Boundary value problems of explicit ordinary differential equations from probabilistic evolution perspective

Abstract

The boundary value problems of ordinary differential equations have many important features in comparison with the initial value problems of ODEs. However, one of the basic tendencies is to evaluate forward and backward evolutions and then to combine them to satisfy the boundary conditions. One other issue is also the possibility of more than one type of problems: eigenvalue problems and inversion problems. In this or that way, the basic issue is the solution of ODEs. We use a recently developed method we call “Probabilistic Evolution Approach” whose mainlines are recalled in this paper. Then the boundary condition are imposed for the resulting infinite number of ODEs. This is perhaps the most elementary but promising approach to the boundary value problems of ODEs and works not only for linear but also for nonlinear ODEs.