Analytical and Numerical Solvability for Certain Sequential Differential Problems via the Approaches of Caputo, Caputo-Fabrizio, and Atangana-Baleanu.

Abstract

This thesis contains important findings in existence and uniqueness theory, as well as Ulam stability results for certain types of sequential differential problems involving various forms of Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives. Boyd and Wong’s fixed point theorems, the sum of weakly sequentially continuous mappings, and O’Regan’s fixed point theory are used to investigate existence and uniqueness. We also evaluate stability using the Mittag-Leffler Hyers-Ulam Rassias concept. Afterwards, explores several categories of nonlinear multi-delay differential equations, carried by the appearance of novel generalized Caputo-style fractional integrals and variations. We explore existence and uniqueness using Fixed Point Theory for Operators with the Volterra Property, which includes the Step-by-Step Contraction Theorem and Burton and Kirk’s Theorem. To enhance our studies, we include numerical evaluations for each provided issue, allowing for a more accurate comprehension of the responses of the aforementioned derivatives. Based on the three-step Adams-Bashforth principles, we create novel numerical techniques and algorithms. Multiple examples illustrate the usefulness and robustness of this suggested method, overall the results displaying excellent accuracy and strong agreement in most problems