Numerical solutions of the Korteweg-de Vries (KdV) equation using spline base functions

ABSTRACTMaster ThesisNUMERICAL SOLUTIONS OF THE KORTEWEG-deVRIES (KdV) EQUATION USING SPLINE BASEFUNCTIONSMuharrem ÖZLÜK,İnönü UniversityGraduate School of Natural and Applied SciencesDepartment of Mathematics64 + ix pages2005Supervisor : Assist.Prof. Turabi GEYİKLİThe Korteweg-de Vries (KdV) equation is an important partial differ-ential equation which arises in the study of many physical systems.In this MSc. Thesis, numerical solutions of the KdV equation based onfinite element methods using B-spline functions are investigated.In the first chapter of this thesis, theoretical background of the KdVequation is discussed. In the second chapter, finite element methods, spline andB-spline functions, Galerkin and Collocation methods and the conservationlaws for the KdV equation are given. In the following chapters, numericalsolutions of KdV equation are obtained with Galerkin and Collocation methodsusing Quadratic, Cubic, Quartic and Quintic B-spline functions. Computedresults are compared with the numerical results given by previous authors.iThe stability analysis of the numerical techniques based on von Neumanntheory is given.As a result, Galerkin and Collocation methods with B-spline functionsgive adequately good results. So it is recommended that B-spline functions canbe used for solving other nonlinear partial differential equations.Keywords: Korteweg-de Vries, KdV, Finite Element Method, B-Spline,Galerkin, Collocation.ii