H.M.Fahad: Survey of fractional calculus with respect to functions & general transmutation relations

Date: Friday, 30 August, 2024 - 15:00 to 16:00 CEST
Title : A historical survey of fractional calculus with respect to functions and general transmutation relations
Speaker : Hafiz Muhammad Fahad, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy
Organizers : Pavan Pranjivan Mehta* and Arran Fernandez**
SISSA, International School of Advanced Studies, Italy
** Eastern Mediterranean University, Northern Cyprus

Keywords: Historical survey, Fractional calculus with respect to functions, Algebraic conjugation, Fractional differential equations

Abstract

Derivatives and integrals of one function with respect to another function are well known from basic calculus, using the chain rule and Riemann–Stieltjes integration. The fractional-order versions of these ideas give rise to a theory of fractional calculus with respect to functions, which is often referred to nowadays as $\psi$-fractional calculus. The history of these operators is longer than most researchers realise, as they have been discovered and re-discovered several times through the decades. In this talk, we trace the full history of fractional calculus with respect to functions. Moreover, we also consider a very general class of fractional calculus operators, given by transmuting the classical fractional calculus along an arbitrary invertible linear operator $S$. Specific cases of $S$, such as shift, reflection, and composition operators, give rise to well-known settings such as that of fractional calculus with respect to functions, and allow simple connections between left-sided and right-sided fractional calculus with different constants of differintegration. We define general transmuted versions of the Laplace transform and convolution of functions, and discuss how these ideas can be used to solve fractional differential equations in more general settings.

This is a joint work with Arran Fernandez [1,2,3].

Biography

Dr. Hafiz Muhammad Fahad is an Assistant Professor in the Department of Mathematics at the National University of Sciences and Technology, Islamabad, Pakistan. He completed his PhD in 2023 from the Eastern Mediterranean University, Northern Cyprus.

Dr. Fahad's research focuses on special functions, operational and fractional calculus. He investigates the mathematical structure of fractional calculus, connections between different operators and their generalisations, and fractional calculus with respect to functions and corresponding differential equations.

His expertise has earned him invitations to prestigious research programmes organised by institutions like the Isaac Newton Institute for Mathematical Sciences in Cambridge, United Kingdom, and the University of Ghent Analysis \& PDE centre in Ghent, Belgium.

Bibliography

[1] Arran Fernandez and Hafiz Muhammad Fahad. “A historical survey of fractional calculus with respect to functions”. In: Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA 2023), (accepted).

[2] Arran Fernandez and Hafiz Muhammad Fahad. “General Transmutation Relations and Their Applications”. In: Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA 2024), (accepted).

[3] Arran Fernandez and Hafiz Muhammad Fahad. “On the importance of conjugation relations in fractional calculus”. In: Computational and Applied Mathematics 41.6 (2022), p. 246