Numerical Solutions for a Class of Singular Neutral Functional Differential Equations
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In this dissertation we study numerical solutions for certain classes of singular neutral functional differential equations (SNFDEs). We begin with an approximation scheme based on piecewise linear approximating spline functions which is used to find numerical solutions for a class of scalar singular neutral equations. The weakly singular kernel that appears in the equation leads to a degradation in the numerical rate of convergence of the approximate solution to the true solution. By modifying the scheme using a graded mesh adapted to the kernel the rate of convergence is restored. Numerical examples and a supporting lemma on the discretization error provide evidence that this is achieved. The approximation scheme is then extended to a system SNFDE. We show that the scheme is convergent provided the mesh is uniform and that there is sufficient smoothness in the true solution. The particular system considered here is a simplified version of an SNFDE that describes the dynamics of a two-dimensional thin airfoil in uniform flow. We provide a numerical example and discuss numerical difficulties. As an application of the approximation scheme we construct a simple forcing function to stabilize the SNFDE.