A Riccati Transformation Method for Solving Linear BVPs. I: Theoretical Aspects

A method of solving linear two-point boundary value problems (BVPs) for first order systems of ordinary differential equations (ODEs) is considered. Of particular interest is the class of singularly perturbed BVPs. The method attempts to split the integration of the rapidly increasing/decreasing solution components, typical for such problems, into different initial value problems (IVPs). The case of separated boundary conditions (BCs) is considered and their ability to detect solution components that (eventually) increase/decrease is exploited. By using the Riccati transformation, the original BVP is then reformulated as three IVPs, one of them being a matrix Riccati equation. The Riccati equation is nonlinear, and a "multiple embedding" setting must be used. The entire method is then fit theoretically into a multiple shooting-like framework, and the original BVP as a chain of subproblems is studied. In this way, a generalized decoupling result is presented which enables the study of stability to be brought back to properties of the differential system itself.