TY - JOUR

T1 - Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions

AU - Restrepo, Joel E.

AU - Ruzhansky, Michael

AU - Suragan, Durvudkhan

N1 - Funding Information:
This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09058317 ). The first and third authors were supported by the Nazarbayev University Program 091019CRP2120 . Joel E. Restrepo also thanks to Colciencias and Universidad de Antioquia (Convocatoria 848 - Programa de estancias postdoctorales 2019) for their support. The second author was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the EPSRC Grant EP/R003025.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/8/15

Y1 - 2021/8/15

N2 - Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula [1, Theorem 4.1] to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.

AB - Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite series of fractional integro-differential operators, which can be widely and efficiently used for analytic and computational purposes. In the case of constant coefficients, the solution can be expressed in terms of the multivariate Mittag-Leffler functions. In particular, the obtained result extends the Luchko-Gorenflo representation formula [1, Theorem 4.1] to a general class of linear fractional differential equations with variable coefficients, to complex fractional derivatives, and to fractional derivatives with respect to a given function.

KW - Fractional calculus

KW - Fractional differential equations

KW - Fractional integro-differential operators

KW - Mittag-Leffler functions

KW - Variable coefficients

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U2 - 10.1016/j.amc.2021.126177

DO - 10.1016/j.amc.2021.126177

M3 - Article

AN - SCOPUS:85102970711

SN - 0096-3003

VL - 403

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 126177

ER -