article · Computation
Mathematical modelling often relies on fractional calculus to describe complex physical and biological dynamics. A new fractional derivative has been formulated that integrates definitions using singular and non-singular kernels into a single framework. This unified definition encompasses established singular kernel forms, including the Riemann-Liouville and Caputo derivatives, alongside non-singular formulations such as the Caputo-Fabrizio, Atangana-Baleanu, and generalized Hattaf derivatives. In addition to the derivative, an associated fractional integral has been rigorously introduced. To support the practical solution of related mathematical problems, a novel numerical scheme has been designed to approximate solutions for a class of fractional differential equations governed by this mixed operator. The practical utility of the framework is demonstrated through an application in computational biology.
Fractional differential equations are valuable for capturing memory effects and intricate behaviours in natural systems. By bringing several distinct singular and non-singular derivative types into one overarching definition, this work provides computational researchers with a versatile framework and numerical tool to model complex dynamic phenomena, particularly within biological settings.
This work represents early-stage fundamental research in applied mathematics. It could potentially benefit software developers and computational biology organisations that design simulation tools for complex biological systems. However, because the abstract details only mathematical formulations and a biological demonstration, the development remains far from market readiness and requires further translation into practical software workflows.
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This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented.
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DOI: 10.3390/computation12010007
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