B-spline Collocation Techniques for Numerical Solutions of Linear and Non-Linear Differential Equations

Abstract

Various physical linear and non-linear models are designed using di erential equations. Speci cally, to compute the numerical solution of non-linear model is a challenging task. The B-spline functions and its di erent types have been used by a number of researchers to obtain the numerical solution of such non-linear initial/boundary value problems. For the generation of curves as well as surfaces, an extensive usage of these basis functions as tools is observed in Computer Aided Geometric Design. The core reason why we prefer B-splines to calculate the numerical solution of differential equation is their local support properties which means that B-spline functions have support in particular interval i.e. they are non-zero in particular interval and zero otherwise. In this thesis, di erent types of B-splines are utilized to compute the numerical solution of di erential equations. The time derivatives are discretized by nite di erence schemes and the B-spline functions interpolate the spatial derivatives. In general, non-linear generalized Burgers-Huxley and Burgers-Fisher equations are solved by hybrid B-spline collocation method, non-linear modi ed Camassa-Holm and Degasperis-Procesi equations by quartic B-spline collocation method, non-linear generalized Newell-Whitehead Segel equation by exponential B-spline collocation method, non-linear Phi-Four and Allen-Cahn equations by modi ed cubic B-spline collocation method and non-linear second order singular boundary value problems by a new Extended cubic B-spline approximation. The unconditional stability and convergence analysis of the proposed methods are also established. The innovative part and the main contribution of this thesis is the formation of numerical scheme based on hybrid B-spline function and its execution to nonlinear partial di erential equations. Some test problems are provided to enhance the accuracy, validity and e ciency of the proposed schemes.