Abstract
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.
Highlights
The use of fractional order derivatives is nowadays widespread in many fields
It is a fact that the theoretical analysis of fractional differential equations (FDEs) is much more advanced than finding their numerical solution
We address the numerical solution of multiterm fractional differential equations (MTFDEs), that is, FDEs in which multiple fractional derivatives are involved
Summary
The use of fractional order derivatives is nowadays widespread in many fields. the possibility to use any real order improves the modeling of several phenomena in physics, engineering and many application areas. We address the numerical solution of multiterm fractional differential equations (MTFDEs), that is, FDEs in which multiple fractional derivatives are involved. These turn out to be very helpful in many fields, to model complex multirate physical processes. We show the effectiveness of the matrix approach when solving MTFDEs, both in terms of execution time and in terms of accuracy, and in comparison with some well-established numerical methods. The tests we present show an excellent accuracy, and the approach can be favorably applied to solve more general multiterm FDEs. In particular, among the available numerical methods for MTFDEs, we consider the product integration (PI) rules.
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