A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets

: In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Several analytical and numerical methods have been used to solve partial differential equations (PDEs) with local fractional derivative operators (LFDOs), such as the Adomian decomposition method [13][14][15], variational iteration method [16][17][18][19][20][21][22], differential transform method [23,24], series expansion method [25], Sumudu transform method [26], Laplace transform method [27], reduced differential transform method [28], Laplace variational iteration method [29], Fourier series method [30], and homotopy perturbation method [31]. Our aim is to present the coupling method of local fractional Laplace transform (LFLT) and homotopy perturbation method (HPM), which we call the local fractional Laplace homotopy perturbation method (LFLHPM), and use it to solve differential Helmholtz and coupled Helmholtz equations on Cantor sets within a local fractional operator. This paper is organized as follows. In Section 2, the basic mathematical tools of local fractional calculus are introduced. The analysis of the proposed method is given in Section 3. Then, in Sections 4 and 5, the proposed method is implemented to solve differential Helmholtz and coupled Helmholtz equations on Cantor sets within a local fractional operator. Finally, concluding remarks are presented in Section 6.

Definition 4. The Mittage-Leffler function and hyperbolic sine are, respectively, defined by
The properties for the local fractional Laplace transform used in the paper are given as follows:

Analysis of LFLHPM
The local fractional homotopy perturbation method (LFHPM) has been developed and applied to solve a class of local fractional partial differential equations [31,34]. Based on it, we suggest a new analytical method.
Let us consider the following partial differential equation with local fractional derivative: where = , is a linear local fractional operator, and ( , ) is the source term.
Applying the local fractional Laplace transform (LFLT) to Equation (6), it gives the following: Using the inversion of LFLT on Equation (7), we have the following: where ( , ) represents the term arising from the source term and the prescribed initial conditions. Now, we apply the LFHPM [31]: Using Equation (9) in Equation (8), it yields the following result: On equating the multipliers of same powers of the parameter p of Equation (10) Applying the LFLT on both sides (12), subject to initial condition (13), we have the following: The inversion of LFLT implies that Now, applying LFHPM, we obtain the following: The result is the same as the one which is obtained by the local fractional variational iteration method [35].

Application of LFLHPM for Coupled Helmholtz Equations
Applying the LFLT on both sides (18), subject to initial condition (19), we have the following: The inversion of LFLT implies that Now, applying LFHPM, we obtain the following: Using Equation (22) in Equation (21), it yields the following result: The result is the same as the one which is obtained by the local fractional variational iteration transform method [36].

Conclusions
In this work, the LFLHPM was successfully applied to finding the approximate solution of Helmholtz and coupled Helmholtz equations involving LFDOs. A comparison was made to show that the method has a small computational size in comparison with the computational size required in other numerical methods, such as the local fractional variational iteration method and the local fractional variational iteration transform method. The method is very powerful and efficient in finding analytical as well as numerical solutions for wide classes of linear and nonlinear local fractional PDEs.
Author Contributions: H.K.J. wrote some sections of the manuscript; D.B. prepared some other sections of the paper and analyzed. All authors have read and approved the final manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest