Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

Abstract

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

Solving Poisson Equation within Local Fractional Derivative Operators

“…Taking the local fractional RDTM of (3.11), by using the basic operation in theorems, yields From equations (3.16), (3.20) and (3.10), approximate solution of the given problem equation (3.11) by using local fractional HPTM is the same results as that obtained by the local fractional RDTM and the local fractional variational iteration method [6].…”

“…Applying the Yang-Laplace transform on both sides of (3.1), subject to the initial conditions From equations (3.6) and (3.10), approximate solution of the given problem equation (3.1) by using local fractional HPTM is the same results as that obtained by the local fractional RDTM and the local fractional variational iteration method [6]. (3.12) Figure 2: Plot of the nondifferentiable solution of (3.11) with the parameter = ln 2/ ln 3.…”

“…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 Nondifferentiable Solution for Local Fractional Tricomi Equation Arising in Fractal Transonic Flow by Local Fractional Variational Iteration Method

“…Local fractional derivative was applied to deal with nondifferentiable phenomena arising in mathematical physics [4][5][6][7][8][9]. When the fractal dimension is equal to 1, we obtain the following differential equation:…”

“…Local fractional variational iteration method first structured in [4] was an efficient tool to solve the local fractional differential equations, such as the fractal heat equation [4], the damped and dissipative wave equation in fractal strings [5], the wave equation on Cantor sets [6], the local fractional Poisson equation [7], the local fractional Laplace equation [8], and the local fractional Helmholtz equation [9]. The aim of this paper is to use the local fractional variational iteration method to deal with the local fractional Tricomi equation which arises in fractal transonic flow.…”