Analytical study on the existence, uniqueness and stability of fractional integro-differential equations involving the modified Atangana-Baleanu operator
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Abstract
The recently constructed Modified Atangana–Baleanu (MAB) fractional derivative, whose kernel is given by the generalised Mittag–Leffler function, is the subject of a class of integro-fractional differential equations that are thoroughly examined in this paper. Classical or even conventional fractional derivatives are unable to fully capture nonlocal dynamics and memory effects, but this derivative operator offers a potent tool for doing so. Complex physical, biological, and engineering systems exhibiting anomalous diffusion or hereditary behaviour can be best described by the MAB operator's versatility. We rigorously prove the existence and uniqueness of solutions to the suggested integro-fractional problem inside a suitable Banach space framework by utilising fixed-point techniques like Schauder's fixed-point theorem and the Banach contraction principle. A tangible example is provided to further bolster the analytical findings, proving the applicability and accuracy of the theoretical framework. Furthermore, the concepts of Hyers-Ulam stability and generalised Hyers-Ulam stability are examined in the stability analysis of the obtained solutions. The results of this work add to the body of knowledge on fractional differential equations and aid in the continuous creation of new fractional operators that better represent nonlocal occurrences. The proven results also pave the way for new research directions, such as numerical simulation, control, and optimisation of systems controlled by modified kernel-based fractional integro-differential equations.
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References
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