Abstract
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.
Highlights
The theory of local fractional calculus was successfully utilized to describe the non-differentiable problems arising in mathematical physics, such as Schrödinger equations [1], the gas dynamic equation [2], the telegraph equation [3], the wave equation [4,5,6,7], Fokker–Planck equations [8,9], Laplace equations [10], Klein–Gordon equations [11], Helmholtz equations [12], and the Goursat problem [13] on Cantor sets
The properties for the local fractional Laplace transform used in the paper are given as follows: 1. LTδ φ(kδ)(μ) = wkδ LTδ φ(μ) − w(k−1)δφ(0) − · · · − φ((k−1)δ)(0)
The result is the same as the one which is obtained by the local fractional variational iteration method [35]
Summary
The theory of local fractional calculus was successfully utilized to describe the non-differentiable problems arising in mathematical physics, such as Schrödinger equations [1], the gas dynamic equation [2], the telegraph equation [3], the wave equation [4,5,6,7], Fokker–Planck equations [8,9], Laplace equations [10], Klein–Gordon equations [11], Helmholtz equations [12], and the Goursat problem [13] on Cantor sets. 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.
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