On solutions of linear fractional differential equations and systems thereof

TL;DR

This paper derives exact invariant solutions for classes of linear fractional differential equations and systems, expressed through special functions, generalizing previous results and connecting to diffusion-wave equations.

Contribution

It provides new exact solutions to linear fractional differential equations using special functions, extending known results and linking to diffusion-wave equations.

Findings

Solutions expressed in Mittag-Leffler, Wright, and Fox H-functions

Invariant solutions obtained via transformations

Contains previously known results as special cases

Abstract

It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling transformations. We then derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.