Fractional integro-differential equations have been recently solved by many methods, such
as Adomian decomposition method, differential transform method, collocation method and
Taylor expansion approach. In this thesis a hybrid collocation method is used which combines
a non-polynomial collocation used on the first subinterval and graded piecewise
polynomial collocation used on the rest of the interval. A theoretical analysis for the convergence
order of the method is presented. ‎To ‎analyze ‎and ‎verify ‎the ‎accuracy ‎of ‎the proposed ‎numerical ‎method, ‎we ‎examined ‎samples ‎and ‎benchmark ‎problems ‎with ‎known ‎exact ‎solutions.‎
 

In this paper, we consider the discrete Legendre spectral Galerkin and discrete Legendr espectral collocation methods to approximate the solution of mixed type Hammerstein integral equation with smooth kernels. The convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained. We have shown that, when the quadrature rule is of certain degree of precision, the rates of convergence for the Legendre spectral Galerkin and Legendr spectral collocation methods are preserved. We obtain super convergence rates for the iterated discrete Legendre Galerkin solution. By choosing the collocation nodes and quadrature points to be same, we also obtain superconvergence rates for the iterated discrete Legendre collocation solution.‎to ‎analyze ‎and ‎verify ‎the ‎accuracy ‎of ‎the method,we ‎examine ‎sample ‎and ‎benchmark ‎problems ‎with ‎known ‎exact ‎solutions.

In this thesis, we discuss the application of spectral Jacobi-collocation methods to a certain class of weakly singular Volterra integral equations. First, we use some function transformation and variable changes to transform the equation into a Volterra integral equation defined on the standard interval [-1,1]. Then the Jacobi-Gauss quadrature formula is used to approximate the integral operator. For the spectral Jacobi-collocation method, a rigorous error analysis is given under the assumption that both the kernel function and the source function are sufficiently smooth. Finally, some numerical examples are provided to illustrate the theoretical results.

In this thesis, we consider a second kind weakly singular nonlinear Volterra–Hammerstein integral equation defined by a compact operator and derive a Nystrom type interpolant of the solution based on Gauss–Radau nodes. We prove the convergence of the interpolant and derive convergence estimates. For equations with nonlinearity of algebraic kind, we improve the rate of convergence by using a smoothing transformation. Some numerical examples are given