Subdivision Schemes and their Applications to Solve Differential Equations

Abstract

Subdivision schemes are important for the generation of smooth curves and surfaces through an iterative process from a finite set of points. The subdivision schemes have been considered well-regarded in many fields of computational sciences. In this dissertation, we have used subdivision schemes for the numerical solution of different types of boundary value problems. In literature three methods such as spline based methods, finite difference methods and finite element methods are commonly used to find the numerical solution of boundary value problems. Subdivision based algorithms for the numerical solution of second order boundary value problems have also been used in the literature. In this dissertation, we develop subdivision based collocation algorithms for the numerical solution of linear and non linear boundary value problems of order three and four. Subdivision based collocation algorithms for the solution of second and third order singularly perturbed boundary value problems are also presented in this dissertation. These algorithms are developed by using basis functions of subdivision schemes. Convergence analysis of these collocation algorithms are also discussed. Accuracy and efficiency of the developed algorithms are shown through comparison with the existing numerical algorithms.