Development of a Hybrid Spectral Collocation Method Based on Transformed Fibonacci Polynomials for Solving Stochastic Fractional Integro-Differential Equations

This thesis introduces a novel and highly efficient numerical scheme, the Fibonacci Spectral Collocation Method (F-SCM), for solving the complex class of nonlinear Stochastic Fractional Integro-Differential Equations (SFIDEs), the core of the methodology lies in projecting the solution onto a finite-dimensional functional space spanned by shifted Fibonacci polynomials, we derive a new operational matrix for the Caputo fractional derivative corresponding to this basis, which, in conjunction with a spectral collocation strategy at Gauss-Lobatto nodes, transforms the original functional equation into a system of algebraic equations. For nonlinear cases, this system is effectively solved for each realization of the Wiener process using a Newton-Raphson iterative method within a Monte Carlo simulation framework, a rigorous numerical analysis is presented to validate the method, the results demonstrate that the F-SCM achieves spectral (exponential) convergence with respect to the number of basis functions and the canonical Monte Carlo convergence rate for the statistical moments, the method's exceptional accuracy, robustness in handling nonlinearities, and pathwise fidelity are confirmed through a series of benchmark problems, establishing it as a powerful and reliable tool for simulating systems with memory and stochastic effects