Numerical solution of boundary value problems in ordinary differential equations with time and space singularities

Abstract

The aim of this work is to apply the existing software, the open domain MATLAB code bvpsuite, developed at the Institute for Analysis and Scientific Computing, Vienna University of Technology, to solve nonlinear singular problems. The interest in solving singular problems in ordinary differential equations was motivated by an international cooperation with I. Rachunkova and S. Stanek from the Palacky University, Olomouc, Czech Republic. The results of the numerical experiments provided in the present work. The difficulty arising above is not only due to the singular point t = 0 in the differential operator on the left-hand side of the differential equation but also due to the fact that the inhomogeneity f(t; x; y) may become singular in the space variables x = 0 and/or y = 0. Since the differential operator exhibits the singularity of the first kind at t = 0, and it is clear that for such problems important basic questions have to be answered before trying to solve such problems numerically. Such questions are the existence and uniqueness (or local uniqueness) of analytical solutions and the well-posedness of the problem, stability and convergence properties of a method used to solve the problem, and the performance of the controlling strategies usually implemented in the software, such as error estimate procedure and adaptation of the grid. Fortunately, there exists an extensive literature providing satisfactory answers to the above issues. Moreover, dependable software is available. The numerical method and its implementation will be discussed and it will be shown how the code bvpsuite can be used to simulate the above model. We point out that the models often are parameter dependent and therefore, a pathfollowing strategy to follow the solution in the solution-parameter space uc(t) needs to be utilized.