωL−1Lsvux,tuxx,t.We simply substitute equation (31) and the initial condition equation (23) into equation (21); by the new modified function, we find
(32)u0ψ,ω=uψ,0=0,u1ψ,ω=ψωσΓσ+1+2ψωσ+2Γσ+3,u2ψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2−ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4,u3ψ,ω=ψΓσ+1tσ+2ψΓσ+3tσ+2−ψΓ2σ+1Γ2σ+1Γ3σ+1t3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3t3σ+2+2ψΓ2σ+1Γ4σ+1Γ2σ+1Γ3σ+1Γ5σ+1t5σ+2ψ4Γ2σ+3Γ4σ+3Γ2σ+1Γσ+3Γ3σ+3Γ5σ+3+2Γ2σ+5Γ4σ+3Γ3σ+3Γ3σ+5Γ5σ+3ω5σ+2+2ψ8Γ2σ+3Γσ+1Γ2σ+3Γ3σ+3+8Γ2σ+5Γ2σ+3Γσ+1Γ3σ+5Γ4σ+5Γ5σ+5ω5σ+4−4ψΓ2σ+5Γ3σ+3Γ3σ+5ω3v+4+16ψΓ2σ+5Γ4σ+7Γ3σ+3Γ3σ+5Γ5σ+7ω5σ+6−ψΓ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7v−8ψΓ2σ+1Γ2σ+3Γ6σ+3Γ2σ+1Γ3σ+1Γσ+3Γ3σ+3Γ7σ+3ω7σ+2−ψ16Γ22σ+3Γ2σ+1Γ2σ+3Γ23σ+3+Γ6σ+5Γ7σ+5ω7v+4−ψ4Γ2σ+1Γ2σ+5Γ2σ+3Γ2σ+1Γ3σ+1Γ3σ+5Γ6σ+5Γ7σ+5ω7v+4−32ψΓ2σ+3Γ2σ+5Γ6σ+7Γσ+1Γ3σ+3Γ3σ+3Γ3σ+5Γ7σ+7ω7σ+6−16ψΓ22σ+5Γ6v+9Γ4σ+3Γ23σ+5Γ7σ+9ω7σ+8⋮⋮⋮See Table1 and Figure 1.Example 6.
Assume the following nonlinear fractional partial differential equation [20]:
(33)Dσuωψ,ω+uψ,ωuωψ,ω=0,ω>0,0<σ≤1.
Follow the first condition,(34)uψ,0=−ψ.Table 1
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (22).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.010000000.10.0034389565051070.0034389566298400.0010000000000000.0010000000000000.0010000000000000.20.0068779130102150.0068779132596800.0020000000000000.0020000000000000.0020000000000000.30.0103168695153220.0103168698895200.0030000000000000.0030000000000000.0030000000000000.40.0137558260204300.0137558265193590.0040000000000000.0040000000000000.0040000000000000.50.0171947825255370.0171947831491990.0050000000000000.0050000000000000.0050000000000000.60.0206337390306450.0206337397790390.0060000000000000.0060000000000000.0060000000000000.70.0240726955357520.0240726964088790.0070000000000000.0070000000000000.0070000000000000.80.0275116520408600.0275116530387190.0080000000000000.0080000000000000.0080000000000000.90.0309506085459670.0309506096685590.0090000000000000.0090000000000000.00900000000000010.0343895650510740.0343895662983980.0100000000000000.0100000000000000.010000000000000Figure 1
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (22).
(a)(b)(c)(d)Now, by enforcement of the Laplace transform to the sides of equation (33), we obtain
(35)sσuψ,s=sσ−1uψ,0−Luψ,ωuψψ,ω.By using the initial condition, we obtain
(36)uψ,s=−ψs−1sσLuψ,ωuψψ,ω.Taking the inverse Laplace transform to the sides of equation (36), we obtain
(37)uψ,ω=−ψ−L−11sσLuψ,ωuψψ,ω.


### 5.3. Laplace Adomian Decomposition

By applying the standard Laplace decomposition method which defines the solutionuψ,ωknown by the series equations (14) and (15) into equation (37), we obtain
(38)∑n=0∞unψ,ω=−ψ−L−11sσL∑n=0∞An.The few ingredients of Adomian polynomials(39)A0=u0u0ψ,A1=u0ψu1+u0u1ψ,A2=u0ψu2+u1ψu1+u2ψu0,A3=u0ψu3+u1ψu2+u2ψu1+u3ψu0.⋮⋮⋮We have the following recurrent relations:
(40)u0ψ,ω=−ψ,u1ψ,ω=−ψΓσ+1ωσ,u2ψ,ω=−2ψΓ2σ+1ω2σ,u3ψ,ω=−4ψΓ2σ+1+2ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ.⋮⋮⋮The fourth term’s approximate solution for equation (33) is given by
(41)uψ,ω=−ψ1+ωσΓσ+1+2ω2σΓ2σ+1+4Γ2σ+1+2Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+⋯.


### 5.4. Modified Laplace Variational Iteration

Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (37), and we get
(42)∂uψ,ω∂ω=∂∂ω−ψ−∂∂ωL−11sσLuψ,ωuψψ,ω.We simply substitute equation (42) and the initial condition equation (34) into equation (21); by the new correction function, we find
(43)u0ψ,ω=−ψ,u1ψ,ω=−ψ−ψΓσ+1ωσ,u2ψ,ω=−ψ−ψ1Γσ+1ωσ+2Γ2σ+1ω2σ+Γ2σ+1Γ2σ+1Γ3σ+1ω3σ,u3ψ,ω=−ψ−ψ1Γσ+1ωσ+2Γ2σ+1ω2σ+1Γ2σ+1+4Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+2Γ3σ+1Γ3σ+1+4Γσ+1Γ2σ+1Γ3σ+1Γ4σv+1ω4σ+4Γ22σ+1+2Γ2σ+1Γ3σ+1Γ3σ+1Γ4σ+1Γ5σ+1ω5σ+4Γ2σ+1Γ5σ+1Γ2σ+1Γ3σ+1Γ2σ+1Γ6σ+1ω6σ+4Γ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮See Table2 and Figure 2.Example 7.
Assume the following nonlinear fractional partial differential equation [20]:
(44)Dσuωψ,ω=uψ2+uuψψ,ω>0,0<σ≤1.
Follow the first condition,(45)uψ,0=ψ2.Table 2
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (33).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.0100.0000000000000000.0000000000000000.0000000000000000.0000000000000000.0000000000000000.1-0.103598315342320-0.103600081349948-0.101010100586678-0.101010133333333-0.1010101010101010.2-0.207196630684640-0.207200162699895-0.202020201173356-0.202020266666667-0.2020202020202020.3-0.310794946026960-0.310800244049843-0.303030301760034-0.303030400000000-0.3030303030303030.4-0.414393261369279-0.414400325399790-0.404040402346711-0.404040533333333-0.4040404040404040.5-0.517991576711599-0.518000406749738-0.505050502933389-0.505050666666667-0.5050505050505050.6-0.621589892053919-0.621600488099685-0.606060603520067-0.606060800000000-0.6060606060606060.7-0.725188207396239-0.725200569449633-0.707070704106745-0.707070933333333-0.7070707070707070.8-0.828786522738559-0.828800650799581-0.808080804693423-0.808081066666667-0.8080808080808080.9-0.932384838080879-0.932400732149528-0.909090905280101-0.909091200000000-0.9090909090909091-1.035983153423190-1.036000813499470-1.010101005866770-1.010101333333330-1.010101010101010Figure 2
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (33).
(a)(b)(c)(d)Now, by enforcement of the Laplace transform to the sides of equation (44), we obtain
(46)uψ,s=uψ,0s+1sσLuψ2+uuψψ.By using the initial condition, we obtain
(47)uψ,s=ψ2s+1sσLuψ2+uuψψ.Taking the inverse Laplace transform to the sides of equation (44), we obtain
(48)uψ,ω=ψ2+L−11sσLuψ2+uuψψ.


### 5.5. Laplace Adomian Decomposition

By applying the standard Laplace decomposition method which defines the solutionuψ,ωknown by the series equations (14) and (15) into equation (48), we obtain
(49)∑n=0∞unψ,ω=ψ2+L−11sσLunψ2+∑n=0∞An.The few components of Adomian polynomials(50)A0=u0u0ψψ,A1=u0ψψu1+u0u1ψψ,A2=u0ψψu2+u1ψψu1+u2ψψu0,A3=u0ψu3ψ+u1ψψu2+u2ψψu1+u3ψψu0.⋮⋮⋮We have the following recurrent relations:(51)u0ψ,ω=ψ2,u1ψ,ω=6ψ2Γσ+1ωσ,u2ψ,ω=24ψ2Γ2σ+1ω2σ+144ψ2Γ2σ+1Γ2σ+1Γ3σ+1ω3σ,u3ψ,ω=ψ296Γ3σ+1+72Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+72ψ2Γ2σ+1Γ2σ+1Γ4σ+1ω4σ+2304ψ2Γ2σ+1Γ5σ+1ω5v+27648ψ2Γ2σ+1Γ3σ+1Γ6σ+1ω6σ+82944ψ2Γ22σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮The fourth term’s approximate solution for equation (44) is given by
(52)uψ,ω=ψ21+6Γσ+1ωσ+24Γ2σ+1t2σ+144Γ2σ+1Γ2σ+1Γ3σ+1t3σ+96Γ3σ+1+72Γ2σ+1Γ2σ+1Γ3σ+1t3σ+72Γ2σ+1Γ2σ+1Γ4σ+1ω4σ+2304Γ2σ+1Γ5σ+1ω5v+27648Γ2σ+1Γ3σ+1Γ6σ+1t6σ+82944x2Γ22v+1Γ4v+1Γ23v+1Γ7v+1t7v+⋯.


### 5.6. Modified Laplace Variational Iteration

(53)∂uψ,ω∂ω=∂∂ωψ2+∂∂ωL−11sσLuψ2+uuψψ.We simply substitute equation (53) and the initial condition equation (45) into equation (21); by the new correction function, we find
(54)u0ψ,ω=ψ2,u1ψ,ω=ψ21+6Γσ+1ωσ,u2ψ,ω=ψ21+6Γσ+1ωσ+72Γ2σ+1ω2σ+216Γ2σ+1Γ2ω+1Γ3σ+1ω3σ,u3ψ,ω=ψ21+1Γσ+1ωσ+72Γ2σ+1ω2σ+216Γ2σ+1+864Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+4684Γσ+1Γ2σ+1+2592Γ2σ+1Γ2σ+1Γ3σ+1Γ3σ+1Γ4σ+1ω4σ+31104Γ22σ+1+6912Γ2σ+1Γ2σ+1Γ3σ+1Γ4σ+1Γ5σ+1ω5σ+279936Γ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮See Table3 and Figure 3.Table 3
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (44).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.0100.00000000000000.00000000000000.00000000000000.00000000000000.00000000000000.10.01264824897620.01229931531540.01063672000000.01061288061930.01063829787230.20.05059299590460.04919726126170.04254688000000.04245152247710.04255319148940.30.11383424078540.11069383783890.09573048000000.09551592557340.09574468085110.40.20237198361840.19678904504690.17018752000000.16980608990820.17021276595740.50.31620622440380.30748288288570.26591800000000.26532201548160.26595744680850.60.45533696314140.44277535135550.38292192000000.38206370229350.38297872340430.70.61976419983140.60266645045600.52119928000000.52003115034390.52127659574470.80.80948793447360.78715618018750.68075008000000.67922435963290.68085106382980.91.02450816706820.99624454054980.86157432000000.85964333016040.861702127659611.26482489761501.22993153154291.06367200000001.06128806192641.0638297872340Figure 3
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (44).
(a)(b)(c)(d)


## 5.1. Laplace Adomian Decomposition

By applying the Laplace’s standard decomposition method to determine the solution known by the series equations (11) and (12) into equation (26), we obtain
(27)∑n=0∞unψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2−L−11sσL∑n=0∞An.The few ingredients of Adomian polynomials(28)A0=u0u0ψ,A1=u0ψu1+u0u1ψ,A2=u0ψu2+u1ψu1+u2ψu0,A3=u0ψu3+u1ψu2+u2ψu1+u3ψu0.⋮⋮⋮We have the following recurrent relations:
(29)u0ψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2,u1ψω=−ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4,u2ψ,ω=2ψΓ2σ+1Γ4σ+1Γ3σ+1Γ3σ+1Γ5σ+1ω5σ+8ψΓ2σ+3Γ2σ+1Γσ+3Γ3σ+3+4ψΓ2σ+3Γσ+1Γ2σ+3Γ3σ+3Γ4σ+3Γ5σ+3ω5σ+2+8ψΓ2σ+5Γσ+1Γ2σ+3Γ3σ+5+16ψΓ2σ+3Γσ+1Γ2σ+3Γ3σ+3Γ4σ+5Γ5σ+5ω5σ+4+16ψΓ2σ+5Γ4σ+7Γ3σ+3Γ3σ+5Γ5σ+7ω5σ+6.The third term approximate solution for equation (22) is given by
(30)uψ,ω=ψΓσ+1ωσ+2xΓσ+3ωσ+2+ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4+⋯.


## 5.2. Modified Laplace Variational Iteration

Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (26), and we get
(31)∂uψ,ω∂ω=∂∂ωψωσΓσ+1+∂∂ω2ψωσ+2Γσ+3−∂∂ωL−1Lsvux,tuxx,t.We simply substitute equation (31) and the initial condition equation (23) into equation (21); by the new modified function, we find
(32)u0ψ,ω=uψ,0=0,u1ψ,ω=ψωσΓσ+1+2ψωσ+2Γσ+3,u2ψ,ω=ψΓσ+1ωσ+2ψΓσ+3ωσ+2−ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3ω3σ+2−4ψΓ2σ+5Γ2σ+3Γ3σ+5ω3σ+4,u3ψ,ω=ψΓσ+1tσ+2ψΓσ+3tσ+2−ψΓ2σ+1Γ2σ+1Γ3σ+1t3σ−4ψΓ2σ+3Γσ+1Γσ+3Γ3σ+3t3σ+2+2ψΓ2σ+1Γ4σ+1Γ2σ+1Γ3σ+1Γ5σ+1t5σ+2ψ4Γ2σ+3Γ4σ+3Γ2σ+1Γσ+3Γ3σ+3Γ5σ+3+2Γ2σ+5Γ4σ+3Γ3σ+3Γ3σ+5Γ5σ+3ω5σ+2+2ψ8Γ2σ+3Γσ+1Γ2σ+3Γ3σ+3+8Γ2σ+5Γ2σ+3Γσ+1Γ3σ+5Γ4σ+5Γ5σ+5ω5σ+4−4ψΓ2σ+5Γ3σ+3Γ3σ+5ω3v+4+16ψΓ2σ+5Γ4σ+7Γ3σ+3Γ3σ+5Γ5σ+7ω5σ+6−ψΓ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7v−8ψΓ2σ+1Γ2σ+3Γ6σ+3Γ2σ+1Γ3σ+1Γσ+3Γ3σ+3Γ7σ+3ω7σ+2−ψ16Γ22σ+3Γ2σ+1Γ2σ+3Γ23σ+3+Γ6σ+5Γ7σ+5ω7v+4−ψ4Γ2σ+1Γ2σ+5Γ2σ+3Γ2σ+1Γ3σ+1Γ3σ+5Γ6σ+5Γ7σ+5ω7v+4−32ψΓ2σ+3Γ2σ+5Γ6σ+7Γσ+1Γ3σ+3Γ3σ+3Γ3σ+5Γ7σ+7ω7σ+6−16ψΓ22σ+5Γ6v+9Γ4σ+3Γ23σ+5Γ7σ+9ω7σ+8⋮⋮⋮See Table1 and Figure 1.Example 6.
Assume the following nonlinear fractional partial differential equation [20]:
(33)Dσuωψ,ω+uψ,ωuωψ,ω=0,ω>0,0<σ≤1.
Follow the first condition,(34)uψ,0=−ψ.Table 1
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (22).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.010000000.10.0034389565051070.0034389566298400.0010000000000000.0010000000000000.0010000000000000.20.0068779130102150.0068779132596800.0020000000000000.0020000000000000.0020000000000000.30.0103168695153220.0103168698895200.0030000000000000.0030000000000000.0030000000000000.40.0137558260204300.0137558265193590.0040000000000000.0040000000000000.0040000000000000.50.0171947825255370.0171947831491990.0050000000000000.0050000000000000.0050000000000000.60.0206337390306450.0206337397790390.0060000000000000.0060000000000000.0060000000000000.70.0240726955357520.0240726964088790.0070000000000000.0070000000000000.0070000000000000.80.0275116520408600.0275116530387190.0080000000000000.0080000000000000.0080000000000000.90.0309506085459670.0309506096685590.0090000000000000.0090000000000000.00900000000000010.0343895650510740.0343895662983980.0100000000000000.0100000000000000.010000000000000Figure 1
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (22).
(a)(b)(c)(d)Now, by enforcement of the Laplace transform to the sides of equation (33), we obtain
(35)sσuψ,s=sσ−1uψ,0−Luψ,ωuψψ,ω.By using the initial condition, we obtain
(36)uψ,s=−ψs−1sσLuψ,ωuψψ,ω.Taking the inverse Laplace transform to the sides of equation (36), we obtain
(37)uψ,ω=−ψ−L−11sσLuψ,ωuψψ,ω.


## 5.3. Laplace Adomian Decomposition

By applying the standard Laplace decomposition method which defines the solutionuψ,ωknown by the series equations (14) and (15) into equation (37), we obtain
(38)∑n=0∞unψ,ω=−ψ−L−11sσL∑n=0∞An.The few ingredients of Adomian polynomials(39)A0=u0u0ψ,A1=u0ψu1+u0u1ψ,A2=u0ψu2+u1ψu1+u2ψu0,A3=u0ψu3+u1ψu2+u2ψu1+u3ψu0.⋮⋮⋮We have the following recurrent relations:
(40)u0ψ,ω=−ψ,u1ψ,ω=−ψΓσ+1ωσ,u2ψ,ω=−2ψΓ2σ+1ω2σ,u3ψ,ω=−4ψΓ2σ+1+2ψΓ2σ+1Γ2σ+1Γ3σ+1ω3σ.⋮⋮⋮The fourth term’s approximate solution for equation (33) is given by
(41)uψ,ω=−ψ1+ωσΓσ+1+2ω2σΓ2σ+1+4Γ2σ+1+2Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+⋯.


## 5.4. Modified Laplace Variational Iteration

Now the new tactic of the modified Laplace variational iteration technique is instituted on Step 2 from equation (37), and we get
(42)∂uψ,ω∂ω=∂∂ω−ψ−∂∂ωL−11sσLuψ,ωuψψ,ω.We simply substitute equation (42) and the initial condition equation (34) into equation (21); by the new correction function, we find
(43)u0ψ,ω=−ψ,u1ψ,ω=−ψ−ψΓσ+1ωσ,u2ψ,ω=−ψ−ψ1Γσ+1ωσ+2Γ2σ+1ω2σ+Γ2σ+1Γ2σ+1Γ3σ+1ω3σ,u3ψ,ω=−ψ−ψ1Γσ+1ωσ+2Γ2σ+1ω2σ+1Γ2σ+1+4Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+2Γ3σ+1Γ3σ+1+4Γσ+1Γ2σ+1Γ3σ+1Γ4σv+1ω4σ+4Γ22σ+1+2Γ2σ+1Γ3σ+1Γ3σ+1Γ4σ+1Γ5σ+1ω5σ+4Γ2σ+1Γ5σ+1Γ2σ+1Γ3σ+1Γ2σ+1Γ6σ+1ω6σ+4Γ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮See Table2 and Figure 2.Example 7.
Assume the following nonlinear fractional partial differential equation [20]:
(44)Dσuωψ,ω=uψ2+uuψψ,ω>0,0<σ≤1.
Follow the first condition,(45)uψ,0=ψ2.Table 2
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (33).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.0100.0000000000000000.0000000000000000.0000000000000000.0000000000000000.0000000000000000.1-0.103598315342320-0.103600081349948-0.101010100586678-0.101010133333333-0.1010101010101010.2-0.207196630684640-0.207200162699895-0.202020201173356-0.202020266666667-0.2020202020202020.3-0.310794946026960-0.310800244049843-0.303030301760034-0.303030400000000-0.3030303030303030.4-0.414393261369279-0.414400325399790-0.404040402346711-0.404040533333333-0.4040404040404040.5-0.517991576711599-0.518000406749738-0.505050502933389-0.505050666666667-0.5050505050505050.6-0.621589892053919-0.621600488099685-0.606060603520067-0.606060800000000-0.6060606060606060.7-0.725188207396239-0.725200569449633-0.707070704106745-0.707070933333333-0.7070707070707070.8-0.828786522738559-0.828800650799581-0.808080804693423-0.808081066666667-0.8080808080808080.9-0.932384838080879-0.932400732149528-0.909090905280101-0.909091200000000-0.9090909090909091-1.035983153423190-1.036000813499470-1.010101005866770-1.010101333333330-1.010101010101010Figure 2
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (33).
(a)(b)(c)(d)Now, by enforcement of the Laplace transform to the sides of equation (44), we obtain
(46)uψ,s=uψ,0s+1sσLuψ2+uuψψ.By using the initial condition, we obtain
(47)uψ,s=ψ2s+1sσLuψ2+uuψψ.Taking the inverse Laplace transform to the sides of equation (44), we obtain
(48)uψ,ω=ψ2+L−11sσLuψ2+uuψψ.


## 5.5. Laplace Adomian Decomposition

By applying the standard Laplace decomposition method which defines the solutionuψ,ωknown by the series equations (14) and (15) into equation (48), we obtain
(49)∑n=0∞unψ,ω=ψ2+L−11sσLunψ2+∑n=0∞An.The few components of Adomian polynomials(50)A0=u0u0ψψ,A1=u0ψψu1+u0u1ψψ,A2=u0ψψu2+u1ψψu1+u2ψψu0,A3=u0ψu3ψ+u1ψψu2+u2ψψu1+u3ψψu0.⋮⋮⋮We have the following recurrent relations:(51)u0ψ,ω=ψ2,u1ψ,ω=6ψ2Γσ+1ωσ,u2ψ,ω=24ψ2Γ2σ+1ω2σ+144ψ2Γ2σ+1Γ2σ+1Γ3σ+1ω3σ,u3ψ,ω=ψ296Γ3σ+1+72Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+72ψ2Γ2σ+1Γ2σ+1Γ4σ+1ω4σ+2304ψ2Γ2σ+1Γ5σ+1ω5v+27648ψ2Γ2σ+1Γ3σ+1Γ6σ+1ω6σ+82944ψ2Γ22σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮The fourth term’s approximate solution for equation (44) is given by
(52)uψ,ω=ψ21+6Γσ+1ωσ+24Γ2σ+1t2σ+144Γ2σ+1Γ2σ+1Γ3σ+1t3σ+96Γ3σ+1+72Γ2σ+1Γ2σ+1Γ3σ+1t3σ+72Γ2σ+1Γ2σ+1Γ4σ+1ω4σ+2304Γ2σ+1Γ5σ+1ω5v+27648Γ2σ+1Γ3σ+1Γ6σ+1t6σ+82944x2Γ22v+1Γ4v+1Γ23v+1Γ7v+1t7v+⋯.


## 5.6. Modified Laplace Variational Iteration

(53)∂uψ,ω∂ω=∂∂ωψ2+∂∂ωL−11sσLuψ2+uuψψ.We simply substitute equation (53) and the initial condition equation (45) into equation (21); by the new correction function, we find
(54)u0ψ,ω=ψ2,u1ψ,ω=ψ21+6Γσ+1ωσ,u2ψ,ω=ψ21+6Γσ+1ωσ+72Γ2σ+1ω2σ+216Γ2σ+1Γ2ω+1Γ3σ+1ω3σ,u3ψ,ω=ψ21+1Γσ+1ωσ+72Γ2σ+1ω2σ+216Γ2σ+1+864Γ2σ+1Γ2σ+1Γ3σ+1ω3σ+4684Γσ+1Γ2σ+1+2592Γ2σ+1Γ2σ+1Γ3σ+1Γ3σ+1Γ4σ+1ω4σ+31104Γ22σ+1+6912Γ2σ+1Γ2σ+1Γ3σ+1Γ4σ+1Γ5σ+1ω5σ+279936Γ22σ+1Γ6σ+1Γ4σ+1Γ23σ+1Γ7σ+1ω7σ.⋮⋮⋮See Table3 and Figure 3.Table 3
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.2, with mesh points ψ=0:1, for equation (44).
ωψσ=0.75σ=1ExactApproximationsApproximationsMLVIMLADMMLVIMLADM0.0100.00000000000000.00000000000000.00000000000000.00000000000000.00000000000000.10.01264824897620.01229931531540.01063672000000.01061288061930.01063829787230.20.05059299590460.04919726126170.04254688000000.04245152247710.04255319148940.30.11383424078540.11069383783890.09573048000000.09551592557340.09574468085110.40.20237198361840.19678904504690.17018752000000.16980608990820.17021276595740.50.31620622440380.30748288288570.26591800000000.26532201548160.26595744680850.60.45533696314140.44277535135550.38292192000000.38206370229350.38297872340430.70.61976419983140.60266645045600.52119928000000.52003115034390.52127659574470.80.80948793447360.78715618018750.68075008000000.67922435963290.68085106382980.91.02450816706820.99624454054980.86157432000000.85964333016040.861702127659611.26482489761501.22993153154291.06367200000001.06128806192641.0638297872340Figure 3
Comparison of LADM and MLVIM solution for the first three approximations with exact solution forω=0.01, with mesh points ψ=0:1, for equation (44).
(a)(b)(c)(d)


## 6. Conclusion

In this paper, we applied the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition methods (LADM) to obtain the exact solution of nonlinear fractional partial differential equations. We studied the convergence of LADM and MLVIM with the exact solution and the efficacy of the methods shown by the table, and we find that the convergence of LADM results better than MLVIM. The solutions obtained by the two methods are derived by infinite approximate series. Numerical examples are offered to clarify the performance and accuracy of the two methods.



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