Numerical Solution of Boundary-Value and Initial-Boundary-Value Problems Using Spline Functions

The following two types of problems in differential equations are investigated: (i) Second and sixth-order linear and nonlinear boundary-value problems in ordinary differential equations using non-polynomial spline functions. (ii) One dimensional nonlinear Initial-boundary-value problems in partial differential equations using B-spline collocation method. Polynomial splines, non-polynomial splines and B-splines are introduced. Some well known results and preliminary discussion about convergence analysis of boundary-value problems and stability theory are described. Quartic non-polynomial spline functions are used to develop numerical methods for computing approximations to the solution of linear, nonlinear and system of second- order boundary-value problems and singularly perturbed boundary-value problems. Convergence analysis of the method is discussed. Numerical methods for computing approximations to the solution of linear and nonlinear sixth-order boundary-value problems with two-point boundary conditions are developed using septic non-polynomial splines. Second-, Fourth- and Sixth-order convergence is obtained. Numerical method based on collocation method using quartic B-spline functions for the numerical solution of one-dimensional modified equal width (MEW) wave equation is developed. The scheme is shown to be unconditionally stable using Von-Neumann approach. Propagation of a single wave, interaction of two waves and Maxwellian initial condition are discussed. Algorithms based on quartic and Quintic B-spline collocation methods are designed for the numerical solution of the modified regularized long wave (MRLW) equation. Stability analysis is performed. Propagation of a solitary wave, interaction of multiple solitary waves, and generation of train of solitary waves are also investigated. Quartic and quintic B-spline functions have been used to develop collocation methods for the numerical solution of Kuramoto-Sivashinsky (KS) equation. Also, using splitting technique, the equation is reduced to a problem of second order in space. Using error norms L2 and L∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the suggested methods is established through comparison with the existing numerical techniques. Performance of the algorithms is tested through application of the methods on benchmark problems.