Numerical solution of linear and nonlinear ordinary differential equations with the tau-collocation method

Abstract

In this thesis, a Tau-Collocation Method for numerical solution of linear and nonlinear ordinary differential equations with variable coefficients and supplementary conditions is presented. The Tau-Collocation Method is a new numerical method developed from the idea of the Tau Method and the Collocation Method. In our study, both single ordinary differential equation and systems of ordinary differential equations are considered. With the operational approach to the Tau-Collocation Method, the differential equations are transformed into system of linear algebraic equations in matrix form. By repeating the matrix structure, the operational approach to the Tau-Collocation Method for single ordinary differential equation is extended for solving systems of differential equations. We can get accurate numerical results by using simple formulation and algorithm. This method is widely applied to solve different kinds of ordinary differential equations. For nonlinear differential equations, we use the iterative schemes to linearize the equations and then solve the linearized problem by the Tau-Collocation Method. We have successfully applied the Tau-Collocation Method to deal with the nonlinear problem which possesses a limit cycle solution. For problems with multiple solutions, a searchlight technique is introduced to trace the multiple solutions. Two typical problems which are relevant to linear elastic fracture mechanics are selected to test the applicability of the Tau-Collocation Method. These two problems are partial differential equations with boundary singularities. The Tau-Collocation Method incorporated with the method of lines and the singularity subtraction technique is applied to solve these problems. Some numerical examples are discussed and the numerical results for treating different kinds of problems are given.

Date of Award	4 Oct 2004
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Kam Moon Lester LIU (Supervisor)