Fundamental solutions of the fractional two-parameter telegraph equation

Abstract

This paper is intended to investigate a fractional telegraph equation of the form

with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.

Keywords:

• fractional telegraph equation
• Riemann–Liouville fractional integrals and derivatives
• generalized Mittag–Leffler function

AMS Subject Classification :

• 45J05
• 26A33
• 33E12

Acknowledgements

The work of S.Y. was supported by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT -Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011. The work of M.M.R. was supported by Fundação para a Ciência e a Tecnologia via grant SFRH/BPD/73537/2010. The second author was also supported by FEDER founds through COMPETE–Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese founds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.