Numerical solving of initial value problems for ordinary differential equations with internal limiting singular points

Abstract

In the present paper, we investigate the numerical methods for solving initial value problems for systems of nonlinear ordinary differential equations with several limiting singular points. Similar tasks have great practical importance in such areas as hydrodynamics, chemical kinetics, combustion theory, computational geometry, and nuclear reactors. However, the exact solution may be constructed to nonlinear initial value problems only in exceptional cases. Numerical methods also do not give the desired result. Explicit methods make it possible to obtain a solution only if there is one limiting singular point on the right boundary of the considered interval of the argument change. Implicit methods also have usage restrictions. They consist of the difficulty or impossibility of overcoming the internal limiting singular points. For solving the considered class of problems, the paper proposes to use the method of solution continuation, which allows passing any limiting singular points effectively. Features of the application of various numerical methods are shown by solving one test initial value problem. The reliability of the obtained numerical results is confirmed by comparison with the finding analytical solution of the considered test problem.