A New Representation for the Solutions of Fractional Differential Equations with Variable Coefficients

Arran Fernandez; Joel E. Restrepo; Durvudkhan Suragan

doi:10.1007/s00009-022-02228-7

A New Representation for the Solutions of Fractional Differential Equations with Variable Coefficients

Arran Fernandez
, Joel E. Restrepo
, Durvudkhan Suragan

Nazarbayev University

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

A recent development in differential equations with variable coefficients by means of fractional operators has been a method for obtaining an exact solution by infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here, we show a new representation of the solution function, as a convergent series of single fractional integrals, which is computationally simpler and which we believe will quickly prove its usefulness in future computational work for applications. In particular, for constant coefficients, the solution is given by the Mittag-Leffler function. We also show some applications in Cauchy problems involving both time-fractional and space-fractional operators and with time-dependent coefficients.

Original language	English
Article number	27
Journal	Mediterranean Journal of Mathematics
Volume	20
Issue number	1
DOIs	https://doi.org/10.1007/s00009-022-02228-7
Publication status	Published - Feb 2023

Funding

The second and third authors were supported by the Nazarbayev University Program 091019CRP2120. The second author was also supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations and the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant Number 01M01021).

Keywords

fractional Cauchy problems
Fractional differential equations
fractional integrals
time-dependent coefficients

ASJC Scopus subject areas

General Mathematics

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10.1007/s00009-022-02228-7

Cite this

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title = "A New Representation for the Solutions of Fractional Differential Equations with Variable Coefficients",

abstract = "A recent development in differential equations with variable coefficients by means of fractional operators has been a method for obtaining an exact solution by infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here, we show a new representation of the solution function, as a convergent series of single fractional integrals, which is computationally simpler and which we believe will quickly prove its usefulness in future computational work for applications. In particular, for constant coefficients, the solution is given by the Mittag-Leffler function. We also show some applications in Cauchy problems involving both time-fractional and space-fractional operators and with time-dependent coefficients.",

keywords = "fractional Cauchy problems, Fractional differential equations, fractional integrals, time-dependent coefficients",

author = "Arran Fernandez and Restrepo, \{Joel E.\} and Durvudkhan Suragan",

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AU - Suragan, Durvudkhan

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PY - 2023/2

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N2 - A recent development in differential equations with variable coefficients by means of fractional operators has been a method for obtaining an exact solution by infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here, we show a new representation of the solution function, as a convergent series of single fractional integrals, which is computationally simpler and which we believe will quickly prove its usefulness in future computational work for applications. In particular, for constant coefficients, the solution is given by the Mittag-Leffler function. We also show some applications in Cauchy problems involving both time-fractional and space-fractional operators and with time-dependent coefficients.

AB - A recent development in differential equations with variable coefficients by means of fractional operators has been a method for obtaining an exact solution by infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here, we show a new representation of the solution function, as a convergent series of single fractional integrals, which is computationally simpler and which we believe will quickly prove its usefulness in future computational work for applications. In particular, for constant coefficients, the solution is given by the Mittag-Leffler function. We also show some applications in Cauchy problems involving both time-fractional and space-fractional operators and with time-dependent coefficients.

KW - fractional Cauchy problems

KW - Fractional differential equations

KW - fractional integrals

KW - time-dependent coefficients

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UR - https://www.scopus.com/pages/publications/85143727368#tab=citedBy

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A New Representation for the Solutions of Fractional Differential Equations with Variable Coefficients

Abstract

Funding

Keywords

ASJC Scopus subject areas

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