D Alemberts solution of fractional wave equations using complex fractional transformation

TL;DR

This paper introduces a complex fractional transformation method to solve fractional wave equations analytically, providing a simpler alternative to traditional techniques and revealing solutions as scaled traveling waves.

Contribution

The paper presents a novel complex fractional transformation approach to solve fractional wave equations analytically, improving ease of solution over existing methods.

Findings

Solutions are obtained in analytical form.

Solutions manifest as scaled traveling waves.

Method is simpler than fractional characteristics approach.

Abstract

Fractional wave equation arises in different type of physical problems such as the vibrating strings, propagation of electro-magnetic waves, and for many other systems. The exact analytical solution of the fractional differential equation is difficult to find. Usually Laplace-Fourier transformation method, along with methods where solutions are represented in series form is used to find the solution of the fractional wave equation. In this paper we describe the D Alembert s solution of the fractional wave equation with the help of complex fractional transform method. We demonstrate that using this fractional complex transformation method, we obtain the solutions easily as compared to fractional method of characteristics; and get the solution in analytical form. We show that the solution to the fractional wave equation manifests as travelling waves with scaled coordinates, depending on the considered fractional order value.