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arXiv · 2108.06361

On tempered fractional calculus with respect to functions and the associated fractional differential equations

Ashwini D. Mali·Kishor D. Kucche·Arran Fernandez·Hafiz Muhammad Fahad
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Abstract

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the $\Psi$-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the $\Psi$-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann--Liouville type and Caputo type cases of these operators. Furthermore, we study some nonlinear fractional differential equations involving $\Psi$-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle, and proving stability results by using Gr\"onwall type inequalities.

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Ashwini D. Mali, Kishor D. Kucche, Arran Fernandez, Hafiz Muhammad Fahad. 2021-08-13. On tempered fractional calculus with respect to functions and the associated fractional differential equations. https://doi.org/10.1002/mma.8441

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