Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions
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 genera...
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| Published in | Mathematics (Basel) Vol. 6; no. 1; p. 7 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Basel
MDPI AG
01.01.2018
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| ISSN | 2227-7390 2227-7390 |
| DOI | 10.3390/math6010007 |
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| 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. |
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| AbstractList | 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. |
| Author | Popolizio, Marina |
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| Cites_doi | 10.1016/j.amc.2014.12.127 10.1137/140971191 10.1142/9781848163300 10.1023/B:NUMA.0000027736.85078.be 10.1137/0517050 10.1090/S0025-5718-1983-0701626-6 10.1007/978-3-642-14574-2 10.1016/j.cam.2008.04.004 10.1080/01630563.2012.748669 10.1016/j.matcom.2013.09.012 10.1155/2011/298628 10.1016/j.cam.2008.03.025 10.1007/978-3-662-43930-2 10.1007/978-3-7091-2664-6 10.1090/S0025-5718-1985-0804935-7 10.1142/9789814355216 10.1016/0960-0779(95)00125-5 10.1007/s10444-012-9274-z 10.1142/S0218127412500733 10.1515/fca-2016-0060 10.1137/1.9780898717778 10.1016/j.cam.2010.07.008 10.1016/j.cam.2008.04.003 10.1023/A:1021973025166 10.1016/S0096-3003(03)00739-2 10.1137/080738374 10.1016/j.camwa.2016.11.028 10.1016/S0377-0427(02)00558-7 10.3390/math6020016 10.24200/sci.2017.4503 10.1016/j.jmaa.2010.08.048 10.1115/1.3167615 10.1016/0377-0427(84)90027-X 10.1016/j.jcp.2014.09.023 10.1016/j.cam.2005.03.023 |
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