Numerical solution of partial differential equations with the tau-collocation method

Abstract

The focus on this thesis is to present our research results in further developing the idea of the Tau-Collocation Method proposed by Liu in 1988. The Tau-Collocation Method, based on the Tau Method, is a numerical method for approximating the solution of ordinary di erential equations (ODEs). In this thesis, we de ne the format of perturbation term for the Tau-Collocation Method so that the idea of the Tau-Collocation Method becomes a complete method. Also, we successfully simulate the recursive formulation of the Tau Method by the Tau-Collocation Method with same perturbation term in both cases. Moreover, we extended the idea and formulation of the Tau-Collocation Method to approximate the solution of 2-dimensional and 3-dimensional linear partial di erential equations (PDEs). The format of perturbation term for the Tau-Collocation Method for PDEs is extended from the case for ODEs. It restricts the selection and number of collocation points so that the problem in redundance of linear dependent equation inside the resultant matrix is overcame. Besides, we discuss the approximation to the solution of nonlinear ODEs and PDEs by the Tau-Collocation Method incorporated with iterative schemes in this thesis. And, we proposed a new technique, the Tau-Collocation Method incorporated with the Adomian's polynomials, to tackle the nonlinear ODEs and PDEs. No linearization process is required when this new technique is applied and this new technique can be applied to the nonlinear ODEs and PDEs with highly nonlinearity. It is a blank new idea in the eld of the spectral methods, included the Tau Method. Also, we proposed the Segmented Tau-Collocation Method for the solution of ODEs and PDEs de ned on a large domain or experienced with a sharp change over the domain. Moreover, the Tau-Collocation Method with the singularity subtraction technique is proposed in this thesis for tackling the PDEs with boundary singularities. Two problems in the eld of linear elastic fracture mechanics are selected to test the e ectiveness of the Tau-Collocation Method incorporated with the singularity subtraction technique.

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