TY - JOUR

T1 - A note on a class of Caputo fractional differential equations with respect to another function

AU - Zaky, M. A.

AU - Hendy, A. S.

AU - Suragan, D.

N1 - Funding Information:
The authors are grateful to the handling editor and the anonymous referees for their constructive feedback and helpful suggestions, which highly improved the presentation of the paper. D. Suragan and Mahmoud A. Zaky were funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09058317 ). Mahmoud A. Zaky is also supported by the National Research Centre of Egypt and Nazarbayev University program 091019CRP2120 . Ahmed S. Hendy wishes to acknowledge the support of the RSF grant, project 22-21-00075.
Publisher Copyright:
© 2022 International Association for Mathematics and Computers in Simulation (IMACS)

PY - 2022/6

Y1 - 2022/6

N2 - The mathematical analysis and solutions for a class of ψ-Caputo fractional differential equations are discussed. Assuming that ψ(t) is strictly monotone and armed by the possibility of converting the ψ-Caputo fractional differential equations with respect to another function ψ to its Caputo counterpart by a mapping transformation, the solutions of the ψ-Caputo fractional differential equations can be deduced from the solution representation for the Caputo version via an inverse transformation. We show that the mapping transformation for such derivatives is extremely useful in practical applications. The representation of solutions for constant order time ψ-Caputo fractional diffusion equation and variable order ψ-Caputo fractional mobile-immobile diffusion equation is investigated and the regularity estimates are deduced accordingly.

AB - The mathematical analysis and solutions for a class of ψ-Caputo fractional differential equations are discussed. Assuming that ψ(t) is strictly monotone and armed by the possibility of converting the ψ-Caputo fractional differential equations with respect to another function ψ to its Caputo counterpart by a mapping transformation, the solutions of the ψ-Caputo fractional differential equations can be deduced from the solution representation for the Caputo version via an inverse transformation. We show that the mapping transformation for such derivatives is extremely useful in practical applications. The representation of solutions for constant order time ψ-Caputo fractional diffusion equation and variable order ψ-Caputo fractional mobile-immobile diffusion equation is investigated and the regularity estimates are deduced accordingly.

KW - Analytical solution

KW - Existence and uniqueness

KW - Fractional derivative with respect to another function

KW - Variable-order derivative

KW - Well-posedness

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U2 - 10.1016/j.matcom.2022.01.016

DO - 10.1016/j.matcom.2022.01.016

M3 - Article

AN - SCOPUS:85124422295

SN - 0378-4754

VL - 196

SP - 289

EP - 295

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

ER -