B-Spline Collocation Approach For Solving Partial Differential Equations
Loading...
Date
2016-01
Authors
Mat Zin, Shazalina
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
The B-spline and trigonometric B-spline functions were used extensively in Computer Aided Geometric Design (CAGD) as tools to generate curves and surfaces. An advantage of these piecewise functions is its local support properties where the functions are said to have support in specific interval. Due to this properties, B-splines have been used to generate the numerical solutions of linear and nonlinear partial differential equations. In this thesis, two types of B-spline basis function are considered. These are B-spline basis function and trigonometric B-spline basis function. The development of these functions for different orders is carried out. A new function called hybrid B-spline basis function is developed where a new parameter incorporated with B-spline and trigonometric B-spline basis functions is introduced. Collocation methods based on the proposed basis functions and finite difference approximation are developed. B-splines are used to interpolate the solution in x-dimension and finite difference approximations are used to discretize the time derivatives. In general, initial-boundary value problems involving one-dimensional wave equation, nonlinear Klein-Gordon equation and Korteweg de Vries equation are solved using the collocation methods. In order to demonstrate the capability of the schemes, some problems are solved and compared with the analytical solutions and the results from literature. Another new finding of this thesis is the collocation methods that are applicable to solve nonlinear partial differential equations with accurate result. The stability of the schemes is analysed using Von Neumann stability analysis and the truncation errors are examined. The proposed
methods have been proved to be unconditionally stable. Cubic and quartic B-spline
collocation methods have been verified as 2 ( ) ( ) O t O h accurate. The main
contribution and innovation of this thesis are the development of hybrid B-spline
basis function and the applicability of the proposed collocation methods to solve
nonlinear partial differential equations.
Description
Keywords
B-spline