# A Numerical Approach for the Analytical Solution of the Fourth-Order Parabolic Partial Differential Equations

Authors: Fenglian Liu; Muhammad Nadeem; Ibrahim Mahariq; Suliman Dawood
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this study, we propose a new iterative scheme (NIS) to investigate the approximate solution of the fourth-order parabolic partial differential equations (PDEs) that arises in transverse vibration problems. We introduce the Mohand transform as a new operator that is very easy to implement coupled with the homotopy perturbation method. This NIS is capable of reducing the linearization, perturbation, and restrictive assumptions that ruin the nature of the numerical problems. Some numerical examples are demonstrated to legitimate the accuracy and authenticity of this NIS. The computational results are obtained in the shape of a series that converges only after a few iterations. The comparison of the graphical representations shows that NIS is a very simple but also an effective approach for other numerical problems involving complex variables.


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## Body

## 1. Introduction

Many physical phenomena of differential equations in complex variables play an important role in science and engineering such as physics, chemical energy, biology, medicine, and engineering [1–3]. These physical phenomena are of great interest in this modern era and are introduced by parabolic PDEs. It is still very difficult to investigate the exact solution of the PDEs in most numerical problems. Therefore, most of the researchers introduced numerous analytical and numerical approaches to provide the approximate solution for these PDEs such as the quintic B-spline collocation method [4], q-HATM [5], quintic B-spline [6], Legendre wavelet method [7], homotopy perturbation transform method [8], and so on [9–11].Consider the fourth-order parabolic PDEs with variable coefficients [12, 13]
(1)∂2Ψ∂η2+αξ,§,θ∂4Ψ∂ξ4+1§βξ,§,θ∂4Ψ∂§4+1θγξ,§,θ∂4Ψ∂θ4=gξ,§,θ,η,where α,β,γ>0, subjected to the following initial conditions
(2)Ψξ,§,θ,0=f1ξ,§,θ,∂Ψ∂ηξ,§,θ,0=f2ξ,§,θ,and boundary conditions
(3)Ψa,§,θ,η=g0§,θ,η,Ψb,§,θ,η=g1§,θ,η,Ψξ,a,θ,η=k0ξ,θ,η,Ψξ,b,θ,η=k1ξ,θ,η,Ψξ,§,a,η=h0ξ,§,η,Ψξ,§,b,η=h1ξ,§,η,∂2Ψ∂ξ2a,§,θ,η=g0¯§,θ,η,∂2Ψ∂ξ2b,§,θ,η=g1¯§,θ,η,∂2Ψ∂§2ξ,a,θ,η=k0¯ξ,θ,η,∂2Ψ∂§2ξ,b,θ,η=k1¯ξ,θ,η,∂2Ψ∂θ2ξ,§,a,η=h0¯ξ,§,η,∂2Ψ∂θ2ξ,§,b,η=h1¯ξ,§,η,where fj, gj, hj, kj, gj¯, hj¯, and kj¯ are continuous functions and j varies from 0 to 1.Wazwaz [14] used the Adomian decomposition method to examine the analytical solution of transverse vibrations of a uniform flexible beam. Aziz et al. [15] studied the fourth-order nonhomogeneous parabolic partial differential equations that govern the behavior of a vibrating beam by using a new three-level method based on the parametric quintic spline in space and finite difference discretization in time. Biazar and Ghazvini [16] used the variational iteration method for the analytical solution of the fourth-order parabolic equations. Dehghan and Manafian [17] applied HPM for the solution of the fourth-order parabolic PDEs. El-Gamel [18] used the sinc-Galerkin method to examine the fourth-order PDEs in one space variable coefficient. Rashidinia and Mohammadi [19] reported new three-level implicit methods for the numerical solution of the fourth-order nonhomogeneous parabolic PDEs with variable coefficients. Mittal and Jain [20] applied the quintic B-spline method, and Birol [21] used the reduced differential transformation method for the fourth-order nonhomogeneous parabolic partial differential equation. Khan and Sultana [22] used the parametric septic spline for the numerical solution of the fourth-order parabolic PDEs.The homotopy perturbation method (HPM) was developed by He [23, 24]. HPM gives the solution in the form of a rapid and consecutive series toward the exact solution. Dehghan and Manafian [17] used HPM to obtain the numerical results for the linear and nonlinear boundary value problems. The convergence rate of HPM can be studied through [25]. Nadeem et al. [13] applied the Laplace transform coupled with the homotopy perturbation method to solve the fourth-order parabolic PDEs with variable coefficients. Luo et al. [26] introduced a combined form of the Mohand transform and the homotopy perturbation method to provide the analytical solution of the delay differential equations. Recently, many integral transformations have been introduced to find the approximate solution of ordinary and partial differential equations such as the Elzaki transform [27, 28], Sumudu transform [29], Aboodh transformation [30], Mohand transform [31], and homotopy perturbation method [24].In this paper, we construct the idea of NIS with the help of the Mohand transform and the homotopy perturbation method for obtaining the approximate solution of partial differential equations. This NIS provides the results in the form of a series that converges to the exact solution very rapidly. This scheme does not require any linearization, variation, and limiting expectations. In particular, this study is organized as follows. In Section (2), we recall some basic definitions of the Mohand transform. In Sections (3) and (4), first, we present the basic idea of HPM and then formulate the idea of NIS for finding the approximate solution of PDEs. We illustrate three examples to present the accuracy and validity of NIS in Section (5). We give a brief discussion of the obtained results in Section (6), and finally, the conclusion is presented in Section (7).


## 2. Fundamental Concepts of the Mohand Transform

In this section, we introduce some basic definitions and preliminary concepts of the Mohand transform, which reveals the idea of its implementations to functions.Definition 1.
Mohand and Mahgoub [31] presented a new scheme Mohand transform M. in order to gain the results of ordinary differential equations, which is defined as
(4)MΨη=Rw=w2∫0∞Ψηe−wηdη,k1≤w≤k2.
On the other hand, ifRw is the Mohand transform of a function Ψη, then Ψη is the inverse of Rwsuch that
(5)M−1Rw=Ψη,M−1istheinverseMohandoperator.Definition 2.
IfΨη=ηn,
(6)Rw=n!wn−1.Definition 3.
IfMΨη=Rw, then it has the following differential properties:
(i)
MΨ′η=wRw−w2F0(ii)
MΨ′′η=w2Rw−w3F0−w2F′0(iii)
MFunη=wnRw−wn+1F0−wnF′0−⋯−wnFn−10


## 3. Basic Idea of HPM

In this segment, we illustrate a nonlinear functional equation to explain the basic view of HPM [32, 33]. Consider
(7)TΨ−gh=0,h∈Ω,with conditions
(8)SΨ,∂Ψ∂n=0,h∈Γ,where T and S are known as the general functional operator and boundary operator, respectively, and gh is a known function with Γ as an interval of the domain Ω. We now divide T into two units such thatT1 represents a linear and T2 a nonlinear operator. As a result, we can express Equation (8) such that
(9)T1Ψ+T2Ψ−gh=0.Assume a homotopyϑh,p:Ω×0,1⟶ℍ in such a way that it is appropriate for
(10)Hϑ,p=1−pT1ϑ−T1Ψ0+pT1ϑ−T2ϑ−gh,or
(11)Hϑ,p=T1ϑ−T1Ψ0+qLΨ0+pT2ϑ−gh=0,where p∈0,1 is the embedding parameter and Ψ0 is an initial guess of Equation (7), which is suitable for the boundary conditions. The theory of HPM states that p is considered a slight variable, and the solution of Equation (7) in the resulting form of pis
(12)ϑ=ϑ0+pϑ1+p2ϑ2+p3ϑ3+⋯=∑i=0∞piϑi.Letp=1, and then the particular solution of Equation (8) is written as
(13)Ψ=limp⟶1ϑ=ϑ0+ϑ1+ϑ2+ϑ3+⋯=∑i=0∞ϑi.The nonlinear terms can be calculated as(14)T2Ψx,t=∑n=0∞pnHnΨ.Then, He’s polynomialsHnΨ can be obtained using the following expression:
(15)HnΨ0+Ψ1+⋯+Ψn=1n!∂n∂pnT2∑i=0∞piΨip=0,n=0,1,2,⋯.The series solution in Equation (14) is mostly convergent due to the convergence rate of the series depending on the nonlinear operator T2.


## 4. Formulation of NIS

This segment presents the formulation of a new iterative scheme (NIS) for obtaining the approximate solution of the fourth-order parabolic PDEs. Let us consider a second-order differential equation of the form(16)Ψ′′ξ,η+Ψξ,η+gΨ=gξ,η,with the following conditions:
(17)Ψξ,0=a,Ψ′ξ,0=b,where Ψ is a function in time domain η, gΨ represents a nonlinear term, and gη is a source term, whereas a and b are constants. Rewrite Equation (16) again as
(18)Ψ′′ξ,η=−Ψξ,η−gΨ+gξ,η.Now, taking MT on both sides of Equation (18), we obtain
(19)MΨ′′ξ,η=M−Ψξ,η−gΨ+gξ,η.Applying the differential properties of MT, we get(20)w2Rw−w3Ψξ,0−w2Ψ′ξ,0=M−Ψξ,η−gΨ+gξ,η.Thus,Rw can be obtained from Equation (20) such that
(21)Rw=wuξ,0+Ψ′ξ,0−1w2MΨξ,η+gΨ−gξ,η.Operating the inverse Mohand transform on Equation (21), we get
(22)Ψξ,η=Gξ,η−M−11w2MΨξ,η+gΨ,where Equation (22) is called the NIS and
(23)Gξ,η=M−1wu0+Ψ′0+1w2gξ,η.Now, we apply HPM on Equation (22). Let
(24)Ψη=∑i=0∞piΨin=Ψ0+p1Ψ1+p2Ψ2+⋯,and nonlinear terms gΨ can be calculated by using the following formula:
(25)gΨ=∑i=0∞piHiΨ=H0+p1H1+p2H2+⋯,where Hn′s is He’s polynomial, which may be computed using the following procedure:
(26)HnΨ0+Ψ1+⋯+Ψn=1n!∂n∂png∑i=0∞piΨip=0,n=0,1,2,⋯.Put Equations (24)–(26) in Equation (22), and comparing the similar factors of p, we get the following consecutive elements:
(27)p0:Ψ0ξ,η=Gξ,η,p1:Ψ1ξ,η=−M−11w2MΨ0ξ,η+H0Ψ,p2:Ψ2ξ,η=−M−11w2MΨ1ξ,η+H1Ψ,p3:Ψ3ξ,η=−M−11w2MΨ2ξ,η+H2Ψ,⋮.In continuing the similar process, we can summarize this series to get the approximate solution such that(28)Ψξ,η=Ψ0+Ψ1+Ψ2+⋯=∑i=0∞Ψi.Thus, Equation (28) is to be considered an approximate solution of differential equations of Equation (16).


## 5. Numerical Examples

In this part, we consider three numerical problems to check the authenticity and validity of NIS. We also demonstrate the solution surface of the illustrated problems for the behavior and a better understanding of this strategy where we see that the solution graphs of the approximate solution and the particular solution coincide with each other only after a few iterations.


### 5.1. Example 1

Consider the one-dimensional fourth-order parabolic PDEs(29)∂2Ψ∂η2+1ξ+ξ4120∂4Ψ∂ξ4=0,with the initial conditions
(30)Ψξ,0=0,Ψηξ,0=1+ξ5120.Applying MT on Equation (29) together with the differential property as defined in Equation (6), we get
(31)w2Rw−w3Ψξ,0−w2Ψηξ,0=−M1ξ+ξ4120∂4Ψ∂ξ4.Thus,Rw yields
(32)Rw=wΨξ,0−Ψηξ,0−1w2M1ξ+ξ4120∂4Ψ∂ξ4.Using the inverse Mohand transform, we get(33)Ψξ,η=Ψξ,0−ηΨηξ,0−M−11w2M1ξ+ξ4120∂4Ψ∂ξ4.Applying MHPTM to get He’s polynomials, we get(34)∑i=0∞piΨin=Ψξ,0−ηΨηξ,0−M−11w2M1ξ+ξ4120∑i=0∞pi∂4Ψi∂ξ4.Observing the similar powers ofp, we get
(35)p0:Ψ0ξ,η=1ξ+ξ4120η,p1:Ψ1ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ0∂ξ4=−1ξ+ξ4120η33!,p2:Ψ2ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ1∂ξ4=1ξ+ξ4120η55!,p3:Ψ3ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ2∂ξ4=−1ξ+ξ4120η77!,p4:Ψ4ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ3∂ξ4=1ξ+ξ4120η99!,⋮.In continuing this process, the approximate solution results can be summarized as(36)Ψξ,η=Ψ0ξ,η+Ψ1ξ,η+Ψ2ξ,η+Ψ3ξ,η+Ψ4ξ,η+⋯=1+ξ5120η−η33!+η55!−η77!+η99!+⋯.This series converges to the particular solution(37)Ψξ,η=1+ξ5120sinη.


### 5.2. Example 2

Consider the two-dimensional fourth-order parabolic PDEs(38)∂2Ψ∂η2+21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=0,with the initial conditions
(39)Ψξ,§,0=0,Ψηξ,§,0=2+ξ66!+§66!.Applying NIM, we get(40)Ψξ,§,η=Ψξ,§,0−ηΨηξ,§,0−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4.This equation provides He’s polynomials(41)∑i=0∞piΨin=Ψξ,§,0−ηΨηξ,§,0−M−11w2M21ξ2+ξ46!∑i=0∞pi∂4Ψi∂ξ4+21§2+§46!∑i=0∞pi∂4Ψi∂§4.Observing the similar powers ofp, we get
(42)p0:Ψ0ξ,§,η=2+ξ66!+§66!η,p1:Ψ1ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=−2+ξ66!+§66!η33!,p2:Ψ2ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=2+ξ66!+§66!η55!,p3:Ψ3ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=−2+ξ66!+§66!η77!,p4:Ψ4ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=2+ξ66!+§66!η99!,⋮.In continuing this process, the approximate solution results can be summarized as(43)Ψξ,§,η=Ψ0ξ,§,η+Ψ1ξ,§,η+Ψ2ξ,§,η+Ψ3ξ,§,η+Ψ4ξ,§,η+⋯=2+ξ66!+§66!η−η33!+η55!−η77!+η99!+⋯.This series converges to the particular solution(44)Ψξ,§,η=2+ξ66!+§66!sinη.


### 5.3. Example 3

Consider the three-dimensional fourth-order parabolic PDEs(45)∂2Ψ∂η2+2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4=0,with the initial conditions
(46)Ψξ,§,θ,0=ξ+§+θ−cosξ+cos§+cosθ,Ψξ,§,θ,0=cosξ+cos§+cosθ−ξ+§+θ.Applying NIM, we get(47)Ψξ,§,θ,η=Ψξ,§,0−ηΨηξ,§,0−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4.This equation provides He’s polynomials(48)p0:Ψ0ξ,§,θ,η=wΨξ,§,0−Ψηξ,§,01−η,p1:Ψ1ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p2:Ψ2ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p3:Ψ3ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p4:Ψ4ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,which gives
(49)Ψ0ξ,§,η=ξ+§+θ−cosξ−cos§−cosθ1−η,Ψ1ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη22!−η33!,Ψ2ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη44!−η55!,Ψ3ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη66!−η77!,Ψ4ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη88!−η99!,⋮.In continuing this process, the approximate solution results can be summarized as(50)Ψξ,η=Ψ0ξ,η+Ψ1ξ,η+Ψ2ξ,η+Ψ3ξ,η+Ψ4ξ,η+⋯=ξ+§+θ−cosξ−cos§−cosθ1−η+η22!−η33!+η44!−η55!+η66!−η77!+η88!−η99!+⋯.This series converges to the particular solution(51)Ψξ,η=ξ+§+θ−cosξ−cos§−cosθe−η.


## 5.1. Example 1

Consider the one-dimensional fourth-order parabolic PDEs(29)∂2Ψ∂η2+1ξ+ξ4120∂4Ψ∂ξ4=0,with the initial conditions
(30)Ψξ,0=0,Ψηξ,0=1+ξ5120.Applying MT on Equation (29) together with the differential property as defined in Equation (6), we get
(31)w2Rw−w3Ψξ,0−w2Ψηξ,0=−M1ξ+ξ4120∂4Ψ∂ξ4.Thus,Rw yields
(32)Rw=wΨξ,0−Ψηξ,0−1w2M1ξ+ξ4120∂4Ψ∂ξ4.Using the inverse Mohand transform, we get(33)Ψξ,η=Ψξ,0−ηΨηξ,0−M−11w2M1ξ+ξ4120∂4Ψ∂ξ4.Applying MHPTM to get He’s polynomials, we get(34)∑i=0∞piΨin=Ψξ,0−ηΨηξ,0−M−11w2M1ξ+ξ4120∑i=0∞pi∂4Ψi∂ξ4.Observing the similar powers ofp, we get
(35)p0:Ψ0ξ,η=1ξ+ξ4120η,p1:Ψ1ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ0∂ξ4=−1ξ+ξ4120η33!,p2:Ψ2ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ1∂ξ4=1ξ+ξ4120η55!,p3:Ψ3ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ2∂ξ4=−1ξ+ξ4120η77!,p4:Ψ4ξ,η=−M−11w2M1ξ+ξ4120∂4Ψ3∂ξ4=1ξ+ξ4120η99!,⋮.In continuing this process, the approximate solution results can be summarized as(36)Ψξ,η=Ψ0ξ,η+Ψ1ξ,η+Ψ2ξ,η+Ψ3ξ,η+Ψ4ξ,η+⋯=1+ξ5120η−η33!+η55!−η77!+η99!+⋯.This series converges to the particular solution(37)Ψξ,η=1+ξ5120sinη.


## 5.2. Example 2

Consider the two-dimensional fourth-order parabolic PDEs(38)∂2Ψ∂η2+21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=0,with the initial conditions
(39)Ψξ,§,0=0,Ψηξ,§,0=2+ξ66!+§66!.Applying NIM, we get(40)Ψξ,§,η=Ψξ,§,0−ηΨηξ,§,0−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4.This equation provides He’s polynomials(41)∑i=0∞piΨin=Ψξ,§,0−ηΨηξ,§,0−M−11w2M21ξ2+ξ46!∑i=0∞pi∂4Ψi∂ξ4+21§2+§46!∑i=0∞pi∂4Ψi∂§4.Observing the similar powers ofp, we get
(42)p0:Ψ0ξ,§,η=2+ξ66!+§66!η,p1:Ψ1ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=−2+ξ66!+§66!η33!,p2:Ψ2ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=2+ξ66!+§66!η55!,p3:Ψ3ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=−2+ξ66!+§66!η77!,p4:Ψ4ξ,§,η=−M−11w2M21ξ2+ξ46!∂4Ψ∂ξ4+21§2+§46!∂4Ψ∂§4=2+ξ66!+§66!η99!,⋮.In continuing this process, the approximate solution results can be summarized as(43)Ψξ,§,η=Ψ0ξ,§,η+Ψ1ξ,§,η+Ψ2ξ,§,η+Ψ3ξ,§,η+Ψ4ξ,§,η+⋯=2+ξ66!+§66!η−η33!+η55!−η77!+η99!+⋯.This series converges to the particular solution(44)Ψξ,§,η=2+ξ66!+§66!sinη.


## 5.3. Example 3

Consider the three-dimensional fourth-order parabolic PDEs(45)∂2Ψ∂η2+2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4=0,with the initial conditions
(46)Ψξ,§,θ,0=ξ+§+θ−cosξ+cos§+cosθ,Ψξ,§,θ,0=cosξ+cos§+cosθ−ξ+§+θ.Applying NIM, we get(47)Ψξ,§,θ,η=Ψξ,§,0−ηΨηξ,§,0−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4.This equation provides He’s polynomials(48)p0:Ψ0ξ,§,θ,η=wΨξ,§,0−Ψηξ,§,01−η,p1:Ψ1ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p2:Ψ2ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p3:Ψ3ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,p4:Ψ4ξ,§,θ,η=−M−11w2M2§+θcosξ−1∂4Ψ∂ξ4+ξ+θ2cos§−1∂4Ψ∂§4+§+ξ2cosθ−1∂4Ψ∂θ4,which gives
(49)Ψ0ξ,§,η=ξ+§+θ−cosξ−cos§−cosθ1−η,Ψ1ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη22!−η33!,Ψ2ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη44!−η55!,Ψ3ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη66!−η77!,Ψ4ξ,§,η=ξ+§+θ−cosξ−cos§−cosθη88!−η99!,⋮.In continuing this process, the approximate solution results can be summarized as(50)Ψξ,η=Ψ0ξ,η+Ψ1ξ,η+Ψ2ξ,η+Ψ3ξ,η+Ψ4ξ,η+⋯=ξ+§+θ−cosξ−cos§−cosθ1−η+η22!−η33!+η44!−η55!+η66!−η77!+η88!−η99!+⋯.This series converges to the particular solution(51)Ψξ,η=ξ+§+θ−cosξ−cos§−cosθe−η.


## 6. Results and Discussion

In this segment, we present the discussion of some graphical representations in Figures1–3. It can be seen that the formulated series converges to the particular solution only after a few iterations very rapidly. Figures 1(a) and 1(b) represent the comparison between the approximate solution and the exact solution of Equations (36) and (37) at 0≤η≤1 and 0≤ξ≤10, respectively. Figures 2(a) and 2(b) show the comparison between the approximate solution and the particular solution of Equations (43) and (44) at 0≤η≤3 and 0≤ξ≤10, respectively, and similarly, Figures 3(a) and 3(b) represent the comparison between the approximate solution and the particular solution of Equations (50) and (51) at 0≤η≤1 and 0≤ξ≤10, respectively. This comparison shows that NIS is easy to implement and does not require any heavy calculation for the computation of the approximate solution of the fourth-order parabolic PDEs with variable coefficients.Figure 1
Surface solutions for the one-dimensional parabolic differential equation.
(a)
Approximate solution ofΨξ,η for Equation (29)(b)
Particular solution ofΨξ,η for Equation (29)Figure 2
Surface solutions for the two-dimensional parabolic differential equation.
(a)
Approximate solution ofΨξ,§,η for Equation (38)(b)
Particular solution ofΨξ,§,η for Equation (38)Figure 3
Surface solutions for the three-dimensional parabolic differential equation.
(a)
Approximate solution ofΨξ,§,θ,η for Equation (45)(b)
Particular solution ofΨξ,§,θ,η for Equation (45)


## 7. Conclusion and Future Work

In this analysis, we successfully employed the NIS to examine the approximate solution of the fourth-order parabolic partial differential equations with variable coefficients. The Mohand transform coupled with HPM has been used to construct the idea of this scheme. This NIS approach is applicable for both the linear and nonlinear partial differential equations. This approach does not require the recurrence relation for the assumption of a variable. This NIS formulates the obtained results of the illustrated problems in the form of a series that converges to the particular solution very rapidly. This approach has an advantage of direct implementation to the numerical problems and confirms the accuracy with full agreement. This NIS is also applicable for the other partial differential equations with fractional derivatives in science and engineering.



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*Source: 3309674-2022-08-16.xml*