Approximate analytical solutions of Goursat problem within local fractional operators

Abstract

The local fractional differential transform method (LFDTM) and local fractional decomposition method (LFDM) are applied to implement the homogeneous and nonhomogeneous Goursat problem involving local fractional derivative operators. The approximate analytical solution of this problem is calculated in form of a series with easily computable components. Examples are studied in order to show the accuracy and reliability of presented methods. We demonstrate that the two approaches are very effective and convenient for finding the analytical solutions of partial differential equations with local fractional derivative operators.

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

“…6,8 Recently, the local fractional calculus [9][10][11] was developed to describe the mathematical problems in the fractal media. For example, the fractal Goursat problem was proposed in Baleanu et al 12 The fractal-space diffusion equation was reported in previous works. 13,14 The fractal acoustic wave equation was discussed in Ray.…”

The Yang Laplace transform- DJ iteration method for solving the local fractional differential equation

“…In the following, we try to solve (3.2) by the DJ method [1]. According to the DJ method, we can construct the following equation:…”

Solving Helmholtz Equation with Local Fractional Derivative Operators

“…In recent years, many of the approximate and analytical methods have been utilized to solve the PDEs with LFDOs such as the Adomian decomposition method [3][4][5], variational iteration method [6][7][8][9][10][11], differential transform method [12,13], series expansion method [14][15][16], Sumudu transform method [17], Fourier transform method [18], function decomposition method [19,20], Laplace transform method [21,22], reduce differential transform method [23,24], homotopy perturbation Sumudu transform [25], and the existence and uniqueness of solutions for local fractional differential equations [26,27].…”