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.
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 , 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 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). 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. Given as a function. Then, the order conformable derivatives [1] is expressed as, . If is -differentiable (-DifF) in some , , and exists, then it is expressed as
Theorem 1 (See [1]). If a function is -DifF at , , then is continuous (ContF) at .
Theorem 2 (See [1]). Suppose that , and are -DifFs at a point . Then, we get(i), (ii), (iii), constant function (iv)(v)(vi)If is DifF, then From the above definition, the conformable derivatives of some functions are expressed as(i)(ii)(iii)(iv), In addition, some helpful formulas are expressed as(i)(ii)(iii)(iv)
Definition 2. The left-conformable derivative beginning from of function of order is expressed as [2]when , we have . If is -DifF in some , then we get
Theorem 3 (Chain Rule) [2]. Suppose that be left -DifFs, where . Assume that . is and . Thus, we obtainIf , then
Theorem 4 (See [2]). Supposed that is infinitely -DifF, for some at a neighborhood of a point . Then, has a conformable power series expansion as
Hence, implies the application of conformable derivatives times.
The -conformable integral of a function is beginning from .
Definition 3 (See [1]). , where this is a usual Riemann improper integral, .
As a result, we have
Theorem 5. , for , where is any ContF in domain.
Lemma 1. Suppose that is DifF and . Then, , we get [2]
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 that and are real valued functions. Then, the F-LT of order beginning from 0 of is expressed as
Remark 1. The F-LT for the conformable derivatives is expressed asThus, we have
Theorem 6. If exists for , then(i)If a constant, then(ii)Let be a constant:(iii)If and are arbitrary constants,
Definition 5. A function is considered as a conformable exponentially bounded if it satisfies , where and are the positive real constants and , sufficiently large .
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:with the initial condition (I.C):where is the two variables function, is a linear operator (LO) with conformable derivative of order , is the LO’s other part, is a nonlinear operator (NLO), and is a nonhomogeneous term. By solving for , 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 (19), we getwhere the term has been arisen from the source term, and the prescribed I.C is denoted by .
The HPM suggests the solution to be decomposed into the infinite series of components [18, 19]:and nonlinear term is decomposed tofor some He’s polynomials [17, 20] which are given by
By the substitution of equations (21) and (22) into equation (20), we havewhich 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 as follows:
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:with I.C: . If , 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 toUsing I.C and inverse con-LT, equation (27) reduces toThe HPM is applied to obtainwhere represents He’s polynomials.
The 1st three parts of areand so on.
The coefficient of power of is equated in equation (33) to getSimilarly, the approximations can further be obtained as follows:and so on.
Substitute the above values in the following equation:The equation (26) exact solution with I.C: for as a special case is found as follows:which is the same solution in [21]. This result is verified graphically in Figure 1.
In Figure 2, the con-LTHPM’s solution for various values of , i.e., is represented. The F-GDEs is considered here with parameter .
Example 2. Consider F-NWSEs as follows:with I.C: . If , 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 toUsing I.C and inverse con-LT, equation (36) reduces toThe HPM is applied to obtainThe coefficients of the power of are compared as follows:Similarly, approximations can further be obtained as follows:and so on.
Substitute the above values into the following equation:The equation (35) exact solution with I, for as a special case, is found as follows:which is the same solution in [22]. This result is verified graphically, as shown in Figure 3.
In Figure 4, the con-LTHPM’s solution for various values of , i.e., is represented.
Example 3. Consider F-DCEs as follows:with I.C: . If , 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 toUsing I.C and inverse con-LT, equation (44) reduces toThe HPM is applied to obtainThe coefficients of the power of are compared here as follows:Similarly, approximations can further be obtained as follows:and so on.
Substitute the above values in the following equation:The equation (43) exact solution with I.C: , for as a special case, is found as follows:which is the same solution with [23]. This result is also verified graphically, as shown in Figure 5.
In Figure 6, the con-LTHPM’s solution for various values of , i.e., is represented.
Example 4. Consider the F-BSEs:with I.C: . If , then equation (51) becomes the classical linear BSEs [24].
Remark 5. Equation (51) contains only two dimensionless parameters , where represents the balance between the rates of interest and the variability of the return on stocks and the dimensionless time to expiry .
By taking con-LT on equation (51) both sides and from the properties of con-LT, equation (51) reduces toUsing I.C and inverse con-LT, equation (52) reduces toThe HPM is applied to obtainThe coefficients of the power of are compared to getSimilarly, approximations can further be obtained as follows:and so on.
Substitute the above values in the following equation:The equation (35) exact solution with I.C: , for as a special case, is found as follows:which is the same solution in [24].
In Figure 7, the con-LTHPM’s solution for various values of , i.e., is represented. In this example, F-BSEs is considered for .
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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Sajad Iqbal involved in actualization, methodology, formal analysis, validation, investigation, and initial draft. Mohammed K. A. Kaabar involved in actualization, methodology, formal analysis, validation, investigation, initial draft, and supervision of the original draft and editing. Francisco Martínez involved in actualization, validation, methodology, formal analysis, investigation, and initial draft. All authors read and approved the final version.