Spectral homotopy analysis method and composite Chebyshev finite difference method for solving integro-differential equations - Atabakan, Zohreh Pashazadeh (2015)
In this thesis, spectral homotopy analysis method (SHAM) is proposed for solving different type of second order integro-differential equations such as linear and nonlinear Volterra, Fredholm and Volterra-Fredholm integrodifferential equations. Linear and nonlinear systems of second order Fredholm integro-differential equations are solved using SHAM. In this method, the Chebyshev pseudo spectral method is used to solve the linear high-order deformation equations. The convergence analysis of the proposed method is proved, the error estimation of the method is done and the rate of convergence is obtained. Many different examples are solved using spectral homotopy analysis method to confirm the accuracy and the efficiency of the introduced method. An efficient and accurate method based on hybrid of block-pulse functions and Chebyshev polynomials using Chebyshev-Gauss-Lobatto points is introduced for solving linear and nonlinear Fredholm and system of Fredholm integro-differential equations. The useful properties of Chebyshev polynomials and finite difference method make it a computationally efficient method to approximate the solution of Fredholm integro-differential equations. In this method, the given problem is converted into a system of algebraic equations using collocation points. The error bound of the method is estimated. Several numerical examples have been provided and compared with well-known approaches and exact solutions to confirm that the introduced method is more accurate and efficient. For future studies, some problems are proposed at the end of this thesis.