Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators

Abstract

The local fractional Laplace variational iteration method (LFLVIM) is employed to handle the diffusion and wave equations on Cantor set. The operators are taken in the local sense. The nondifferentiable approximate solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

Non-differentiable Solutions for Local Fractional Nonlinear Riccati Differential Equations

“…where ζ is the fractal dimension of the local fractional operator [1][2][3][4][5][6][7][8][9][10][11][12][13], ̟ 0 (µ) = 0 and ̟ 2 (µ) = 0. Our main aim is to study non-differentiable solutions of LFNRDE.…”

Solving Poisson Equation within Local Fractional Derivative Operators

“…We notice that recently local fractional Poisson equation was analyzed in [5]. Recently, the Poisson equation (PE) with local fractional derivative operators (LFDOs) was presented in [6] as follows: where ( , ) is an unknown function, ( ) and ( ) are given functions, and the local fractional derivative operators (LFDOs) of ( ) of order at = 0 are given by In recent years, a many of approximate and analytical methods have been utilized to solve the ordinary and partial differential equations with local fractional derivative operators such as local fractional Adomian decomposition method [7][8][9][10][11], local fractional variational iteration method [6,7,[12][13][14][15], local fractional function decomposition method [8,16], local fractional series expansion method [11,17], local fractional Laplace decomposition method [18,19], local fractional Laplace variational iteration method [20][21][22][23], local fractional homotopy perturbation method [24], local fractional reduce differential transform method [24], local fractional differential transform method [26,27], and local fractional Laplace transform method [28]. Our main purpose of the paper is to utilize the local fractional RDTM and local fractional HPTM to solve the PE with LFDOs.…”

The Laplace series solution for local fractional Korteweg-de Vries equation

“…For example, the diffusion equation [5][6][7][8][9] is a parabolic local fractional PDE. The local fractional wave equation [10,11] is hyperbolic local fractional PDE. The local fractional Laplace equation [12][13][14] is elliptic local fractional PDE.…”