Supervisor: Peeter Oja (dots., TÜ matemaatika instituut)
Opponents: Jaan Janno (prof., Tallinna Tehnikaülikool)
Natalja Budkina (Dr. math., Riga Technical University)
Boundary value problems arise in several branches of scientific and engineering problems. Traditional methods for approximate solution of boundary value problems are finite difference method which only gives a discrete solution, and collocation method with polynomial splines. The latter one has been quite well studied. It is known, that in interpolation in some cases the rational splines may have better results compared to polynomial ones. In such circumstances, it is natural to pose the question about rational spline collocation method for boundary value problems. In recent years the shape preserving problems have been considered by several authors. Polynomial splines do not preserve the monotonicity or convexity. On the other hand, the linear/linear rational spline is constant or strictly monotone and the quadratic/linear rational spline is always convex (or concave). Therefore, it is a reasonable approximate solution only if the exact solution of the problem has same properties. The main purpose of the thesis is to study the linear/linear and quadratic/linear rational spline collocation method for boundary value problems and also compare them to quadratic and cubic spline cases, respectively. The collocation method with rational splines is, in nature, a nonlinear method because it leads to a nonlinear system with respect to the spline parameters. Also the rational spline interpolation has been studied because the results of rational spline collocation method are based on certain convergence properties of interpolating rational splines. Numerical examples support the obtained theoretical results.