On the numerical solutions of fredholm volterra integro-differential and integral equations with weakly singular kernel

In this thesis, we aim to a solve fractional integro-differential equation with a weakly singular kernel using the Chelyshkov Wavelets Collocation Method. Chapter One presents the historical background and fundamental Integro-differential equations, concepts of fractional calculus and wavelet methods, as well as their roles in fractional integro-differential equations. The aim is to present the reasoning behind the wavelet-based method and to clarify its place within the scope of this thesis. Chapter Two presents some information about Fractional integration and differentiation. This includes the Gamma function, Riemann-Liouville fractional integral and Caputo-derivative. Chapter Three provides an overview of Chelyshkov polynomials and wavelets. A thorough literature review reveals that Chelyshkov wavelets have not previously been applied to the solution of fractional-order integro-differential equations with weakly singular kernels, thereby establishing the originality of this thesis. Furthermore, the chapter introduces the concept of an operational matrix for fractional integration, demonstrating how the fractional integration of a function can be achieved through a matrix-based approach. Chapter Four presents a numerical method for solving fractional integrals with a weakly singular kernel using properties of block pulse functions. Chapter Five presents the wavelet collocation method for solving fractional-order integro-differential equations. In this formulation, the Volterra component involves a weakly singular kernel, which is addressed using the technique introduced in the preceding chapter. The Fredholm component is handled through the construction of a kernel matrix, a vector formed from psi-functions, and an operational matrix of fractional integration. Chapter Seven presents tables showing the absolute errors between the approximate and exact solutions and L-infinity errors. This thesis includes only seven representative examples, with the approximate solutions obtained using Mathematica and Pandas software. The accuracy of the method has been verified through these seven numerical examples. Furthermore, two examples have been compared with CAS wavelet using the absolute error and L-infinity errors.