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

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. +en, 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.


Introduction
e initial idea of fractional derivative in history goes back to the late 17 th century, when the French mathematician L'Hopital concerned with the meaning of (d n y/dx n ), 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. e 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. e 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. ey 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. ey 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. e 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.
e main purpose of our study is to construct the Laplace transform homotopy perturbation method (LTHPM) for F-DEs using conformable derivatives. e article is outlined as follows: some fundamental notions of conformable derivatives are revisited in Section 2. 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). e conclusion of this work is discussed in Section 5.

Basic Notions
is section presents some essential concepts of conformable derivatives, which will be necessary in our investigation.
As a result, we have 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.
en, the F-LT of order α beginning from 0 of f is expressed as Remark 1. e F-LT for the conformable derivatives is expressed as us, we have (ii) Let q be a constant: (iii) If c and q are arbitrary constants, where M and c are the positive real constants and α ∈ (0, 1], ∀ sufficiently large t.

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: with the initial condition (I.C): where u is the two variables function, M α � (z α /zt α ) 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 g(x, t) is a nonhomogeneous term. By solving for M α u(x, t), equation (14) can be written as By applying the con-LT to equation (16), we obtain From Remark 1, the above equations can be rewritten as By substituting I.C, equation (18) can be written as Furthermore, by applying inverse con-LT to equation (20) where the term has been arisen from the source term, and the prescribed I.C is denoted by G(x, t).
e HPM suggests the solution u(x, t) to be decomposed into the infinite series of components [18,19]: and nonlinear term Nu(x, t) is decomposed to for some He's polynomials H n (u) [17,20] which are given by By the substitution of equations (21) and (22) into equation (20), we have Mathematical Problems in Engineering which is the coupled con-FLT and HPM via He's polynomials. e approximation can be easily obtained by a comparison of all coefficients like powers of p as follows: with I.C: v(x, 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 Using I.C and inverse con-LT, equation (27) e HPM is applied to obtain where H n (v) represents He's polynomials. e 1 st three parts of H n (v) are and so on.
e coefficient of power of p is equated in equation (33) to get Similarly, the approximations can further be obtained as follows: and so on. Substitute the above values in the following equation: v e equation (26) exact solution with I.C: v(x, 0) � ae −x for α � 1 as a special case is found as follows: which is the same solution in [21]. is result is verified graphically in Figure 1.

Remark 3.
is 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 Using I.C and inverse con-LT, equation (36) reduces to

Mathematical Problems in Engineering
e coefficients of the power of p are compared as follows: Similarly, approximations can further be obtained as follows: and so on. Substitute the above values into the following equation: (41) e equation (35) exact solution with I v(x, 0) � e 2x , for α � 1 as a special case, is found as follows: which is the same solution in [22]. is result is verified graphically, as shown in Figure 3. In Figure 4, the con-LTHPM's solution for various values of α, i.e., α � 0.7, 0.9 is represented. 6 Mathematical Problems in Engineering Example 3. Consider F-DCEs as follows: with I.C: v(x, 0) � (1/10)e cos x− 11 . If α � 1, then equation (43) becomes an example of the classical DCEs [23].

Remark 4.
e 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 Using I.C and inverse con-LT, equation (44) reduces to v(x, t) � 1 10 e cos x− 11 e HPM is applied to obtain e coefficients of the power of p are compared here as follows: e cos x− 11 ,    Mathematical Problems in Engineering Similarly, approximations can further be obtained as follows: and so on. Substitute the above values in the following equation: e equation (43) exact solution with I.C: v(x, 0) � (1/10)e cos x−11 , for α � 1 as a special case, is found as follows: which is the same solution with [23]. is result is also verified graphically, as shown in Figure 5.
Remark 5. Equation (51) contains only two dimensionless parameters k � (2r/σ 2 ), where k represents the balance between the rates of interest and the variability of the return on stocks and the dimensionless time to expiry (σ 2 T/2).
By taking con-LT on equation (51) both sides and from the properties of con-LT, equation (51) reduces to zv(x, t) zx . (52) Using I.C and inverse con-LT, equation (52) e HPM is applied to obtain e coefficients of the power of p are compared to get Similarly, approximations can further be obtained as follows: and so on. Substitute the above values in the following equation: which is the same solution in [24]. In Figure 7, 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.

Conclusion
e 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. is 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.

Data Availability
No data were used to support this study.