A new numerical approach for solving initial value problems with exponential growth integral curves

Abstract

Some physical processes raised in aerodynamics, hydrodynamics, and theory of catalytic reactions are modeled by stiff initial value problems. This paper considers a class of stiff problems such as singularly perturbed problems for ordinary differential equations. Singularly perturbed problems have regions of rapid change in the solution, which are called contrast structures. Most often, boundary layer exhibits in such problems, but several pieces of research devoted to inner layers too. There are many numerical and asymptotic methods for solving singularly perturbed initial value problems with contrast structures. However, it is still an open field for researches because the universal method does not exist. Problems with an exponential growth rate of integral curves have a particular interest for researchers because most of the methods for solving singularly perturbed problems cannot handle these. We are going to discuss methods for solving singularly perturbed initial value problems with contrast structures. Also, we are proposing a new technique based on the method of solution continuation for problems with the exponential growth of integral curves. This technique allows explicit numerical methods to get the solution of the considered problem when it seems impossible for them. The efficiency of the new approach will be shown on the test initial value problem.