Numerical solutions of coupled differential equations with B-Spline finite element methods

This thesis consists of four chapters. In the first chapter, after giving some general information about the finite element methods which will be used in the thesis, fundamental concepts about Galerkin, Petrov-Galerkin, subdomain and collocation methods together with spline functions and B-spline functions are presented.The second, third and fourth chapters of this thesis make up its original parts. In the second chapter, coupled Burgers' equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods with different degrees B-spline functions. These methods are applied to three model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, the error norms L_{2 } and L_{? }are given in the form of tables. The stability analysis of the finite element approximation obtained by applying each method is also investigated.In the third chapter, after giving general information about the coupled Korteweg-de Vries (KdV) equation, the three test problems are solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? }and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated.In the fourth chapter, the coupled modified Korteweg-de Vries (mKdV) equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. These methods are applied to five model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? } and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated.KEY WORDS: Coupled Burgers' Equation, Coupled KdV Equation, Coupled mKdV Equation, Finite Element Methods, B-Spline Functions, Galerkin Method, Petrov-Galerkin Method, Subdomain Method, Collocation Method, Stability Analysis.