# Solving Fractional-Order Diffusion Equations in a Plasma and Fluids via a Novel Transform

Authors: Pongsakorn Sunthrayuth; Haifa A. Alyousef; S. A. El-Tantawy; Adnan Khan; Noorolhuda Wyal
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Motivated by the importance of diffusion equations in many physical situations in general and in plasma physics in particular, therefore, in this study, we try to find some novel solutions to fractional-order diffusion equations to explain many of the ambiguities about the phenomena in plasma physics and many other fields. In this article, we implement two well-known analytical methods for the solution of diffusion equations. We suggest the modified form of homotopy perturbation method and Adomian decomposition methods using Jafari-Yang transform. Furthermore, illustrative examples are introduced to show the accuracy of the proposed methods. It is observed that the proposed method solution has the desire rate of convergence toward the exact solution. The suggested method’s main advantage is less number of calculations. The proposed methods give series form solution which converges quickly towards the exact solution. To show the reliability of the proposed method, we present some graphical representations of the exact and analytical results, which are in strong agreement with each other. The results we showed through graphs and tables for different fractional-order confirm that the results converge towards exact solution as the fractional-order tends towards integer-order. Moreover, it can solve physical problems having fractional order in different areas of applied sciences. Also, the proposed method helps many plasma physicists in modeling several nonlinear structures such as solitons, shocks, and rogue waves in different plasma systems.


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## Body

## 1. Introduction

The integer-order differentiation operators are used to study local phenomena, whereas fractional-order operators are used to studying nonlocal phenomena [1]. The mathematical groundwork for fractional-order derivatives was laid by the collective struggles of various mathematicians, such as Riemann, Liouville, Caputo, Podlubny, Miller, and Ross. Afterward, numerous mathematicians dedicated their efforts to this area. Fractional calculus (FC) can be described as very successful in many phenomena in applied sciences, fluid mechanics, physics of plasmas [2, 3], and other biology utilising mathematical tools of FC. [4, 5]. Other numerous applications of FC in the field of science and technology are related to solid mechanics [6, 7], anomalous transport [8], continuum and statistical mechanics [9], economics [10], relaxation electrochemistry [11], diffusion procedures [12], complex networks [13, 14], and optimal control problems [15, 16].Fractional differential equations (FDEs) have gotten a lot of attention from researchers in the last decade due to their ability to improve real-world challenges in different areas of engineering and physics. Mathematicians used various methods described based on the above applications to solve different important fractional-order differential equations (FDEs), mainly partial differential equations of fractional-orders (FPDEs). FPDEs are fundamental mathematical approaches that can be utilized to model different physical models more accurately than integer-order models. Nonlinear FPDEs define various phenomena in engineering, plasma physics, and applied sciences. Especially, nonlinear FPDEs are the preeminent tools to be used in various areas, for example, electrochemistry [17], mathematical social dynamics [18, 19], signal processing [20], informatics [21], traffic model [22], theory of solitons in plasma physics [23], biology [24], and much more [25–28]. Moreover, many authors reduced the fluid plasma equations to FDEs for studying the impact of derivative time-frational on the profile of nonlinear structures in a plasma [2, 3]. For instance, El-Wakil et al. [2] reduced the basic equations of a collisionaless unmagnetized nonthermal plasma having cold inertial electrons and inertialess nonthermal electrons as well as stationary positive ions to a normal KdV equation. After that, the authors use a suitable transformation to convert the normal KdV equation to a time-fractional KdV equation in order to investigate the time-fractional on the profile of electron-acoustic (EA) solitons. Furthermore, the basic equations of an ultrarelativistic plasma were reduced to the normal cylindrical Kadomtsev–Petviashvili (CKP) and cylindrical modified KP (CmKP) equations using a reductive perturbation techniques [3]. Posteriorly, the mentioned equations were converted into space-time fractional CKP and CmKP equations using one of the proper transformations in order to study the influence of space-time fractional domain of the characteristics of the ion-acoustic waves (IAWs) in the ultrarelativistic plasma. The authors made a comparison between the integer- and fractional-order models and found that the fractional-order model gives description to the IAWs in the ultrarelativistic plasmas better than the integer-order model [3]. In literature, different methods are implemented for solving FDEs, such as Iterative Laplace Transform method (ILTM) [29, 30], Approximate-analytical method (AAM) [31], Homotopy Analysis method (HAM) [32], Variational Iteration method (VIM) [33], Elzaki Transform Decomposition method (ETDM) [34], the Differential Transformation method (DTM) [35], and the homotopy perturbation method (HPM) [36, 37].In literature, there is lot of transformations [38–40], but in this article, we implement the Homotopy Perturbation Jafari-Yang Transform Method (HPYTM) and Jafari-Yang transform decomposition method (YTDM) for the analysis of fractional-order diffusion equations. Xiao-Jun Jafari-Yang introduce the Jafari-Yang transformation and applied for the analysis of different differential problems with constant coefficients, while the Adomian decomposition method [41, 42], on the other hand, is a renowned method to solve linear and nonlinear and nonhomogeneous and homogeneous differential, integro, ordinary, and partial differential equations. It gives analysis in the form of series that converges towards the exact solutions quickly. In 1998, He introduced the homotopy perturbation technique [43, 44]. Later, the nonlinear nonhomogeneous partial differential equations are solved using the HPM (homotopy perturbation method), a semianalytical technique [45–48]. The solution is assumed to be the sum of an infinite sequence that converges rapidly to the exact results. This approach was investigated to analyze both linear and nonlinear problems. In the current work, we proposed a novel approximate analytical method known as (HPYTM). The newly developed technique is the combination of Jafari-Yang transform and HPM. It is investigated that the present methods are very effective in finding fractional diffusion equations analytical solution. The fractional problem results using the proposed methods are also devoted to the fractional view analysis of the problems. It is confirmed that the current techniques can be modified to solve other fractional partial differential equations.A type of PDE that expresses the phenomenon of atoms or molecule’s movement from a region of higher concentration to an area of lower concentration is known as diffusion equations. Adolf Fick, a physiologist, was the first to present Fick’s law of diffusion. Fick’s law was then transformed into the diffusion equation. Scholar modified diffusion equations such as slow diffusion and the hybrid classical wave equation by generalizing the classical diffusion law [49]. Several implementations of diffusion equation are phase transition, electromagnetism, filtration, biochemistry, geochemistry, dynamics of biological groups, cosmology, plasma physics, and acoustics [50]. There are many investigations related to the Diffusion-type equation and its applications in plasma physics and fluids [51–53]. Motivated by these investigation, in this article, we implement YTDM and HPYTM for solving diffusion equations of the form.
(1)
Fractional-order diffusion equation in one dimension as(1)∂δν∂τδ=∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν0<δ≤1,τ≥0having initial values
(2)νψ,0=gψ(2)
Fractional-order diffusion equation in two dimension as(3)∂δν∂τδ=∂2ν∂ψ2+∂2ν∂ϕ20<δ≤1,τ≥0having initial values
(4)νψ,ϕ,0=gψ,ϕ(3)
Fractional-order diffusion equation in three dimension as(5)∂δν∂τδ=∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ20<δ≤1,τ≥0having initial values
(6)νψ,ϕ,φ,0=gψ,ϕ,φ


## 2. Preliminaries

We covered several fundamental fractional calculus definitions as well as Laplace transform theory properties in this section.Definition 1.
In Caputo manner, the fractional derivative is given as [54].
(7)Dφσνψ,φ=1Γk−σ∫0φφ−ρk−σ−1νkψ,ρdρ,k−1<σ≤k,k∈N.Definition 2.
Yang and Jafari introduced the Jafari-Yang Laplace transform in 2018.Y. determines the Jafari-Yang transform for a function ξτ and is given as [55].
(8)Yντ=Tu=∫0∞e−τ/uντdτ,τ>0,u∈−τ1,τ2.
The inverse transform is given as(9)Y−1Tu=ντ.Definition 3.
For derivative of ordern, the Jafari-Yang transform is determined as [55].
(10)Yνnτ=Tuun−∑k=0n−1uk0νk−n−1,∀n=1,2,3,⋯.Definition 4.
For fractional-order derivatives, the Jafari-Yang transform is given as [55].
(11)Tuστ=Tνuσ−∑k=0n−1νk0uk−σ+1,0<σ≤n.Definition 5.
The Mittag-Leffler function, a generalisation of the exponential function, is as [54].
(12)Eσφ=∑q=0∞φqΓσq+1σ∈C,Reσ>0.
Equation (12) further generalization is of the form
(13)Eσ,βφ=∑q=0∞φqΓσq+β;σ,β∈C,Reσ>0,Reβ>0.


## 3. Homotopy Perturbation Jafari-Yang Transform Method

To explain the basic ideas of this approach the following equation is considered:(14)τσνψ,τ=Lψνψ,τ+Nψνψ,τ,0<σ≤2,with some initial sources
(15)νψ,0=ξψ,∂∂τνψ,0=ζψ,whereDτσ=∂σ/∂τσ Caputo‘s derivative, and Lψ and Nψ are the linear and nonlinear operators respectively.Implement Jafari-Yang transform to (14), we have
(16)Dτσνψ,τ=YLψνψ,τ+Nψνψ,τ,(17)1uσTu−uν0−u2ν′0=YLψνψ,τ+Nψνψ,τ.Equation (17) implies that
(18)Tν=uν0+u2ν′0+uσYLψνψ,τ+Nψνψ,τ.We now have by using the inverse Jafari-Yang transform(19)νψ,τ=ν0+ν′0+Y−1uσYLψνψ,τ+Nψνψ,τ.Now, perturbation technique having parameterε is given as
(20)νψ,τ=∑k=0∞εkνkψ,τ,where perturbation parameter is ε and ε∈0,1.The decomposition of nonlinear terms is defined as(21)Nψνψ,τ=∑k=0∞εkHnν,where Hn are of the form ν0,ν1,ν2,⋯,νn, and can be determined as
(22)Hnν0,ν1,⋯,νn=1γn+1DεkN∑k=0∞εiνiε=0,where Dεk=∂k/∂εk.Using (21) and (22) in (19) and making the homotopy, we get
(23)∑k=0∞εkνkψ,τ=ν0+ν′0+ε×Y−1uσYL∑k=0∞εkνkψ,τ+∑k=0∞εkHkν.When the coefficient ofε on both sides is compared, we get
(24)ε0:ν0ψ,τ=ν0+ν′0,ε1:ν1ψ,τ=Y−1uσYLψν0ψ,τ+H0ν,ε2:ν2ψ,τ=Y−1uσYLψν1ψ,τ+H1ν,⋮εk:νnψ,τ=Y−1uσYLψνk−1ψ,τ+Hk−1ν.As a result, we can quickly determine componentνkψ,τ, which leads us to the convergent series. We get by taking ε⟶1,(25)νψ,τ=limM⟶∞∑k=1Mνkψ,τ.The result is in the form of a series that rapidly converges to the exact solution of the problem.


## 4. Idea of YTDM

Consider the fractional order partial differential equation(26)Dτσνψ,τ=G¯1ν,φ+N1ν,φ,0<σ≤2,with some initial sources
(27)νψ,0=ξψ,∂∂τνψ,0=ζψ,where Dτσ=∂σ/∂τσ denotes the derivative having fractional-order σ in Caputo manner, and G¯1 and N1 denote linear and nonlinear functions.On taking Jafari-Yang transformation of (26), we get
(28)YDτσνψ,τ=YG¯1ν,φ+N1ν,φ.By using Jafari-Yang transform property of differentiation, we have(29)1uσTu−uν0−u2ν′0=YG¯1ν,φ+N1ν,φ.Equation (29) implies that
(30)Tν=uν0+u2ν′0+uσYG¯1ν,φ+N1ν,φ.We now have by using the inverse Jafari-Yang transform(31)νψ,τ=ν0+ν′0+Y−1uσYG¯1ν,φ+N1ν,φ.The infinite series ofνψ,τ(32)νψ,τ=∑k=0∞νkψ,τ.The nonlinear term decomposition ofN1 by Adomian polynomials is expressed as
(33)N1ν,φ=∑k=0∞Ak.All nonlinear terms can be denoted by means of Adomian polynomials as(34)Ak=1k!∂k∂ℓkN1∑j=0∞ℓjνj,∑j=0∞ℓjφjℓ=0,putting Equations (32) and (34) into (31), gives
(35)∑k=0∞νkψ,τ=ν0+ν′0+Y−1uσYG¯1∑k=0∞νk,∑k=0∞φk+∑k=0∞Ak.The following terms are described as(36)ν0ψ,τ=ν0+τν′0,ν1ψ,τ=Y−uσY+G¯1ν0,φ0+A0.The general for k≥1 is determined as
(37)νk+1ψ,τ=Y−uσY+G¯1νk,φk+Ak.


## 5. Applications

The solutions to various fractional order diffusion equations are obtained by implementing HPYTM and YTDM in this section.Example 1.
Consider one-dimension fractional-order diffusion equation [54]
(38)∂σν∂τσ=∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν0<σ≤1,τ>0,with initial source
(39)νψ,0=eψ.
On taking Jafari-Yang transformation of Equation (38), we get
(40)Y∂σν∂τσ=Y∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν.
We get by using the Jafari-Yang transform’s differential property(41)1uσTu−uν0=Y∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν,(42)Tu=uν0+uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν.
By applying inverse Jafari-Yang transform to Equation (42)
(43)νψ,τ=ν0+Y−1uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν,νψ,τ=eψ+Y−1uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν.
Now, applying the abovementioned homotopy perturbation technique as in (23), we obtain
(44)∑k=0∞εkνkψ,τ=eψ+εY−1uσY∑k=0∞εkνkψ,τψψ−∑k=0∞εkνkψ,τψ+∑k=0∞εkHkν+∑k=0∞εkνkψ,τ.where Hkν are He’s polynomial that represents the nonlinear terms. He’s polynomials first few components are given by
(45)H0ν=ν0ν0ψψ−ν0,2H1ν=ν1ν0ψψ+ν0ν1ψψ−2ν0ν1,H2ν=ν2ν0ψψ+ν1ν1ψψ+ν0ν2ψψ−2ν0ν2+ν22.⋮
Comparing the same power coefficient ofε, we obtain
(46)ε0:ν0ψ,τ=eψ,ε1:ν1ψ,τ=Y−1uσY∂2ν0∂ψ2−∂ν0∂ψ+ν0∂2ν0∂ψ2−ν02+ν0=eψτσΓσ+1,ε2:ν2ψ,τ=Y−1uσY∂2ν1∂ψ2−∂ν1∂ψ+ν1∂2ν0∂ψ2+ν0∂2ν1∂ψ2−2ν0ν1+ν1=eψτ2σΓ2σ+1,ε3:ν3ψ,τ=Y−1uσY∂2ν2∂ψ2−∂ν2∂ψ+ν2∂2ν0∂ψ2+ν1∂2ν1∂ψ2+ν0∂2ν2∂ψ2−2ν0ν2−ν22+ν2ε3:ν3ψ,τ=eψτ3σΓ3σ+1.⋮
By takingε⟶1, we obtain the convergence series type result is given as
(47)νψ,τ=ν0+ν1+ν2+ν3+ν4+⋯=eψ+eψτσΓσ+1+eψτ2σΓ2σ+1+eψτ3σΓ3σ+1+⋯,νψ,τ=eψ1+τσΓσ+1+τ2σΓ2σ+1+τ3σΓ3σ+1+⋯,νψ,τ=eψ∑k=0∞tσkΓkσ+1=eψEστσ.
Whenσ=1, the HPYTM solution is
(48)νψ,τ=eψ∑k=0∞tkk!,
The analytical results by YTDM.
On taking Jafari-Yang transformation of Equation (38), we obtain
(49)Y∂σν∂τσ=Y∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν.Using the Jafari-Yang transform the differential property, we get
(50)1uσTu−uν0=Y∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν,(51)Tu=uν0+uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν.
By inverse Jafari-Yang transform of Equation (51)
(52)νψ,τ=ν0+Y−1uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν,νψ,τ=eψ+Y−1uσY∂2ν∂ψ2−∂ν∂ψ+ν∂2ν∂ψ2−ν2+ν,
Assuming the unknownνψ,τ function has the following infinite series form solution:
(53)vψ,τ=∑k=0∞vkψ,τ.
The Adomian polynomialsννψψ=∑k=0∞Ak and ν2=∑k=0∞Bk, as well as the nonlinear term, have been described. Applying specific function, Equation (52) may be determined as
(54)∑k=0∞νkψ,τ=νψ,0+Y−uσY∂2ν∂ψ2−∂ν∂ψ+∑k=0∞Ak−∑k=0∞Bk+ν,∑k=0∞νkψ,τ=eψ+Y−uσY∂2ν∂ψ2−∂ν∂ψ+∑k=0∞Ak−∑k=0∞Bk+ν.
According to Equation (34), the nonlinear function can be find with the help of Adomian polynomials is given as
(55)A0=ν0ν0ψψ,A1=ν1ν0ψψ+ν0ν1ψψ,A2=ν2ν0ψψ+ν1ν1ψψ+ν0ν2ψψ,B0=ν02,B1=2ν0ν1,B2=2ν0ν2+ν22.
Thus, on comparing both sides of Equation (54)
(56)ν0ψ,τ=eψ,1
Fork=0(57)ν1ψ,ϕ,φ,τ=−3sinψsinϕsinφτσΓσ+1.
Fork=1(58)ν2ψ,τ=eψτ2σΓ2σ+1.
Fork=2(59)ν3ψ,τ=eψτ3σΓ3σ+1.
The remaining YTDM solution elementsρk for k≥2 are similarly simple to get. As a result, the solution in series form is as
(60)νψ,τ=∑k=0∞νkψ,τ=ν0ψ,τ+ν1ψ,τ+ν2ψ,τ+ν3ψ,τ+⋯νψ,τ=eψ+eψτσΓσ+1+eψτ2σΓ2σ+1+eψτ3σΓ3σ+1+⋯.
Whenσ=1, the solution by YTDM is
(61)νψ,τ=eψ∑k=0∞tkk!.
In closed form, the exact solution is(62)νψ,τ=eψ+τ.The exact solution shown in Figures1(a) and 1(b) shows HPYTM and YTDM solution at σ=1 and 0≤ψ≥1. Figure 2(a) shows the solution graph for different values of σ=1,0.9,0.8,0.7 and 0≤ψ≥1 of Example 1 and Figure 2(b), respectively, at τ∈0,1 and 0≤ψ≥1 while Figure 2(c) shows the error graph. Also, Table 1 shows the comparison of the exact solution and our methods solution with the aid of absolute error at various fractional order. From the figures and table, it is clear that HPYTM and YTDM solution shows strong contact with the exact solutions of the problem.Figure 1
Example1 solution graph. (a) Exact solution and (b) analytical solution at σ=1.
(a)(b)Figure 2
Example1. (a) Analytical solution at various fractional orders of σ (b) ϕ=0.5 and (c) error.
(a)(b)(c)Table 1
νψ,τ comparison of exact solution, our methods’ solution, and Absolute Error (AE) of Example 1.
τ=0.01Absolute errorAbsolute errorExact solutionOur methods’ solutionAbsolute errorψσ=0.8σ=0.9σ=1σ=1σ=105.2118400000E-045.0962000000E-051.01005016700001.01005016700000.00000000E+000.15.7599800000E-045.6322000000E-051.11627807000001.11627807000000.00000000E+000.26.3657500000E-046.2245000000E-051.23367806000001.23367806000000.00000000E+000.37.0352500000E-046.8792000000E-051.36342511400001.36342511400000.00000000E+000.47.7751600000E-047.6027000000E-051.50681778500001.50681778500000.00000000E+000.58.5928700000E-048.4022000000E-051.66529119500001.66529119500000.00000000E+000.69.4965800000E-049.2858000000E-051.84043139900001.84043139800001.00000000E-090.71.0495350000E-031.0262400000E-042.03399125900002.03399125800001.00000000E-090.81.1599150000E-031.1341700000E-042.24790798700002.24790798600001.00000000E-090.91.2819060000E-031.2534600000E-042.48432253300002.48432253300000.00000000E+001.01.4167240000E-031.3852800000E-042.74560101500002.74560101400001.00000000E-09+Example 2.
Consider one-dimension fractional-order gas dynamic equation [56]
(63)∂σν∂τσ+ν∂ν∂ψ+ν2−ν=00<σ≤1,τ>0,with initial source
(64)νψ,0=e−ψ.
On taking Jafari-Yang transformation of Equation (63), we get
(65)Y∂σν∂τσ=Y−ν∂ν∂ψ+ν2−ν.
We get by using the Jafari-Yang transform’s differential property(66)1uσTu−uν0=Y−ν∂ν∂ψ+ν2−ν,(67)Tu=uν0+uσY−ν∂ν∂ψ+ν2−ν.
By applying inverse Jafari-Yang transform to Equation (67)
(68)νψ,τ=ν0−Y−1uσYν∂ν∂ψ−ν2+ν,νψ,τ=e−ψ−Y−1uσYν∂ν∂ψ−ν2+ν.
Now, applying the abovementioned homotopy perturbation technique as in (23), we obtain
(69)∑k=0∞εkνkψ,τ=e−ψ−εY−1uσY∑k=0∞εkHkν+∑k=0∞εkνkψ,τ.where Hkν are He’s polynomial that represents the nonlinear terms. He’s polynomials first few components are given by
(70)H0ν=ν0ν0ψ−ν02,H1ν=ν1ν0ψ+ν0ν1ψ−2ν0ν1,H2ν=ν2ν0ψ+ν1ν1ψ+ν0ν2ψ−2ν0ν2+ν22.⋮
Comparing the same power coefficient ofε, we obtain
(71)ε0:ν0ψ,τ=e−ψ,ε1:ν1ψ,τ=Y−1uσYν0∂ν0∂ψ−ν02+ν0=e−ψτσΓσ+1,ε2:ν2ψ,τ=Y−1uσYν1∂ν0∂ψ+ν0∂ν1∂ψ−2ν0ν1+ν1=e−ψτ2σΓ2σ+1,ε3:ν3ψ,τ=Y−1uσYν2∂ν0∂ψ+ν1∂ν1∂ψ+ν0∂ν2∂ψ−2ν0ν2−ν22ν2,ε3:ν3ψ,τ=e−ψτ3σΓ3σ+1⋮
By takingε⟶1, we obtain the convergence series type result is given as
(72)νψ,τ=ν0+ν1+ν2+ν3+ν4+⋯=e−ψ+e−ψτσΓσ+1+e−ψτ2σΓ2σ+1+e−ψτ3σΓ3σ+1+⋯νψ,τ=e−ψ1+τσΓσ+1+τ2σΓ2σ+1+τ3σΓ3σ+1+⋯,νψ,τ=e−ψ∑k=0∞tσkΓkσ+1=eψEστσ.
Whenσ=1, the HPYTM solution is
(73)νψ,τ=exp−ψ∑k=0∞tkk!.
The analytical results by YTDM
On taking Jafari-Yang transformation of Equation (63), we obtain
(74)Y∂σν∂τσ=Y−ν∂ν∂ψ+ν2−ν,using the Jafari-Yang transform the differential property, we get
(75)1uσTu−uν0=Y−ν∂ν∂ψ+ν2−ν,(76)Tu=uν0+uσY−ν∂ν∂ψ+ν2−ν.
By inverse Jafari-Yang transform of Equation (76)
(77)νψ,τ=ν0−Y−1uσYν∂ν∂ψ−ν2+ν,νψ,τ=e−ψ−Y−1uσYν∂ν∂ψ−ν2+ν.
Assuming the unknownνψ,τ function has the following infinite series form solution:
(78)νψ,τ=∑k=0∞νkψ,τ.
The Adomian polynomialsννψψ=∑k=0∞Ak and ν2=∑k=0∞Bk, as well as the non-linear term, have been described. Applying specific function, Equation (77) may be determined as
(79)∑k=0∞νkψ,τ=νψ,0−Y−uσY∑k=0∞Ak−∑k=0∞Bk+ν,∑k=0∞νkψ,τ=e−ψ−Y−uσY∑k=0∞Ak−∑k=0∞Bk+ν.
According to Equation (34), the nonlinear function can be found with the help of Adomian polynomials is given as
(80)A0=ν0ν0ψ,A1=ν1ν0ψψ+ν0ν1ψ,A2=ν2ν0ψ+ν1ν1ψ+ν0ν2ψ,B0=ν02,B1=2ν0ν1,B2=2ν0ν2+ν22.
Thus, on comparing both sides of Equation (79)
(81)ν0ψ,τ=e−ψ.
Fork=0(82)ν1ψ,τ=e−ψτσΓσ+1
Fork=1(83)ν2ψ,τ=e−ψτ2σΓ2σ+1.
Fork=2(84)ν3ψ,τ=e−ψτ3σΓ3σ+1.
The remaining YTDM solution elementsρk for k≥3 are similarly simple to get. As a result, the solution in series form is as
(85)νψ,τ=∑k=0∞νkψ,τ=ν0ψ,τ+ν1ψ,τ+ν2ψ,τ+ν3ψ,τνψ,τ=e−ψ+e−ψτσΓσ+1+exp−ψτ2σΓ2σ+1+e−ψτ3σΓ3σ+1+⋯.
Whenσ=1, the solution by YTDM is
(86)νψ,τ=e−ψ∑k=0∞tkk!.
In closed form, the exact solution is(87)νψ,τ=e−ψ+τ.Figure3(a) shows the solution graph for different values of σ=1, 0.9, 0.8, 0.7 and 0≤ψ,ϕ≥1 of Example 2 and Figure 3(b), respectively, at ϕ=0.5,τ∈0,1 and 0≤ψ≥1 while Figure 3(c) shows the error graph. It is verified from the figures that HPYTM and YTDM solution is closely related with the exact solution.Example. 3.
Consider two-dimension fractional-order diffusion equation [54]
(88)∂σν∂τσ=∂2ν∂ψ2+∂2ν∂ϕ20<σ≤1,τ≥0,with initial source
(89)νψ,ϕ,0=1−ϕeψ.
Taking Jafari-Yang transformation to Equation (88), we get
(90)Y∂σν∂τσ=Y∂2ν∂ψ2+∂2ν∂ϕ2,
We get by using the Jafari-Yang transform’s differential property(91)1uσTu−uν0=Y∂2ν∂ψ2+∂2ν∂ϕ2,Tu=uν0+uσY∂2ν∂ψ2+∂2ν∂ϕ2.
By applying inverse Jafari-Yang transform to Equation (91)
(92)νψ,ϕ,τ=ν0+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2,νψ,ϕ,τ=1−ϕeψ+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2.
Now, using the abovementioned homotopy perturbation technique as in (23), we get
(93)∑k=0∞εkνkψ,ϕ,τ=1−ϕeψ+εY−1uσY∑k=0∞εkνkψ,ϕ,τψψ+∑k=0∞εkνkψ,ϕ,τϕϕ.
Comparing the same power coefficient ofε, we obtain
(94)ε0:ν0ψ,ϕ,τ=1−ϕeψ,ε1:ν1ψ,ϕ,τ=Y−1uσY∂2ν0∂ψ2+∂2ν0∂ϕ2=1−ϕeψτσΓσ+1,ε2:ν2ψ,ϕ,τ=Y−1uσY∂2ν1∂ψ2+∂2ν1∂ϕ2=1−ϕeψτ2σΓ2σ+1,ε3:ν3ψ,ϕ,τ=Y−1uσY∂2ν2∂ψ2+∂2ν2∂ϕ2=1−ϕeψτ3σΓ3σ+1,ε4:ν4ψ,ϕ,τ=Y−1uσY∂2ν3∂ψ2+∂2ν3∂ϕ2=1−ϕeψτ4σΓ4σ+1.⋮
By takingε⟶1, we obtain convergence series type result is
(95)νψ,ϕ,τ=ν0+ν1+ν2+ν3+ν4+⋯=1−ϕeψ+1−ϕeψτσΓσ+1+1−ϕeψτ2σΓ2σ+1+1−ϕeψτ3σΓ3σ+1+1−ϕeψτ4σΓ4σ+1+⋯νψ,ϕ,τ=1−ϕeψ1+τσΓσ+1+τ2σΓ2σ+1+τ3σΓ3σ+1+τ4σΓ4σ+1+⋯,νψ,ϕ,τ=1−ϕeψ∑k=0∞tσkΓkσ+1=1−ϕeψEσtσ.
Whenσ=1, the HPYTM solution is
(96)νψ,ϕ,τ=1−ϕeψ∑k=0∞tkk!.
The analytical results by YTDM
Taking Jafari-Yang transformation of Equation (88), we obtain
(97)Y∂σν∂τσ=Y∂2ν∂ψ2+∂2ν∂ϕ2.
We get by using the Jafari-Yang transform’s differential property(98)1uσTu−uν0=Y∂2ν∂ψ2+∂2ν∂ϕ2,(99)Tu=uν0+uσY∂2ν∂ψ2+∂2ν∂ϕ2.
By inverse Jafari-Yang transform of Equation (99)
(100)νψ,ϕ,τ=ν0+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2,νψ,ϕ,τ=1−ϕeψ+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2.
Assuming the unknownνψ,ϕ,τ function has the following infinite series form solution:
(101)νψ,ϕ,τ=∑k=0∞νkψ,ϕ,τ,∑k=0∞νkψ,ϕ,τ=eψ+Y−uσY∂2ν∂ψ2+∂2ν∂ϕ2.
Thus, on comparing both sides of Equation (101)
(102)ν0ψ,ϕ,τ=1−ϕeψ,
Fork=0(103)ν1ψ,ϕ,τ=1−ϕeψτσΓσ+1.
Fork=1(104)ν2ψ,ϕ,τ=1−ϕeψτ2σΓ2σ+1.
Fork=2(105)ν3ψ,ϕ,τ=1−ϕeψτ3σΓ3σ+1.
Fork=3(106)ν4ψ,ϕ,τ=1−ϕeψτ4σΓ4σ+1.
The remaining YTDM solution elementsρk for k≥3 are similarly simple to get. As a result, the solution in series form is as
(107)νψ,ϕ,τ=∑k=0∞νkψ,ϕ,τ=ν0ψ,ϕ,τ+ν1ψ,ϕ,τ+ν2ψ,ϕ,τ+ν3ψ,ϕ,τ+ν4ψ,ϕ,τ+⋯νψ,ϕ,τ=1−ϕeψ+1−ϕeψτσΓσ+1+1−ϕeψτ2σΓ2σ+1+1−ϕeψτ3σΓ3σ+1+1−ϕeψτ4σΓ4σ+1+⋯νψ,ϕ,τ=1−ϕeψ1+τσΓσ+1+τ2σΓ2σ+1+τ3σΓ3σ+1+τ4σΓ4σ+1+⋯,νψ,ϕ,τ=1−ϕeψ∑k=0∞tσkΓkσ+1=1−ϕeψEσtσ.
Whenσ=1, the solution by YTDM is
(108)νψ,ϕ,τ=1−ϕeψ∑k=0∞tkk!.
In closed form, the exact solution is:(109)νψ,ϕ,τ=1−ϕeψ+τ.Figure 3
Example2. (a) Analytical solution at various fractional orders of σ (b) ϕ=0.5 and (c) error.
(a)(b)(c)The exact solution shown in Figures4(a) and 4(b) shows HPYTM and YTDM solutions at σ=1 and 0≤ψ,ϕ≥1. Figure 5(a) shows the error graph for different values of σ=1,0.8,0.6,0.5 and 0≤ψ,ϕ≥1 of Example 3 and Figure 5(d) respectively, at ϕ=0.5,τ∈0,1 and 0≤ψ≥1 while Figure 5(e) shows the error graph. From the figures and Table 2, it is clear that HPYTM and YTDM solution shows strong agreement with the exact solutions of the problem.Example 4.
Consider three-dimension fractional-order diffusion equation [54]
(110)∂σν∂τσ=∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,0<σ≤1,τ≥0,with initial source
(111)νψ,ϕ,φ,0=sinψsinϕsinφ.
Taking Jafari-Yang transformation of Equation (110), we get
(112)Y∂σν∂τσ=Y∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2.
We get by using the Jafari-Yang transform’s differential property(113)1uσTu−uν0=Y∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,(114)Tu=uν0+uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,
By applying inverse Jafari-Yang transform to Equation (113)
(115)νψ,ϕ,φ,τ=ν0+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,νψ,ϕ,φ,τ=sinψsinϕsinφ+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2.
Now, using the abovementioned homotopy perturbation technique as in (23), we obtain
(116)∑k=0∞εkνkψ,ϕ,φ,τ=sinψsinϕsinφ+εY−1uσY∑k=0∞εkνkψ,ϕ,φ,τψψ+∑k=0∞εkνkψ,ϕ,φ,τϕϕ+∑k=0∞εkνkψ,ϕ,φ,τφφ.
Comparing the same power coefficient ofε, we get
(117)ε0:ν0ψ,ϕ,φ,τ=sinψsinϕsinφ,ε1:ν1ψ,ϕ,φ,τ=Y−1uσY∂2ν0∂ψ2+∂2ν0∂ϕ2=−3sinψsinϕsinφτσΓσ+1,ε2:ν2ψ,ϕ,φ,τ=Y−1uσY∂2ν1∂ψ2+∂2ν1∂ϕ2=−32sinψssinϕsinφτ2σΓ2σ+1,ε3:ν3ψ,ϕ,φ,τ=Y−1uσY∂2ν2∂ψ2+∂2ν2∂ϕ2=−33sinψsinϕsinφτ3σΓ3σ+1,ε4:ν4ψ,ϕ,φ,τ=Y−1uσY∂2ν3∂ψ2+∂2ν3∂ϕ2=−34sinψsinϕsinφτ4σΓ4σ+1.⋮
By takingε⟶1, we obtain the convergence series type result is given as
(118)νψ,ϕ,φ,τ=ν0+ν1+ν2+ν3+ν4+⋯=sinψsinϕsinφ−3sinψsinϕsinφτσΓσ+1+−32sinψsinϕsinφτ2σΓ2σ+1+−33sinψsinϕsinφτ3σΓ3σ+1+−34sinψsinϕsinφτ4σΓ4σ+1+⋯νψ,ϕ,φ,τ=sinψsinϕsinφ1−3τσΓσ+1+−3τσ2Γ2σ+1+−3τσ3Γ3σ+1+−3τσ4Γ4σ+1+⋯.
Whenσ=1, then the HPYTM solution in a closed form:
(119)νψ,ϕ,φ,τ=sinψsinϕsinφ1−3τ+−3τ22!+−3τ33!+−3τ44!+⋯,.
The analytical results by YTDM
Taking Jafari-Yang transformation of Equation (110), we obtain
(120)Y∂σν∂τσ=Y∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2.
We get by using the Jafari-Yang transform’s differential property(121)1uσTu−uν0=Y∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,(122)Tu=uν0+uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,
by inverse Jafari-Yang transform of Equation (121)
(123)νψ,ϕ,φ,τ=ν0+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2,νψ,ϕ,φ,τ=expψ+ϕ+Y−1uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2.
Assuming the unknownνψ,ϕ,φ,τ function has the following infinite series form solution:
(124)νψ,ϕ,φ,τ=∑k=0∞νkψ,ϕ,φ,τ,(125)∑k=0∞νkψ,ϕ,φ,τ=sinψsinϕsinφ+Y−uσY∂2ν∂ψ2+∂2ν∂ϕ2+∂2ν∂φ2.
Thus, on comparing both sides of Equation (124)
(126)ν0ψ,ϕ,φ,τ=sinψsinϕsinφ.
Fork=0(127)ν1ψ,ϕ,φ,τ=−3sinψsinϕsinφτσΓσ+1,
Fork=1(128)ν2ψ,ϕ,φ,τ=−32sinψsinϕsinφτ2σΓ2σ+1.
Fork=2(129)ν3ψ,ϕ,φ,τ=−33sinψsinϕsinφτ3σΓ3σ+1.
Fork=3(130)ν4ψ,ϕ,φ,τ=−34sinψsinϕsinφτ4σΓ4σ+1.
The remaining YTDM solution elementsρk for k≥3 are similarly simple to get. As a result, the solution in series form is as
(131)νψ,ϕ,φ,τ=ν0ψ,ϕ,φ,τ+ν1ψ,ϕ,φ,τ+ν2ψ,ϕ,φ,τ+ν3ψ,ϕ,φ,τ+⋯νψ,ϕ,φ,τ=sinψsinϕsinφ−3sinψsinϕsinφτσΓσ+1+−32sinψsinϕsinφτ2σΓ2σ+1+−33sinψsinϕsinφτ3σΓ3σ+1+−34sinψsinϕsinφτ4σΓ4σ+1+⋯νψ,ϕ,φ,τ=sinψsinϕsinφ1−3τσΓσ+1+−3τσ2Γ2σ+1+−3τσ3Γ3σ+1+−3τσ4Γ4σ+1+⋯.
Whenσ=1, then the closed form solution by YTDM
(132)νψ,ϕ,φ,τ=sinψsinϕsinφ1−3τ+−3τ22!+−3τ33!+−3τ44!+⋯.
In closed form, the exact solution is(133)νψ,ϕ,φ,τ=exp−3τsinψsinϕsinφ.Figure 4
Example3solufiguretion graph. (a) Exact solution and (b) analytical solution at σ=1.
(a)(b)Figure 5
Example3. (a) Analytical solution at various fractional orders of σ (b) ϕ=0.5 and (c) error.
(a)(b)(c)Table 2
νψ,ϕ,τ comparison of exact solution, our methods’ solution, and Absolute Error (AE) of Example 3.
τ=0.01Exact solutionOur methods’ solutionAE of our methodsAE of our methodsAE of our methodsψσ=1σ=1σ=1σ=0.9σ=0.800.5050250835000000.5050250835000000.0000000000E+002.5480900000E-052.6059160000E-040.10.5581390350000000.5581390351000001.0000000000E-102.8160900000E-052.8799840000E-040.20.6168390300000000.6168390298000002.0000000000E-103.1122300000E-053.1828710000E-040.30.6817125570000000.6817125572000002.0000000000E-103.4395800000E-053.5176210000E-040.40.7534088925000000.7534088927000002.0000000000E-103.8013200000E-053.8875720000E-040.50.8326455975000000.8326455976000001.0000000000E-104.2011000000E-054.2964300000E-040.60.9202156995000000.9202156991000004.0000000000E-104.6428800000E-054.7482850000E-040.71.0169956300000001.0169956290000001.0000000000E-095.1311000000E-055.2476600000E-040.81.1239539940000001.1239539930000001.0000000000E-095.6708000000E-055.7995600000E-040.91.2421612660000001.2421612670000001.0000000000E-096.2673000000E-056.4095200000E-041.01.3728005080000001.3728005070000001.0000000000E-096.9263000000E-057.0836100000E-04In Figures6(a) and 6(b), we consider fixed order σ=1 for piecewise approximation values of ψ,ϕ in the domain 0≤ψ,ϕ≥10 and =1 . Figure 7(a) shows error graph and Figure 7(b) represents HPYTM and YTDM solution at σ=0.6 of Example 4. It is verified from the Figures 6(a) and 6(b) and Table 3 that HPYTM and YTDM solution is closely related with the exact solution.Figure 6
Example4 solution graph. (a) Exact solution. (b) Analytical solution at σ=1.
(a)(b)Figure 7
(a) Error graph and solution graph (b) atσ=0.5 of Example 4.
(a)(b)Table 3
νψ,ϕ,,φ,τ comparison of exact solution, our methods’ solution, and Absolute Error (AE) of Example 4.
τ=0.01Exact solutionOur methods’ solutionAE of our methodsAE of our methodsAE of our methodsψσ=1σ=1σ=1σ=0.9σ=0.800.000000000000000.000000000000000.000000000E+000.000000000E+000.000000000E+000.10.0390847564600000.0390847564700008.4147098480E-125.8916348320E-066.0179783350E-050.20.0777789909600000.0777789909800001.6829419700E-111.1724408770E-051.1975827680E-040.30.1156960835000000.1156960835000000.0000000000E+001.7440075190E-051.7814016480E-040.40.1524571790000000.1524571790000000.0000000000E+002.2981414070E-052.3474221550E-040.50.1876949728000000.1876949728000008.4147098480E-112.8293115510E-052.8899866580E-040.60.2210573804000000.2210573804000000.0000000000E+003.3322251000E-053.4036760840E-040.70.2522110558000000.2522110559000008.4147098480E-113.8018332270E-053.8833574600E-040.80.2808447217000000.2808447217000008.4147098480E-114.2334573540E-054.3242369270E-040.90.3066722799000000.3066722800000008.4147098480E-114.6227807340E-054.7219102240E-041.00.3294356701000000.3294356702000008.4147098480E-114.9659157730E-055.0724039260E-04


## 6. Conclusion

In the present article, different analytical techniques are used to show the fractional view analysis of diffusion equations. In Caputo, manner fractional derivative is considered. The suggested techniques are tested to solve fractional-order diffusion equations. It is observed that the suggested techniques are the best tool for investigating fractional partial differential equations. The close relation between the exact and analytical results is confirmed by the plotted graphs. The given methods give series form solution which have higher convergence rate towards the exact results. It is also shown that both methods give same solution for the proposed problems. Finally, both proposed methods are very methodical and efficient and may be used to investigate nonlinear physical problems related to physics of plasmas such as modeling nonlinear unmodulated and modulated structures. Moreover, the obtained results/solutions can be useful in investigating the diffusion characteristics of some plasmas and fluids.



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