Numerical Investigation of Iterative Method for Fractional Differential Equations

Abstract

This thesis presents a hybrid numerical framework that combines the Laplace trans- form with the Reproducing Kernel Hilbert Space (RKHS) method within the Atangana- Baleanu fractional setting to solve nonlinear fractional differential systems with memory

effects. The proposed approach reformulates the governing equations into an equivalent integral representation, enabling the construction of approximate solutions in the form of uniformly convergent series. This ensures both accuracy and numerical stability, while the Laplace transform facilitates the treatment of Atangana–Baleanu fractional derivatives

and the RKHS structure provides a rigorous functional-analytic basis for the approxima- tion process. A convergence analysis is established to validate the method, and several

numerical experiments demonstrate its effectiveness in capturing the dynamical behavior of fractional systems and the impact of the fractional order. Overall, the method offers a robust and flexible tool that can be extended to more complex and higher-dimensional fractional models.