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Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

DOI: 10.1155/2014/535048

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Abstract:

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative. 1. Introduction Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger’s, the telegraph, the Advection, the Burgers, the KdV, the Boussinesq, and the Fisher equations and others [1]. Recently, the fractional calculus theory was recognized to be a good tool for modeling complex problems demonstrating its applicability in numerical scientific disciplines. Boundary value problems for the fractional differential equations have been the focus of several studies due to their frequent appearance in various areas, such as fractional diffusion and wave [2], fractional telegraph [3], fractional KdV [4], fractional Schr?dinger [5], fractional evolution [6], fractional Navier-Stokes [7], fractional Heisenberg [8], fractional Klein-Gordon [9], and fractional Fisher equations [10]. Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations (see, e.g., [1–36] and the references therein). The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16, 17], the local fractional decomposition [18], and the generalized local fractional Fourier transform [19] methods. Recently, the wave equation on Cantor sets (local fractional wave equation) was given by [35] where the operators are local fractional ones [16–19, 35, 36]. Following (1), a wave equation on Cantor sets was proposed as follows [36]: where is a fractal wave function. In this paper, our purpose is to compare the local fractional variational iteration and decomposition methods for solving the local fractional differential equations. For illustrating the concepts we adopt one example for solving the wave equation on Cantor sets with local fractional operator. Bearing these ideas in mind, the paper is organized as follows. In Section 2, we present basic definitions and

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