On the Evolution of Fractional Diffusive Waves

TL;DR

This paper investigates fractional diffusion-wave equations by analyzing their fundamental solutions and simulating responses to different initial conditions, bridging diffusion and wave phenomena through fractional calculus.

Contribution

It provides a detailed analysis and simulation of fractional diffusion-wave processes, including fundamental solutions and their behavior with various initial inputs.

Findings

Derived fundamental solutions for fractional diffusion-wave equations.

Simulated responses for delta and box initial functions.

Clarified the role of fractional calculus in modeling these phenomena.

Abstract

In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time derivatives in the above standard equations with pseudo-differential operators interpreted as derivatives of non integer order (nowadays misnamed as of fractional order) we are lead to generalized processes of diffusion that may be interpreted as slow diffusion and interpolating between diffusion and wave propagation. In mathematical physics we may refer these interpolating processes to as fractional diffusion-wave phenomena. In this work we analyze and simulate both the situations in which the input function is a Dirac delta generalized function and a box function, restricting ourselves to the Cauchy problem. In the firsrst case we get the fundamental solutions (or Green functions) of the problem whereas in the latter case the solutions are obtained by a space convolution of the Green function with the input function. In order to clarify the matter for the non-specialist readers, we briefly recall the basic and essential notions of the fractional calculus (the mathematical theory that regards the integration and differentiation of noninteger order) with a look at the history of this discipline.