Approximate solutions of time and time-space fractional wave equations with variable coefficients

Abstract

We consider the Cauchy problem for fractional evolution equations with time fractional derivative as Caputo fractional derivative of order and space variable coefficients on an unbounded domain. The space derivatives that appear in the equations are of integer or fractional order such as the left and the right Liouville fractional derivative as well as the Riesz fractional derivative. In order to solve this problem we introduce and develop generalized uniformly continuous solution operators and use them to obtain the unique solution on a certain Colombeau space. In our solving procedure, instead of the originate problem we solve a certain approximate problem, but therefore we also prove that the solutions of these two problems are associated. At the end, we illustrate the applications of the developed theory by giving some appropriate examples.

Keywords:

• Fractional evolution equations
• fractional wave equations
• generalized Colombeau solution operators
• fractional Duhamel principle
• fractional derivatives
• Mittag–Leffler type functions

AMS Subject Classifications:

• 35R11
• 46F30
• 26A33

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Ministry of Science, Republic of Serbia [grant number OI174024,III44006]; Provincial secretariat for science and technological development, Autonomous Province of Vojvodina [grant number APV 114-451-1084/2014].