Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

• Published in: Mathematics (Basel) Vol. 6; no. 1; p. 7
• Main Author: Popolizio, Marina
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2018
• Subjects: Differential equations; fractional calculus; fractional differential equations; Mathematical models; matrix function; Mittag–Leffler function; multiterm differential equations; Numerical methods
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math6010007

Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.