Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

Abstract

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

Solving Helmholtz Equation with Local Fractional Derivative Operators

“…According to the rule of LFVIM, the correction local fractional functional for Equation 16is constructed as [6][7][8][9][10]30,31]:…”

“…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].…”

An efficient hybrid technique for the solution of fractional-order partial differential equations

“…In recent decades, many of the numerical and analytical techniques have been implemented to solve fractional-order PDEs, such as the fractional variational iteration method [23,34,42,44,45], fractional differential transform method [25,36,46], fractional series expansion method [9,29], fractional Sumudu variational iteration method [20,31], fractional natural decomposition method [32,38], fractional Sumudu decomposition method [17,30,33], fractional Sumudu homotopy perturbation method [28], fractional reduce differential transform method [24,26,41], fractional Adomian decomposition method [16,21,47], fractional Laplace decomposition method [27], fractional Laplace homotopy perturbation method [14], fractional Laplace variational iteration method [13,15,18,35,37], variational iteration method [4][5][6][7][8] and local mesh less УДК 517.95 2020 Mathematics Subject Classification: 34K37, 45J99, 34A08.…”

Exact Solution of Two-Dimensional Fractional Partial Differential Equations

“…The analytical solutions for these dynamical equations play an important role in many phenomena in optics; fluid mechanics; plasma physics and hydrodynamics [7][8][9][10]. In recent years, many authors have investigated partial differential equations of fractional order by various techniques such as homotopy analysis technique [11,12], variational iteration method [13][14][15], homotopy perturbation method [16], homotopy perturbation transform method [17], Laplace variational iteration method [18][19][20], reduce differential transform method [21], Laplace decomposition method [22] and other methods [23][24][25][26][27].…”