B-splines for initial and boundary value problems - Goh, Joan Yah Ru (2013)
Due to the difficulty of solving the initial and boundary value problems analytically, a large number of methods have been developed to approximate the solution of these problems. There has been intere t in thi area of late and there is scope for the investigation and development of new methods and approache . The objective of this work is the development and application of B-spline method for the solution of initial value problems and boundary value problems. In this work, interpolation methods based on cubic B-spline and extended cubic B-spline were considered for solving linear two-point boundary value problems of order two. Extended cubic B-spline is an extension of cubic B-spline possessing one additional free parameter, A which makes the refinement of the produced curve possible. In order to create the best fit curve, the most suitable value of A was found by minimizing the generated error. A higher order Bspline, quartic B- pline, which has the same degree as extended cubic B-spline was also taken into account in solving these problems. As the order is increased, there are infinitely many solutions. However, the closest fit of the approximation curve could still be obtained with the help of Gauss-Jordan elimination method and optimization approach which is applied on extended cubic B-spline. These methods were tested on linear two-point boundary value problems, singular boundary value problems and also nonlinear two-point boundary value problems. The results showed that these methods are well approximate the exact solutions.