Chebyshev Spectral Collocation Method For Solving Third Order Singularly Perturbed Boundary Value Problems

Abstract

Even though spectral methods have engrossed much of the attention in current research of numerical methods for solving differential equa tions, few experience is obtainable in applying spectral methods to solve boundary value problems. In this thesis Chebyshev spectral collocation method has been studied to solve a certain kind of third order singu larly perturbed boundary value problems (TSPBVPs). First of all, the given problem is transformed into a system of two ordinary differential equations (ODEs) subject to appropriate initial and boundary condi tions. Then, the Chebyshev spectral collocation method in barycentric form with sinh transformation is applied to solve the system of ODEs. Based on the asymptotic analysis, the location and width of bound ary layer of the given problem, which are chosen as parameters in the sinh transformation, can be determined. According to a spectral collo cation method, the scheme is constructed by using the differentiation matrix DN to approximate the differential operator d dx. DN is found by taking the derivative of the interpolation polynomial PN(x), which is interpolated by selecting the first kind of Chebyshev-Gauss-Lobatto (C-G-L) points. Lastly, a brief comparison of the above method to the rational spectral collocation method is also presented using some examples. Numerical experiments are reported to illustrate the accuracy and effectiveness of the method.