Solution Of Fractional Wave Equation Research Articles

Analytical solutions of fractional wave equation with memory effect using the fractional derivative with exponential kernel

Analytical solutions of the fractional wave equation via Caputo-Fabrizio fractional derivative are presented in this paper. For this analysis, three cases are considered, the classical, the damped and the damped with a source term defined by fractional wave equations. We show that these solutions are special cases of the time fractional equations with exponential law. Illustrative examples are presented.

Caputo–Riesz-Feller fractional wave equation: analytic and approximate solutions and their continuation

In this article a time-space fractional wave equation is studied. In the new proposed fractional model, the second-order time derivative is replaced with a fractional derivative in Caputo sense, and the second-order space derivative is replaced with a Riesz-Feller derivative defined on infinite space domain. The fundamental solution of fractional wave equation is obtained in terms of Mittag-Leffler function in two parameters, that is by using the joint Laplace–Fourier transform method. We prove the continuation of the solution of the generalized Riesz wave equation as the skewness parameter tends to zero to the one of the corresponding fractional wave equation with classical Riesz derivative, this is accomplished by using the Lebesgue’s dominated convergence theorem. The optimal homotopy analysis method (OHAM) is employed to obtain semi-analytic solution of a newly proposed initial-value fractional wave problem, considering three numerical simulations. The continuation of the optimal solution and its dependence on the fractional derivative parameters are investigated. The study reveals that the OHAM is reliable and effective in case of fractional Riesz-Feller operator represents the fractional Laplacian operator.

Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula

In this article, a numerical study is introduced for solving the fractional wave equations by using an efficient class of finite difference methods. The proposed scheme is based on the Hermite formula. The stability and the convergence analysis of the proposed methods are given by a recently proposed procedure similar to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Finally, a numerical example is presented to confirm the theoretical results.