Radial Basis Function Collocation Methods for Oscillatory Integral Equations

Abstract

This thesis is aimed to find meshless approximate solution of Fredholm, Volterra, and perturbed Volterra oscillatory integral equations. Numerical approximations are tailored in such a way to make them more appropriate to counter solution challenges caused by high level of oscillations and existence of stationary points. Numerical approximation procedures in the form of meshless methods are proposed for one and two dimensional os cillatory integral equations. The numerical procedures have the ability to produce solution on both uniform and scattered nodes according to requirement of the problem. As a priori arrangment, barycentric interpolation is used to establish connectivity between the local and global nodes. Accuracy and efficiency of the proposed procedure is validated through numerical benchmark test problems with and without stationary points. The first model considered in the thesis is one-dimensional highly oscillatory Fredholm integral equations having kernel function with and without stationary points. Two variants of meshless methods are proposed and implemented on uniform and Chebeyshev colloca tion points. The proposed meshless methods are the amalgamation of Levin technique with global and local radial basis function collocated on uniform and Chebyshev nodes. Chebyshev and radial basis functions differentiation matrices are utilized to solve highly oscillatory Fredholm and Voltera integral equations. These procedures are comparatively more accurate and responsive to the problems having kernel function involving stationary points. Application of the meshless procedures are further extended to two-dimensional oscillatory Fredholm integral equation for the kernels with and without stationary points. New algorithms based on global and local meshless formulations are proposed to obtain their numerical solution. Comparison of the meshless methods among themselves and with the methods reported in the literature is carried out. A delaminating quadrature method is introduced for two-dimensional oscillatory Fred holm integral equation model having phase function with stationary phase ponit(s). Inner and outer integral quadrature rules are implemented to solve the resultant PDE(s) and ODE(s) emerged from the oscillatory integrals. The proposed meshless methods are im plemented on two types of domain subdivisions. The collocation points in sub domain and whole domain are named as the local and the global nodes, respectively. Barycentric interpolation is incorporated to link the local and global nodes in the sub domain and the main domain. The proposed methods are aimed to produce more accurate results in the second case as compared to the first case. Theorem on error bound is established and is subsequently verified through numerical convergence. In the last segment of the thesis, numerically challenging singularly perturbed one dimensional highly oscillatory Volterra integral equation model is considered. The kernel function of the model in hand is free of stationary points. The challenging aspect in such a model is the numerical solution of the reduced Volterra integral equation of first kind, which is obtained from the Volterra integral equation of second kind (by taking a small limiting value of the perturbation parameter). Convergence analysis is provided to validate convergence of numerical solution of Volterra and singularly perturbed Volterra integral equations.