# A New Modified Technique of Adomian Decomposition Method for Fractional Diffusion Equations with Initial-Boundary Conditions

Authors: Saadia Masood; Hajira; Hassan Khan; Rasool Shah; Saima Mustafa; Qasim Khan; Muhammad Arif; Fairouz Tchier; Gurpreet Singh
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In general, solving fractional partial differential equations either numerically or analytically is a difficult task. However, mathematicians have tried their best to make the task easy and promoted various techniques for their solutions. In this regard, a very prominent and accurate technique, which is known as the new technique of the Adomian decomposition method, is developed and presented for the solution of the initial-boundary value problem of the diffusion equation with fractional view analysis. The suggested model is an important mathematical model to study the behavior of degrees of memory in diffusing materials. Some important results for the given model at different fractional orders of the derivatives are achieved. Graphs show the obtained results to confirm the accuracy and validity of the suggested technique. These results are in good contact with the physical dynamics of the targeted problems. The obtained results for both fractional and integer orders problems are explained through graphs and tables. Tables and graphs support the physical behavior of each problem and the best of physical analysis. From the results, it is concluded that as the fractional order derivative is changed, the graphs or paths of dynamics are also changed. Therefore, we now choose the best solution or dynamic of the problem at a particular derivative order. It is analyzed that the present technique is one of the best techniques to handle the solutions of fractional partial differential equations having initial and boundary conditions (BCs), which are very rare in literature. Furthermore, a small number of calculations are done to achieve a very high rate of convergence, which is the novelty of the present research work. The proposed method provides the series solution with twice recursive formulae to increase the desired accuracy and is preferred among the best techniques to find the solution of fractional partial differential equations with mixed initials and BCs.


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## Body

## 1. Introduction

Fractional Calculus (FC) is the study of derivatives and integration of fractional orders. The idea was first initiated by L’Hospital, who wrote a letter to Leibniz about the noninteger order of derivatives in 1665. After that, the devoted work done by Euler, Lagrange, Abel, and Liouville gives more extension to the field, which is very popular nowadays because of its essential applications in the areas of biology, physics, fluid mechanics, and other sciences [1–3].Fractional-order differential equations (FPDEs) have a great contribution in the modeling of a variety of complex natural and nonlinear phenomena. FPDEs have made a significant contribution to various scientific research fields, including a diverse range of processes and systems, memories, and various branches of mathematics. The modeling of FPDEs, whether they are with respect to time or space, is more convergent, and many natural phenomena are accurately described by them. The researchers in the fields of anomalous diffusion, dielectric polarization, control theory, and other problems of physical phenomena are interested in FPDEs [4–10]. Since the fractional order of the derivative works more accurately than integer order in describing the properties related to hereditary and where the future state is influenced by the past state, FPDEs give the highest contribution to explaining such types of systems. Psychology, biology, acoustics, chemistry [11], physics, colored noises [12], and continuum mechanics [13] are some of the scientific phenomena and problems that FPDEs are used to model. Special applications of FPDEs can be found in various branches of physics and hydrology, including [14–22]. FPDEs have grown in popularity because, by definition, the fractional derivative is global, whereas the integer order derivative is local [23–25].In recent years, the development of numerical and analytical methodologies for the solution of FPDEs is a hot topic among the researchers. Obtaining numerical or analytical solutions to FPDEs is never a simple task for mathematicians. Many researchers, on the other hand, have devised a number of novel techniques for dealing with FPDE solutions. Some of the important techniques include the Haar wavelet method (HWM) [26], the Laplace transform method [27], the Elzaki transform decomposition method (ETDM) [28], the Adomian decomposition method (ADM) [29], the finite difference technique [30, 31], the natural transform decomposition approach [32], the Legendre base method [33], the homotopy analysis method [34], the differential transform method [35], the variational iteration approach [36], and the Bernstein polynomial [37].Many researchers have studied the time-fractional diffusion equations (TFDEs) because of their various applications in science and engineering and other branches of applied sciences. TFDEs are broadly found in physical, biological, and engineering processes [38–41]. The TFDEs are used by Nigmatulin [39] to describe diffusion in media with spectral geometry. The TFDEs are being investigated by many researchers, both analytically and numerically [42, 43]. The solution method of the Laplace transform and a similar method is used to obtain the invariant solution of TFDEs by Gorenlfo et al. [44, 45], and Lin and Zu applied a finite difference scheme in the Legendre spectral method and in space for TFDEs [46]. Dhaigude and Nikam [13] and Schneider and Wyss [43] worked on TFDEs and wave equations to obtain the solution. Moreover, the existence and uniqueness of the targeted problems are shown in [47]. Here, the researchers either used only one from initial boundary conditions (IBCs) to solve the problems. In the current research article, a modified method of ADM is implemented to solve TFDEs with both IBCs suggested by Ali in [48]. The same procedure is applied to the problems of having both initial and BCs in [49] with the homotopy perturbation method, and the results are excellent. Ali applied this new technique in [50] with a variational iteration method to initial-boundary value problems. The procedure becomes accurate because there is a new initial approximate solution with each new iteration. For justification, some examples are discussed in this paper.ADM was first introduced by Adomian [51] in the 1980s, and it was observed that the technique is beneficial for nonlinear equations. Wazwaz [52] applied the same method to solve different kinds of differential equations. Niu and Wang [53] used the decomposition method to find the solution of fractional heat-like and wave-like equations. Niu and Wang [53] applied this method to boundary value problems to calculate a one-step optimal homotopy analysis method. Pandir and Yildirim used the homotopy perturbation method and ADM in conformable sense [54]. The mathematicians have made several modifications to ADM which have improved the accuracy of the technique, and some of them are [55–57]. Here, a modified technique is implemented to solve the initial and boundary value problems (IBVPs) of TFDEs. In the literature, various authors have used numerical and analytical techniques for the solution of the initial value problems of FPDEs and their systems. However, only a few attempts were made to solve IBVPs of FPDEs and their systems. In this regard, Elaf Jaafar Ali has made a contribution and developed a new technique to solve FPDEs. He modified the existing techniques of ADM to solve IBVPs of FPDEs.In this paper, we will work on the solution of TFDEs by using a new technique of the Adomian decomposition method (ADM). In literature, some important numerical and analytical techniques have been used for the solutions of time and space fractional diffusion equations [25, 54]. In this article, the work of Elaf Jafaar Ali is further extended to solve IBVPs of TFDEs. The general description of the proposed method is implemented to solve some examples of the suggested problems. The analytical solutions of FPDEs with initial and boundary conditions are very difficult to investigate. In the current work, the analytical solutions of TFDEs are obtained in a very simple and straightforward procedure and provide the closed-form solutions. The less computational work and simplicity are the uniqueness of the present modified technique. The obtained results are displayed through graphs. The graphical representations have shown that there is a close contact between the exact and the approximate solutions of the problems. The solutions are obtained for various fractional-order problems. The fractional order solutions provide useful information about the dynamics of the suggested problems. It is observed that the proposed technique has a very effective procedure for solving FPDEs and their systems with IBCs. However, some limitations are observed while using the present technique, that is, if the FPDEs or their systems have a higher number of IBCs, then the proposed method required a large number of calculations to achieve the results. Mostly, the suggested methods have smaller accuracy at a greater time value and their accuracy increased at a smaller time value. For higher nonlinear problems, the solution components are not easy to compute, so very few terms are calculated to achieve the required solution.


## 2. Preliminaries

Some definitions that are related to our study are considered in this section.


### 2.1. Definition

The integral operator of Reimann-Liouville having orderδ is given by [40]
(1)Iσδhσ=1Γδ∫0τσ−νδ−1dν,δ>0,hσ,δ=0,where Γ is the gamma function and can be written as
(2)Γω=∫0∞e−σσω−1dτ,ω∈ℂ.


### 2.2. Definition

The expression for Caputo for fractional orderδ is as follows:
(3)Dδhσ=∂δhσ∂τδ=Im−δ∂δhσ∂τδ,m−1<δ≤m,m∈ℕ,∂δhσ∂τδ,wherem∈ℕ, σ>0, g∈ℂτ, and τ≥1.


### 2.3. Lemma

Forj−1<δ≤j with j∈ℕ and h∈ℂτ with τ≥−1, then [58]
(4)IδIb=Iδ+bhσ,b,δ≥0,Iδσλ=Γλ+1Γδ+λ+1σδ+λ,δ>0,λ>−1,σ>0,IδDδhσ=hσ−∑k=0j−1hk0+σkk!,whereσ>0, j−1<δ≤j.


### 2.4. Definition

The Mittag-Leffler functionEδρ for δ>0 is
(5)Eδρ=∑m=0∞ρmΓmδ+1,δ>0,ρ∈ℂ.


## 2.1. Definition

The integral operator of Reimann-Liouville having orderδ is given by [40]
(1)Iσδhσ=1Γδ∫0τσ−νδ−1dν,δ>0,hσ,δ=0,where Γ is the gamma function and can be written as
(2)Γω=∫0∞e−σσω−1dτ,ω∈ℂ.


## 2.2. Definition

The expression for Caputo for fractional orderδ is as follows:
(3)Dδhσ=∂δhσ∂τδ=Im−δ∂δhσ∂τδ,m−1<δ≤m,m∈ℕ,∂δhσ∂τδ,wherem∈ℕ, σ>0, g∈ℂτ, and τ≥1.


## 2.3. Lemma

Forj−1<δ≤j with j∈ℕ and h∈ℂτ with τ≥−1, then [58]
(4)IδIb=Iδ+bhσ,b,δ≥0,Iδσλ=Γλ+1Γδ+λ+1σδ+λ,δ>0,λ>−1,σ>0,IδDδhσ=hσ−∑k=0j−1hk0+σkk!,whereσ>0, j−1<δ≤j.


## 2.4. Definition

The Mittag-Leffler functionEδρ for δ>0 is
(5)Eδρ=∑m=0∞ρmΓmδ+1,δ>0,ρ∈ℂ.


## 3. Adomian Decomposition Method

This method was discovered by Adomian in 1994 for the solution of linear and nonlinear differential and integrodifferential equations [29]. To understand the method, let us consider an equation of the following form:
(6)Fϑσ=gσ,where F is a nonlinear differential operator and g is the known function. We will split the linear term in Fϑσ into the form Lϑ+Rϑ, where L is the invertible operator, chosen as the highest order derivative, R represents the linear operator, and then, Equation (6) has the representation as follows:
(7)Rϑ+Rϑ+Nϑ=g,where Nϑ is the nonlinear term of Fϑσ. Apply R−1 to Equation (7) on both sides.
(8)ϑ=φ+R−1g−R−1Rϑ−R−1Nϑ,where the constant of integration is φ, and Rφ=0.The following infinite series shows the solution of ADM as(9)ϑ=∑n=0∞ζn.TheNϑ is a nonlinear term represented by An, defined as follows:
(10)Nϑ=∑n=0∞An.Using the following to calculateAn,
(11)An=1n!dndψnN∑k=0∞ψkυk,n=0,1,⋯.Equation (6) has a solution in the form of a series as follows:
(12)ϑ0=φ+R−1g,n=0,ϑn+1=R−1Rϑn−R−1An,n≥0.


## 4. Modification of ADM

To understand the main idea of the proposed technique, we will take the following one-dimensional equation [48]:
(13)Dτδϑσ,τ=∂2ϑσ,τ∂σ2+wσ,τ,0<σ<1,τ>0,0<δ<1,having the IBCs as follows:
(14)ϑσ,τ=ℓ0σ,∂ϑσ,0∂τ=ℓ1σ,0≤σ≤1,ϑ0,τ=ℏ0τ,ϑ1,τ=ℏ1τ,τ>0.The source term is represented bywσ,τ.The new initial solutionϑn∗ calculated for Equation (13) can be written in operator form as
(15)ϑn∗=ϑnσ,τ+1−σℏ0τ−ϑn0,τ+σℏ1τ−ϑn1,τ.In operator form, Equation (13) can be written as
(16)Rϑ=∂2ϑσ,τ∂σ2+wσ,τ.R is
(17)R=∂δ∂τδ,so R−1 is
(18)R−1.=Iδ.dτ.ApplyingR−1 to Equation (16), we obtained
(19)ϑσ,τ=ϑσ,0+R−1∂2ϑσ,τ∂σ2+wσ,τ,where n=0,1,⋯.The initial approximation can be written as(20)ϑ0σ,τ=ϑσ,0+R−1wσ,τ,and hence, the iteration formula is
(21)ϑn+1σ,τ=R−1∂2ϑ∗σ,τ∂σ2.The initial solutionsun∗ of Equation (13) satisfied both the IBCs, as given in the following:
(22)atτ=0,ϑn∗σ,0=ϑnσ,0,σ=0,ϑn∗0,τ=ℏ0τ,σ=1,ϑn∗1,τ=ℏ1τ.The proposed technique works effectively for two-dimensional problems.


## 5. Numerical Results

In this section, some illustrative examples are solved by the new technique of ADM.


### 5.1. Example

Consider TFDE of the following form [59]:
(23)∂δϑσ,τ∂τδ=∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ,0≤σ≤2,0<δ≤1,having the IBCs as follows:
(24)ϑσ,0=0,ϑ0,τ=ϑ2,τ=0.The problem has the analytical solution atδ=1 as follows:
(25)ϑσ,τ=σ42−στ3+δ.Applying the suggested method of ADM to Equation (23), we have
(26)ϑn∗σ,τ=ϑnσ,τ+1−σ0−ϑn0,τ+σ0−ϑn2,τ,wheren=0,1,⋯.ApplyingR to Equation (23), we have
(27)Rϑ=∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ,where R=∂δ/∂τδ and R−1 is
(28)R−1.=Iδ.dτ.Operating Equation (23) by mathcalR−1, we have
(29)ϑσ,τ=uσ,0+R−1∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ.Using ADM solution, the initial approximation becomes
(30)ϑ0σ,τ=ϑσ,0+R−1Γ4+δ6σ42−στ3−4σ26−5στ3+σ=0+Γ4+δΓ4σ42−στ3+δ6Γ4+δ−4σ26−5σΓ4+δτ3+2δΓ4+2δ=σ42−στ3+δ−24σ2+20σ3Γ4+δτ3+2δΓ4+2δ,ϑ0σ,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ.With the help of initial approximationϑn∗, the formula for iterations is
(31)ϑn+1σ,τ=R−1∂2ϑn∗∂σ2.Use the IBCs in Equation (26), for n=0:
(32)ϑ0∗σ,τ=ϑ0σ,τ+1−σ0−ϑ00,τ+σ0−ϑ02,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ=σ42−στ3+δ−24σ2+20σ3Γ4+δτ3+2δΓ4+2δ+1−σ0−0+σ0−0−−16Γ4+δτ3+2δΓ4+2δ,ϑ0∗σ,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ−16σΓ4+δτ3+2δΓ4+2δ.From Equation (31), we have
(33)ϑ1σ,τ=R−1∂ϑ20∗∂σ2=R−124σ2−20σ3τ3+δ−48−120σΓ4+δτ3+2δΓ4+2δ,ϑ1σ,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ.Forn=1, Equation (26) becomes
(34)ϑ1∗σ,τ=ϑ1σ,τ+1−σ0−ϑ10,τ+σ0−ϑ12,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ1−σ0+48Γ4+δτ3+3δΓ4+3δ+σ0+64Γ4+δτ3+2δΓ4+3δ−192Γ4+δτ3+3δΓ4+3δ,ϑ1∗σ,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−120σΓ4+δτ3+3δΓ4+3δ+64σΓ4+δτ3+2δΓ4+2δ.From Equation (31), we have
(35)ϑ2σ,τ=L−1∂2ϑ1∗∂σ2=L−148Γ4+δtau3+2δΓ4+2δ−120σΓ4+δτ3+2δΓ4+2δ,ϑ2σ,τ=48Γ4+δτ3+3δΓ4+3δ−120σΓ4+δτ3+3δΓ4+3δ.Forn=2, Equation (26) becomes
(36)ϑ2∗σ,τ=ϑ2σ,τ+1−σ0−ϑ20,τ+σ0−ϑ22,τ=48−120σΓ4+δτ3+3δΓ4+3δ+1−σ0−ϑ20,τ+σ0−ϑ22,τ,=48−120σΓ4+δτ3+3δΓ4+3δ+1−σ0−48Γ4+δτ3+3δΓ4+3δ+σ0+192Γ4+δτ3+3δΓ4+3δ=−120σΓ4+δτ3+3δΓ4+3δ+48σΓ4+δτ3+3δΓ4+3δ+192σΓ4+δτ3+3δΓ4+3δ,ϑ2∗σ,τ=120σΓ4+δτ3+3δΓ4+3δ.From Equation (31), we have
(37)ϑ3σ,τ=L−1∂2ϑ2∗∂σ2,=R−10=0.⋮Thus, the series form of ADM solution is(38)ϑσ,τ=ϑ0σ,τ+ϑ1σ,τ+ϑ2σ,τ+ϑ3σ,τ+⋯=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ+24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ+48Γ4+δτ3+3δΓ4+3δ−120σΓ4+δτ3+3δΓ4+3δ+0+⋯,ϑσ,τ=σ42−στ3+δ.


### 5.2. Example

Consider the TFDE of the following form [59]:
(39)∂δϑσ,τ∂τδ=∂2ϑσ,τ∂σ2+3Γ1/24τσ4σ−1−4σ25σ−3τ3/2,0≤σ≤1,0<δ≤1,having the IBCs as follows:
(40)ϑσ,0=0,ϑ0,τ=ϑ1,τ=0.With analytical solution atδ=1/2 as follows:
(41)ϑσ,τ=σ4σ−1τ3/2.Apply the suggested method of ADM to Equation (39), we have
(42)ϑn∗σ,τ=ϑnσ,τ+1−σ0−ϑn0,τ+σ0−ϑn1,τ,wheren=0,1,⋯.ApplyingR to Equation (39), we have
(43)Rϑ=∂2ϑσ,τ∂σ2+Γ124τσ4σ−1−4σ25σ−3τ3/2,where R=∂δ/∂τδ and R−1 is defined as
(44)R−1.=Iδ.dτ.Operating Equation (39) by R−1, we have
(45)ϑσ,τ=ϑσ,0+R−13Γ1/24τσ4σ−1−4σ25σ−3τ3/2.Using ADM solution, the initial approximation becomes
(46)ϑ0σ,τ=ϑσ,0+R−13Γ1/24τσ4σ−1−4σ25σ−3τ3/2=0+3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−4σ25σ−3Γ5/2τ3/2+δΓ5/2+δ,ϑ0σ,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ.With the help of initial approximationun∗, the formula for iterations is
(47)ϑn+1σ,τ=R−1∂2ϑn∗∂σ2.Forn=0, put the IBCs into Equation (42).
(48)ϑ0∗σ,τ=ϑ0σ,τ+1−σ0−ϑ00,τ+σ0−ϑ01,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ+1−σ0−0+σ0+8Γ5/2τ3/2+δΓ5/2+δ,ϑ0∗σ,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ+8σΓ5/2τ3/2+δΓ5/2+δ.From Equation (47), we have
(49)ϑ1σ,τ=R−1∂2ϑ0∗∂σ2=R−13Γ1/2Γ2τδ+120σ3−12σ24Γδ+2−430σ−6Γ5/2τ3/2+δΓ5/2+δ,ϑ1σ,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−120σΓ5/2τ3/2+2δΓ5/2+2δ+24Γ5/2τ3/2+2δΓ5/2+2δ.Forn=1 Equation (42), we get
(50)ϑ1∗σ,τ=ϑ1σ,τ+1−σ0−ϑ10,τ+σ0−ϑ11,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−120σΓ5/2τ3/2+2δΓ5/2+2δ+24Γ5/2τ3/2+2δΓ5/2+2δ+1−σ0−24Γ5/2τ3/2+2δΓ5/2+2δ+σ0−24Γ1/2Γ2τ2δ+14Γ2δ+2+96Γ5/2τ3/2+2δΓ5/2+2δ,ϑ1∗σ,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−24Γ1/2Γ2τ2δ+14Γ2δ+2.From Equation (47), we have
(51)ϑ2σ,τ=R−1∂2ϑ1∗∂σ2=R−13120σ−24Γ1/2τ2δ+14Γ2δ+2,ϑ2σ,τ=3120σ−24Γ1/2τ3δ+14Γ3δ+2.Equation (42), for n=2, is
(52)ϑ2∗σ,τ=ϑ2σ,τ+1−σ0−ϑ20,τ+σ0−ϑ21,τ=3120σ−24Γ1/2τ3δ+14Γ3δ+2+1−σ0−72Γ1/2τ3δ+1Γ3δ+2+σ0−288Γ1/2τ3δ+1Γ3δ+2,ϑ2∗σ,τ=144σΓ1/2Γ2τ3δ+14Γ3δ+2−144Γ1/2Γ2τ3δ+14Γ3δ+2.From Equation (47), we get
(53)ϑ3σ,τ=R−1∂2ϑ2∗∂σ2,=R−10=0.⋮The series form of ADM solution is(54)ϑσ,τ=ϑ0σ,τ+ϑ1σ,τ+ϑ2σ,τ+ϑ3σ,τ+⋯3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−430σ−6Γ5/2τ3/2+2δΓ5/2+2δ3120σ−24Γ1/2τ3δ+14Γ3δ+2+0+⋯.


## 5.1. Example

Consider TFDE of the following form [59]:
(23)∂δϑσ,τ∂τδ=∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ,0≤σ≤2,0<δ≤1,having the IBCs as follows:
(24)ϑσ,0=0,ϑ0,τ=ϑ2,τ=0.The problem has the analytical solution atδ=1 as follows:
(25)ϑσ,τ=σ42−στ3+δ.Applying the suggested method of ADM to Equation (23), we have
(26)ϑn∗σ,τ=ϑnσ,τ+1−σ0−ϑn0,τ+σ0−ϑn2,τ,wheren=0,1,⋯.ApplyingR to Equation (23), we have
(27)Rϑ=∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ,where R=∂δ/∂τδ and R−1 is
(28)R−1.=Iδ.dτ.Operating Equation (23) by mathcalR−1, we have
(29)ϑσ,τ=uσ,0+R−1∂2ϑσ,τ∂σ2+Γ4+δ6σ42−στ3−4σ26−5στ3+σ.Using ADM solution, the initial approximation becomes
(30)ϑ0σ,τ=ϑσ,0+R−1Γ4+δ6σ42−στ3−4σ26−5στ3+σ=0+Γ4+δΓ4σ42−στ3+δ6Γ4+δ−4σ26−5σΓ4+δτ3+2δΓ4+2δ=σ42−στ3+δ−24σ2+20σ3Γ4+δτ3+2δΓ4+2δ,ϑ0σ,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ.With the help of initial approximationϑn∗, the formula for iterations is
(31)ϑn+1σ,τ=R−1∂2ϑn∗∂σ2.Use the IBCs in Equation (26), for n=0:
(32)ϑ0∗σ,τ=ϑ0σ,τ+1−σ0−ϑ00,τ+σ0−ϑ02,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ=σ42−στ3+δ−24σ2+20σ3Γ4+δτ3+2δΓ4+2δ+1−σ0−0+σ0−0−−16Γ4+δτ3+2δΓ4+2δ,ϑ0∗σ,τ=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ−16σΓ4+δτ3+2δΓ4+2δ.From Equation (31), we have
(33)ϑ1σ,τ=R−1∂ϑ20∗∂σ2=R−124σ2−20σ3τ3+δ−48−120σΓ4+δτ3+2δΓ4+2δ,ϑ1σ,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ.Forn=1, Equation (26) becomes
(34)ϑ1∗σ,τ=ϑ1σ,τ+1−σ0−ϑ10,τ+σ0−ϑ12,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ1−σ0+48Γ4+δτ3+3δΓ4+3δ+σ0+64Γ4+δτ3+2δΓ4+3δ−192Γ4+δτ3+3δΓ4+3δ,ϑ1∗σ,τ=24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−120σΓ4+δτ3+3δΓ4+3δ+64σΓ4+δτ3+2δΓ4+2δ.From Equation (31), we have
(35)ϑ2σ,τ=L−1∂2ϑ1∗∂σ2=L−148Γ4+δtau3+2δΓ4+2δ−120σΓ4+δτ3+2δΓ4+2δ,ϑ2σ,τ=48Γ4+δτ3+3δΓ4+3δ−120σΓ4+δτ3+3δΓ4+3δ.Forn=2, Equation (26) becomes
(36)ϑ2∗σ,τ=ϑ2σ,τ+1−σ0−ϑ20,τ+σ0−ϑ22,τ=48−120σΓ4+δτ3+3δΓ4+3δ+1−σ0−ϑ20,τ+σ0−ϑ22,τ,=48−120σΓ4+δτ3+3δΓ4+3δ+1−σ0−48Γ4+δτ3+3δΓ4+3δ+σ0+192Γ4+δτ3+3δΓ4+3δ=−120σΓ4+δτ3+3δΓ4+3δ+48σΓ4+δτ3+3δΓ4+3δ+192σΓ4+δτ3+3δΓ4+3δ,ϑ2∗σ,τ=120σΓ4+δτ3+3δΓ4+3δ.From Equation (31), we have
(37)ϑ3σ,τ=L−1∂2ϑ2∗∂σ2,=R−10=0.⋮Thus, the series form of ADM solution is(38)ϑσ,τ=ϑ0σ,τ+ϑ1σ,τ+ϑ2σ,τ+ϑ3σ,τ+⋯=σ42−στ3+δ−24σ2Γ4+δτ3+2δΓ4+2δ+20σ3Γ4+δτ3+2δΓ4+2δ+24σ2Γ4+δτ3+2δΓ4+2δ−20σ3Γ4+δτ3+2δΓ4+2δ−48Γ4+δτ3+3δΓ4+3δ+120σΓ4+δτ3+3δΓ4+3δ+48Γ4+δτ3+3δΓ4+3δ−120σΓ4+δτ3+3δΓ4+3δ+0+⋯,ϑσ,τ=σ42−στ3+δ.


## 5.2. Example

Consider the TFDE of the following form [59]:
(39)∂δϑσ,τ∂τδ=∂2ϑσ,τ∂σ2+3Γ1/24τσ4σ−1−4σ25σ−3τ3/2,0≤σ≤1,0<δ≤1,having the IBCs as follows:
(40)ϑσ,0=0,ϑ0,τ=ϑ1,τ=0.With analytical solution atδ=1/2 as follows:
(41)ϑσ,τ=σ4σ−1τ3/2.Apply the suggested method of ADM to Equation (39), we have
(42)ϑn∗σ,τ=ϑnσ,τ+1−σ0−ϑn0,τ+σ0−ϑn1,τ,wheren=0,1,⋯.ApplyingR to Equation (39), we have
(43)Rϑ=∂2ϑσ,τ∂σ2+Γ124τσ4σ−1−4σ25σ−3τ3/2,where R=∂δ/∂τδ and R−1 is defined as
(44)R−1.=Iδ.dτ.Operating Equation (39) by R−1, we have
(45)ϑσ,τ=ϑσ,0+R−13Γ1/24τσ4σ−1−4σ25σ−3τ3/2.Using ADM solution, the initial approximation becomes
(46)ϑ0σ,τ=ϑσ,0+R−13Γ1/24τσ4σ−1−4σ25σ−3τ3/2=0+3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−4σ25σ−3Γ5/2τ3/2+δΓ5/2+δ,ϑ0σ,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ.With the help of initial approximationun∗, the formula for iterations is
(47)ϑn+1σ,τ=R−1∂2ϑn∗∂σ2.Forn=0, put the IBCs into Equation (42).
(48)ϑ0∗σ,τ=ϑ0σ,τ+1−σ0−ϑ00,τ+σ0−ϑ01,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ+1−σ0−0+σ0+8Γ5/2τ3/2+δΓ5/2+δ,ϑ0∗σ,τ=3Γ1/2Γ2τδ+1σ4σ−14Γδ+2−20σ3Γ5/2τ3/2+δΓ5/2+δ+12σ2Γ5/2τ3/2+δΓ5/2+δ+8σΓ5/2τ3/2+δΓ5/2+δ.From Equation (47), we have
(49)ϑ1σ,τ=R−1∂2ϑ0∗∂σ2=R−13Γ1/2Γ2τδ+120σ3−12σ24Γδ+2−430σ−6Γ5/2τ3/2+δΓ5/2+δ,ϑ1σ,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−120σΓ5/2τ3/2+2δΓ5/2+2δ+24Γ5/2τ3/2+2δΓ5/2+2δ.Forn=1 Equation (42), we get
(50)ϑ1∗σ,τ=ϑ1σ,τ+1−σ0−ϑ10,τ+σ0−ϑ11,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−120σΓ5/2τ3/2+2δΓ5/2+2δ+24Γ5/2τ3/2+2δΓ5/2+2δ+1−σ0−24Γ5/2τ3/2+2δΓ5/2+2δ+σ0−24Γ1/2Γ2τ2δ+14Γ2δ+2+96Γ5/2τ3/2+2δΓ5/2+2δ,ϑ1∗σ,τ=3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−24Γ1/2Γ2τ2δ+14Γ2δ+2.From Equation (47), we have
(51)ϑ2σ,τ=R−1∂2ϑ1∗∂σ2=R−13120σ−24Γ1/2τ2δ+14Γ2δ+2,ϑ2σ,τ=3120σ−24Γ1/2τ3δ+14Γ3δ+2.Equation (42), for n=2, is
(52)ϑ2∗σ,τ=ϑ2σ,τ+1−σ0−ϑ20,τ+σ0−ϑ21,τ=3120σ−24Γ1/2τ3δ+14Γ3δ+2+1−σ0−72Γ1/2τ3δ+1Γ3δ+2+σ0−288Γ1/2τ3δ+1Γ3δ+2,ϑ2∗σ,τ=144σΓ1/2Γ2τ3δ+14Γ3δ+2−144Γ1/2Γ2τ3δ+14Γ3δ+2.From Equation (47), we get
(53)ϑ3σ,τ=R−1∂2ϑ2∗∂σ2,=R−10=0.⋮The series form of ADM solution is(54)ϑσ,τ=ϑ0σ,τ+ϑ1σ,τ+ϑ2σ,τ+ϑ3σ,τ+⋯3Γ1/2Γ2τ2δ+120σ3−12σ24Γ2δ+2−430σ−6Γ5/2τ3/2+2δΓ5/2+2δ3120σ−24Γ1/2τ3δ+14Γ3δ+2+0+⋯.


## 6. Results and Discussion

In Figure1, the 3D graph of exact and approximate solutions to Example 5.1 is presented. The comparison showed that the graphs of exact and obtained solutions are in good agreement and confirms the validity of the proposed method. In Figure 2, the 2D plot of the exact and approximate solution is constructed and again confirms the validity of the suggested technique. Figure 3 represents the fractional order solutions of Example 5.1 at δ=0.3,0.6,0.8,1. The solutions at different fractional orders of the derivative provide the useful information about the dynamics of Example5.1. In Figures 4 and 5, 3D and 2D graphs of exact and obtained solutions are highlighted. From both the presentations, greater accuracy has been observed and the graphs of the derived results are found to be identical to the exact solution of Example 5.2. In Figure 6, the solution of Example 5.2 at different time levels is calculated and obtained useful dynamics for Example 5.2. Table 1 shows the solutions at fractional orders δ=0.3, 0.5, and 0.7 of Example 5.1. For this purpose, the modified approach of ADM is applied to obtain the solutions. The results are listed in the table, which has confirmed that the suggested method gives the solutions that are in close contact with the analytical solution of the problem. The absolute errors are given for the given analytical and ADM solutions in the table. According to the table, the proposed techniques have the desired degree of accuracy in terms of exact problem solution. In Table 2, the solutions of Example 5.2 are given at different time levels, that is, τ=0.3,0.5,0.7. It is verified from Table 2 that the method provides excellent results at different time levels.Figure 1
3D plots of the exact and approximate solution forδ=1 of Example 5.1.Figure 2
2D plots of the exact and approximate solution forδ=1 of Example 5.1.Figure 3
(a) 3D and (b) 2D plots for different fractional value ofδ for Example 5.1.
(a)(b)Figure 4
3D plots of the exact and approximate solution forδ=1 of Example 5.2.Figure 5
2D plots of the exact and approximate solution forδ=1 of Example 5.2.Figure 6
2D plots for approximate solution of Example5.2 at τ=0.25,0.5,0.75,1.Table 1
Absolute error of Example 5.1 atτ=0.8.
σExact−ADMExact−ADMExact−ADMδ=0.3δ=0.5δ=0.70.26.26×10−114.41×10−101.58×10−100.45.3×10−101.2×10−106.0×10−110.67.0×10−118.0×10−102.8×10−100.84.3×10−93.3×10−91.0×10−1014.7×10−92.0×10−102.0×10−101.21.9×10−95.1×10−92.5×10−91.41.6×10−85.0×10−92.0×10−91.62.0×10−96.0×10−91.0×10−91.82.6×10−88.6×10−95.0×10−1022.0×10−81.0×10−82.0×10−9Table 2
Absolute error of Example5.2 at δ=1/2.
σExact−ADMExact−ADMExact−ADMτ=0.3τ=0.5τ=0.70.18.748×10−213.6348×10−205.8352×10−200.21.5×10−222.1×10−223.9×10−220.31.67×10−223.20×10−204.59×10−200.41.60×10−206.1×10−211.443×10−190.51.30×10−221.3×10−203.00×10−190.61.27×10−202.1×10−201.11×10−190.74.2×10−207.0×10−214.40×10−190.81.5×10−201.98×10−191.17×10−190.93.9×10−200.0003.55×10−1910.0001.722×10−190.000


## 7. Conclusion

In this article, the Adomian decomposition method is implemented along with some new modifications to solve fractional partial differential equation boundary value problems. The proposed technique was found to be very efficient in handling the solution of fractional-order boundary value problems. In particular, the suggested procedure is used to solve some illustrative examples of time-fractional diffusion equations. The solutions are calculated for both fractional and integer order problems, and the present method is observed to be very simple and useful for the solutions to such problems. A comparison between exact and analytical solutions is made with the help of plots and tables. The graphical representation is presented to confirm the validity of the present technique. The solution graphs have confirmed that the derived results are in close contact with the problem’s actual solution. Figures1 and 4 represent 3D solution plots of Example 5.1 and 5.2, respectively, at δ=1. Both the graphs displayed a very convincing contact between the exact and approximate solutions. In Figures 2 and 5, 2D solution plots are also constructed to confirm the validity of the proposed method. The fractional-order solutions of Example 5.1 and 5.2 are represented in Figures 3 and 6. From the graphical representation of fractional-order problems, it is confirmed that very accurate and useful information is obtained as compared to the integer order of the problems. It is concluded that the solutions at fractional-order derivatives are very useful to analyze the dynamics of the targeted problems of IBVPs. The dual use of initial conditions has made the procedure suitable for using both IBCs simultaneously, which was not the case in the earlier related literature. The solution obtained at each fractional order is found to be converging to the integer order of the targeted problems. Moreover, the proposed method is more accurate and competent to find the solution of nonlinear fractional partial differential equations and, in the future, can be modified for other important fractional nonlinear partial differential equations with higher dimensions.



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