Legendre Spectral-collocation Method For A Class Of Singular Boundary Value Problems

In gas dynamics, fluid mechanics, elasticity and reaction diffusion processes often encounter singular boundary value problems in ordinary differential equations. Usually，we find the numerical solutions with domain decomposition, non-uniform finite difference method and the spline finite difference method for singular boundary value problems of ordinary differential equations, etc.. But the accuracy of numerical solution with these methods is not very well. Therefore, we need to study high accuracy numerical method for singular boundary value problems of ordinary differential equations.In this paper, a class of singular boundary value problems of ordinary differential equation with regular singularity points is studied. The numerical method we used here is Legendre spectral-collocation method.This paper consists of five parts. In Chapter1, we recall the history of numerical methods for ordinary differential equations briefly. In particular, the numerical methods for singular boundary value problems of ordinary differential equations. We describe the motivation, the difficulties and the main results of this work.In Chapter2, we recall some results on Lagrange interpolation polynomial, Legendre polynomial, Legendre-Gauss-Lobatto nodes and differential matrix, etc..In Chapter3, we use Legendre spectral-collocation method to solve the second order linear and nonlinear singular boundary value problems of ordinary differential equation. For a class of singular points, which is located in the endpoint or interior point of interval, we can use a unified algorithm format to find numerical solutions. Numerical results show the high efficiency of proposed algorithms. For nonlinear problem, we obtain an initial value of fixed point iteration with Newton iteration, to accelerate convergence of numerical solutions. In Chapter4, the high order ordinary differential equation with singular boundary value problems is studied. The condition number of p-order differential operator is O(N2p)(Nis the number of distribution points). The high order equation is firstly reduced to a first order differential equations systems by order reduction method, then be solved by Legendre spectral-collocation method. This improved the condition number of differential matrix and makes the better stability of the algorithm. In addition, when using the iterative algorithm to solve the linear equations, the numbers of iterations can be reduced.The final section is for some concluding remarks.