# A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations

Authors: Sajad Iqbal; Mohammed K. A. Kaabar; Francisco Martínez
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.


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## Body

## 1. Introduction

The initial idea of fractional derivative in history goes back to the late 17th century, when the French mathematician L'Hopital concerned with the meaning of dny/dxn, when n = 1/2. Since then, various definitions of this concept have been formulated from two conceptions: one of a global nature and the other one of a local nature. The global formulation of the fractional derivative is linked to the appearance of the fractional calculus itself, and the Caputo and Riemann–Liouville definitions are the best known [1, 2].Khalil et al. suggested recently an alternative fractional derivative of local type, named as conformable derivative [3] to solve some issues concerning the challenge of solving fractional differential equations (F-DEs) of nonlocal type. The physical and geometrical interpretations of conformable derivatives have been discussed in [4, 5], respectively.In several research studies, the methods of homotopy perturbation and Adomian decomposition have been employed to solve various types of F-DEs [6, 7]. Furthermore, a new numerical technique that is relied on the homotopy perturbation method (HPM) and Laplace transform (LT) has been proposed in [8] to solve F-DEs. Madani et al. employed the Laplace transform homotopy perturbation method (LTHPM) to solve one-dimensional in-homogeneous partial differential equations (PDEs) with a variable coefficient [9]. In [10], LTHPM is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations.Fall et al. [11] implemented the homotopy perturbation method to obtain the analytical solution of time-fractional Black–Scholes (T-BSEs) and the generalized fractional BSEs. They have displayed both the solution graphically and discussed the effect of the order ρ of the generalized fractional BSEs in the diffusion processes.In [12], Yavuz and Ozdemir proposed a novel definition of the Adomian decomposition method (ADM) to get an accurate and quick solution of T-FBSEs and generalized fractional BSEs utilizing initial condition. They obtained an approximate analytical solution to these equations.Moreover, Ahmed and Elbadri [13] carried out the Sumudu decomposition method (SDM) coupled with the Adomian decomposition method to solve the fractional Newell-Whitehead-Segal equations (F-NWSEs). From their numerical results, they concluded that the proposed technique is straightforward to enforce and provide exact results.For solving the Newell-Whitehead-Segal equations [14], the authors presented two alternative exponential finite difference methods: implicit exponential finite difference scheme and fully implicit exponential finite difference scheme. Both techniques are demonstrated to be consistent. Furthermore, the local truncation errors are accurate to the first order in time and the second order in space. The approaches described are more accurate than other methods because they are computationally consistent, stable, and convergent.Yavuz et al. constructed the conformable ADM and HPM, using the aforementioned conformable derivatives to solve various F-DEs such as diffusion equation, Black–Scholes equation (BSEs), and cable equation [15, 16]. In [12], the authors proposed a novel definition of ADM to get an accurate and quick solution of the Black–Scholes equation of time-fractional order (F-BSEs) and the generalized Black–Scholes equation of fractional order (G-FBSEs) utilizing initial condition.The main purpose of our study is to construct the Laplace transform homotopy perturbation method (LTHPM) for F-DEs using conformable derivatives.The article is outlined as follows: some fundamental notions of conformable derivatives are revisited in Section2. Subsequently, HPM is constructed using con-LT in Section 3. Finally, we apply in Section 4 the algorithm which is important to partial derivative equations (F-PDEs). The conclusion of this work is discussed in Section 5.


## 2. Basic Notions

This section presents some essential concepts of conformable derivatives, which will be necessary in our investigation.Definition 1.
Givenf:0,∞⟶R as a function. Then, the αth order conformable derivatives [1] is expressed as(1)Tαft=limε⟶0ft+εt1−α−ftε,∀t>0, α∈0,1. If f is α-differentiable (α-DifF) in some 0,a, a>0, and Tαft exists, then it is expressed as(2)Tαf0=limt⟶0+Tαft.Theorem 1 (See [1]).
If a functionf:0,∞⟶R is α-DifF at t0>0, α∈0,1, then f is continuous (ContF) at t0.Theorem 2 (See [1]).
Suppose thatα∈0,1, and f,h are α-DifFs at a point t>0. Then, we get(i)
Tαmf+wh=mTαf+wTαh, ∀m,w∈R(ii)
Tαtv=vtv−α, ∀v∈R(iii)
Tαϑ=0, ∀ constant function ft=ϑ(iv)
Tαfh=fTαh+hTαf(v)
Tαf/h=hTαf−fTαh/h2(vi)
Iff is DifF, then Tαft=t1−αdf/dtt
From the above definition, the conformable derivatives of some functions are expressed as(i)
Tα1=0(ii)
Tαsinat=at1−αcosat(iii)
Tαcosat=at1−αsinat(iv)
Tαeat=aeat, a∈R
In addition, some helpful formulas are expressed as(i)
Tα1/αtα=1(ii)
Tαe1/αtα=e1/αtα(iii)
Tαsin1/αtα=cos1/αtα(iv)
Tαcos1/αtα=−sin1/αtαDefinition 2.
The left-conformable derivative beginning froma of function f:a,∞⟶R of order α∈0,1 is expressed as [2](3)Tαaft=limε⟶0ft+εt−a1−α−ftε,when a=0, we have Tαft. If f is α-DifF in some a,b, then we get(4)Tαafa=limt⟶a+Tαaft.Theorem 3 (Chain Rule) [2].
Suppose thatf,h:a,∞⟶R be left α-DifFs, where α∈0,1. Assume that st=fht. st is α−DifF∀t≠a and ht≠0. Thus, we obtain(5)Tαast=Tαafht⋅Tαaht.
Ift=a, then(6)Tαasa=limt⟶a+Tαafht⋅Tαaht⋅htα−1.Theorem 4 (See [2]).
Supposed thatf is infinitely α-DifF, for some α∈0,1 at a neighborhood of a point t0. Then, f has a conformable power series expansion as(7)ft=∑k=0∞kTαt0t0αkk!t−t0kα,t0<t<t0+R1/α.Hence,kTαt0t0 implies the application of conformable derivatives k times.Theα-conformable integral of a function f is beginning from a≥0.Definition 3 (See [1]).
Iαaft=∫atfx/x1−α⋅dx, where this is a usual Riemann improper integral, α∈0,1.
As a result, we haveTheorem 5.
TαaIαaft=ft, for t≥a, where f is any ContF in Iα′s domain.Lemma 1.
Suppose thatf:a,b⟶R is DifF and α∈0,1. Then, ∀a>0, we get [2](8)IαaTαaft=ft−fa.Finally, we recall the fractional Laplace transform (F-LT) [2]. In [3], a sufficient condition was accomplished to solve constant coefficient F-DEs via the con-LT method.Definition 4.
Suppose thatα∈0,1 and f:0,∞⟶R are real valued functions. Then, the F-LT of order α beginning from 0 of f is expressed as(9)Lαfts=∫0∞e−stα/αftdtt1−α.Remark 1.
The F-LT for the conformable derivatives is expressed as(10)LαTαfts=sLαfts−f0.
Thus, we haveTheorem 6.
IfFαs=Lαfts exists for s>0, then(i)
Ifc a constant, then(11)Lαcs=cs.(ii)
Letq be a constant:(12)Lαtqs=αq/αΓ1+q/αs1+q/α.(iii)
Ifc and q are arbitrary constants,(13)Lαtqectα/αs=αq/αΓ1+q/αs−c1+q/α.Definition 5.
A functionf is considered as a conformable exponentially bounded if it satisfies ft≤Mectα/α, where M and c are the positive real constants and α∈0,1, ∀ sufficiently large t.


## 3. Conformable Laplace Transform HPM

A conformable version of HPM using LT has been introduced by Madani and Fathizadeh in [17]. In their research work, these authors discuss the effectiveness and ease of this technique to solve ordinary or partial differential equations (PDEs). In our case, a con-PDEs is given as follows:(14)Mαux,t+Nux,t+Rux,t=gx,t,with the initial condition (I.C):(15)ux,0=fx,where u is the two variables function, Mα=∂α/∂tα is a linear operator (LO) with conformable derivative of order α∈0,1, R is the LO’s other part, N is a nonlinear operator (NLO), and gx,t is a nonhomogeneous term. By solving for Mαux,t, equation (14) can be written as(16)Mαux,t=g−Nu−Ru.By applying the con-LT to equation (16), we obtain(17)LαMαux,t=Lαg−Nu−Ru.From Remark1, the above equations can be rewritten as(18)sLαux,ts−ux,0=Lαg−Nu−Ru.By substituting I.C, equation (18) can be written as(19)Lαux,ts=fxs+1sLαg−1sLαNu−1sLαRu.Furthermore, by applying inverse con-LT to equation (19), we get(20)ux,t=Gx,t−Lα−11sLαNux,t+Rux,t,where the term has been arisen from the source term, and the prescribed I.C is denoted by Gx,t.The HPM suggests the solutionux,t to be decomposed into the infinite series of components [18, 19]:(21)ux,t=∑n=0∞pnunx,t,and nonlinear term Nux,t is decomposed to(22)Nux,t=∑n=0∞pnHnu,for some He’s polynomials Hnu [17, 20] which are given by(23)Hnu0,u1,…,un=1n!∂n∂pnN∑i=0∞piui,n=0,1,2,3,….By the substitution of equations (21) and (22) into equation (20), we have(24)∑n=0∞pnunx,t=Gx,t−pLα−11sLαR∑n=0∞pnunx,t+∑n=0∞pnHnu,which is the coupled con-FLT and HPM via He’s polynomials. The approximation can be easily obtained by a comparison of all coefficients like powers of p as follows:(25)p0:u0x,t=Gx,t,p1:u1x,t=−Lα−11sLαRu0x,t+H0u,p2:u2x,t=−Lα−11sLαRu1x,t+H1u,p3:u1x,t=−Lα−11sLαRu2x,t+H2u,⋮.


## 4. Illustrative Examples

The effectiveness of the con-LTHPM is shown in this section through four different problems: the time-fractional nonlinear homogeneous gas dynamics equation (F-GDEs), the time-fractional linear Newell-Whitehead-Segel equation (F-NWSEs), the time-fractional diffusion-convection equation (F-DCEs), and the time-fractional linear Black–Scholes option pricing equation.Example 1.
Consider F-GDEs as follows:(26)∂αvx,t∂tα+vx,t∂vx,t∂x−vx,t1−vx,t=0,t≥0,0≤x≤1,0<α≤1,with I.C: vx,0=ae−x. If α=1, then equation (26) becomes the classical nonlinear homogeneous GDEs [21].Remark 2.
Note that the mathematical expression of the classical GDEs is based on the physical laws of conservation.
By taking con-LT on equation (26) both sides and from the properties of con-LT, equation (26) reduces to(27)Lαvx,ts=vx,0s+1sLαv−v2−v∂v∂x.
Using I.C and inverse con-LT, equation (27) reduces to(28)vx,t=ae−x+Lα−11sLαv−v2−v∂v∂x.
The HPM is applied to obtain(29)∑n=0∞pnvnx,t=ae−x+pLα−11sLα∑n=0∞pnvnx,t−∑n=0∞pnHnv,where Hnv represents He’s polynomials.
The 1st three parts of Hnv are(30)H0v=v02+v0∂v0∂x,H1v=2v0v1+v0∂v1∂x+v1∂v0∂x,H2v=2v0v2+v0∂v2∂x+v1∂v1∂x+v2∂v0∂x+v12,and so on.
The coefficient of power ofp is equated in equation (33) to get(31)p0:v0x,t=ae−x,p1:v1x,t=Lα−11sLαv0−H0v=Lα−11sLαv0−v02−v0∂v0∂x=ae−xtαα,p2:v2x,t=Lα−11sLαv1−H1v=Lα−11sLαv1−2v0v1−v0∂v1∂x−v1∂v0∂x=ae−x12!tαα2,p3:v3x,t=Lα−11sLαv2−H2v=Lα−11sLαv1−2v0v2−v0∂v2∂x−v1∂v1∂x−v2∂v0∂x+v12=ae−x13!tαα3.
Similarly, the approximations can further be obtained as follows:(32)p4:v4x,t=ae−x14!tαα4,p5:v5x,t=ae−x15!tαα5,and so on.
Substitute the above values in the following equation:(33)vx,t=v0x,t+v1x,t+v2x,t+v3x,t+⋯=ae−x+e−xtαα+e−x12!tαα2+e−x13!tαα3+e−x14!tαα4+⋯=ae−x1+tαα+12!tαα2+13!tαα3+14!tαα4+15!tαα5+⋯=aetα/α−x.
The equation (26) exact solution with I.C: vx,0=ae−x for α=1 as a special case is found as follows:(34)vx,t=aet−x,which is the same solution in [21]. This result is verified graphically in Figure 1.
In Figure2, the con-LTHPM’s solution for various values of α, i.e., α=0.7,0.9 is represented. The F-GDEs is considered here with parameter a=1.Figure 1
Comparison of the numerical solution and exact solution forα=1.Figure 2
Con-LTHPM’s solutions of the F-GDEs.Example 2.
Consider F-NWSEs as follows:(35)∂αvx,t∂tα=∂2vx,t∂x2−3vx,t,with I.C: vx,0=e2x. If α=1, then equation (35) becomes an example of the classical linear NWSEs [22].Remark 3.
This equation has been applied in a wide variety of problems. For example, Faraday instability, nonlinear optics, Rayleigh-Bernard convection, or chemical reactions.
By taking con-LT on equation (45) both sides and from the properties of con-LT, equation (35) reduces to(36)Lαvx,ts=vx,0s+3+1s+3Lα∂2vx,t∂x2.
Using I.C and inverse con-LT, equation (36) reduces to(37)vx,t=e2x−3tα/α+Lα−11s+3Lα∂2vx,t∂x2.
The HPM is applied to obtain(38)∑n=0∞pnvnx,t=e2x−3tα/α+pLα−11s+3Lα∑n=0∞pn∂2vnx,t∂x2.
The coefficients of the power ofp are compared as follows:(39)p0:v0x,t=e2x−3tα/α,p1:v1x,t=Lα−11s+3Lα∂2v0x,t∂x2=4tααe2x−3tα/α,p2:v2x,t=Lα−11s+3Lα∂2v1x,t∂x2=1612!tαα2e2x−3tα/α,p3:v3x,t=Lα−11s+3Lα∂2v2x,t∂x2=6413!tαα3e2x−3tα/α.
Similarly, approximations can further be obtained as follows:(40)p4:v4x,t=25614!tαα4e2x−3tα/α,p5:v5x,t=51215!tαα5e2x−3tα/α,and so on.
Substitute the above values into the following equation:(41)vx,t=v0x,t+v1x,t+v2x,t+v3x,t+⋯=e2x−3tα/α+e2x−3tα/α4tαα+e2x−3tα/α12!4tαα2+e2x−3tα/α13!4tαα3+e2x−3tα/α14!4tαα4+⋯=e2x−3tα/α1+4tαα+12!4tαα2+13!4tαα3+14!4tαα4+15!4tαα5+⋯=e2x+3tα/α.
The equation (35) exact solution with Ivx,0=e2x, for α=1 as a special case, is found as follows:(42)vx,t=e2x+t,which is the same solution in [22]. This result is verified graphically, as shown in Figure 3.
In Figure4, the con-LTHPM’s solution for various values of α, i.e., α=0.7,0.9 is represented.Figure 3
Comparison of the numerical solution and exact solution forα=1.Figure 4
Con-LTHPM’s solutions of the F-NWSEs.Example 3.
Consider F-DCEs as follows:(43)∂αvx,t∂tα=∂2vx,t∂x2+cosx−sin2x−1vx,t,with I.C: vx,0=1/10ecosx−11. If α=1, then equation (43) becomes an example of the classical DCEs [23].Remark 4.
The DCEs describe physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.
By taking con-LT on equation (43) both sides and from the properties of con-LT, equation (43) reduces to(44)Lαvx,ts=vx,0s+1sLα∂2vx,t∂x2+cosx−sin2x−1vx,t.
Using I.C and inverse con-LT, equation (44) reduces to(45)vx,t=110ecosx−11+Lα−11sLα∂2vx,t∂x2+cosx−sin2x−1vx,t.
The HPM is applied to obtain(46)∑n=0∞pnvnx,t=110ecosx−11+pLα−11sLα∑n=0∞pn∂2vnx,t∂x2+cosx−sin2x−1∑n=0∞pnvnx,t.
The coefficients of the power ofp are compared here as follows:(47)p0:v0x,t=110ecosx−11,p1:v1x,t=Lα−11sLα∂2v0x,t∂x2+cosx−sin2x−1v0x,t=−110ecosx−11tαα,p2:v2x,t=Lα−11sLα∂2v1x,t∂x2+cosx−sin2x−1v1x,t=110ecosx−1112!tαα2,p3:v3x,t=Lα−11sLα∂2v2x,t∂x2+cosx−sin2x−1v2x,t=−110ecosx−1113!tαα3.
Similarly, approximations can further be obtained as follows:(48)p4:v4x,t=110ecosx−1114!tαα4,p5:v5x,t=−110ecosx−1115!tαα5,and so on.
Substitute the above values in the following equation:(49)vx,t=v0x,t+v1x,t+v2x,t+v3x,t+⋯=110ecosx−111−tαα+12!tαα2−13!tαα3+14!tαα4−15!tαα5+⋯=110ecosx−11−tα/α.
The equation (43) exact solution with I.C: vx,0=1/10ecosx−11, for α=1 as a special case, is found as follows:(50)vx,t=110ecosx−11−t,which is the same solution with [23]. This result is also verified graphically, as shown in Figure 5.
In Figure6, the con-LTHPM’s solution for various values of α, i.e., α=0.7,0.9 is represented.Figure 5
Comparison of the numerical solution and exact solution forα=1.Figure 6
Con-LTHPM’s solutions of the F-DCEs.Example 4.
Consider the F-BSEs:(51)∂αvx,t∂tα=∂2vx,t∂x2+k−1∂vx,t∂x−kvx,t,t>0,x∈R,0<α≤1,with I.C: vx,0=maxex−1,0. If α=1, then equation (51) becomes the classical linear BSEs [24].Remark 5.
Equation (51) contains only two dimensionless parameters k=2r/σ2, where k represents the balance between the rates of interest and the variability of the return on stocks and the dimensionless time to expiry σ2T/2.
By taking con-LT on equation (51) both sides and from the properties of con-LT, equation (51) reduces to(52)Lαvx,ts=vx,0s+k+1s+kLα∂2vx,t∂x2+k−1∂vx,t∂x.
Using I.C and inverse con-LT, equation (52) reduces to(53)vx,t=maxex−1,0e−ktα/α+Lα−11s+kLα∂2vx,t∂x2+k−1∂vx,t∂x.
The HPM is applied to obtain(54)∑n=0∞pnvnx,t=maxex−1,0e−ktα/α+pLα−11s+kLα∑n=0∞pn∂2vnx,t∂x2+k−1∑n=0∞pn∂vnx,t∂x.
The coefficients of the power ofp are compared to get(55)p0:v0x,t=maxex−1,0e−ktα/α,p1:v1x,t=Lα−11s+kLα∂2v0x,t∂x2+k−1∂v0x,t∂x=ktααex−ktα/α,p2:v2x,t=Lα−11s+kLα∂2v1x,t∂x2+k−1∂v1x,t∂x=12!Ktαα2ex−ktα/α,p3:v3x,t=Lα−11s+kLα∂2v2x,t∂x2+k−1∂v2x,t∂x=13!ktαα3ex−ktα/α.
Similarly, approximations can further be obtained as follows:(56)p4:v4x,t=14!ktαα4ex−ktα/α,p5:v5x,t=15!ktαα5ex−ktα/α,and so on.
Substitute the above values in the following equation:(57)vx,t=v0x,t+v1x,t+v2x,t+v3x,t+⋯=maxex−1,0e−ktα/α+ex−ktα/αktαα+12!ktαα2+13!ktαα3+14!ktαα4+⋯=maxex−1,0e−ktα/α+ex1−e−ktα/α.
The equation (35) exact solution with I.C: vx,0=ex−1,0, for α=1 as a special case, is found as follows:(58)vx,t=ex−1,0e−kt+ex1−e−kt,which is the same solution in [24].
In Figure7, the con-LTHPM’s solution for various values of α, i.e., α=0.5,0.7,0.9,1 is represented. In this example, F-BSEs is considered for 0<α≤1.Figure 7
Con-LTHPM’s solutions of the F-BSEs.


## 5. Conclusion

The approximate analytical solutions have been obtained in this work for time F-PDEs, using a numerical method based on the con derivatives, a concept widely used in the field of applications in recent years. We have constructed a con-version of the HPM via LT. This technique’s efficiency and approximation have been verified through four important problems. From the illustrative examples, the results using this technique coincide the corresponding exact solution. As a result, our technique can be applicable to many initial-value problems and F-PDEs including linear and nonlinear ones. In fact, future research will focus on studying this conformable methodology of important classic problems such as the in-homogeneous fractional cable equation, the Burgers equation with fractional order of space and time, or the in-homogeneous nonlinear Klein–Gordon equation.



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