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

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  • Restrepo, Joel E.
  • Ruzhansky, Michael
  • Suragan, Durvudkhan

Abstract

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.

Suggested Citation

  • Restrepo, Joel E. & Ruzhansky, Michael & Suragan, Durvudkhan, 2021. "Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    • Handle: RePEc:eee:apmaco:v:403:y:2021:i:c:s0096300321002678
    DOI: 10.1016/j.amc.2021.126177
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    Citations

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    Cited by:

    1. Restrepo, Joel E. & Suragan, Durvudkhan, 2021. "Hilfer-type fractional differential equations with variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    2. Irgashev, B.Yu., 2023. "Initial boundary value problem for a high-order equation with two lines of degeneracy with the Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    3. Zaky, M.A. & Hendy, A.S. & Suragan, D., 2022. "A note on a class of Caputo fractional differential equations with respect to another function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 289-295.
    4. Du, Feifei & Lu, Jun-Guo, 2021. "Explicit solutions and asymptotic behaviors of Caputo discrete fractional-order equations with variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    5. Fernandez, Arran & Restrepo, Joel E. & Suragan, Durvudkhan, 2022. "On linear fractional differential equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    6. Fahad, Hafiz Muhammad & Fernandez, Arran, 2021. "Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    7. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.

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