Asymptotic solutions of the boundary value problems for the singularly perturbed differential algebraic equations with a turning point

TL;DR

The paper tackles boundary-value problems for singularly perturbed differential-algebraic equations with turning points, where the rank of $A(t,\varepsilon)$ can change. It develops a boundary-function based asymptotic construction that decomposes the solution into a regular part and two boundary-layer contributions, yielding a uniform expansion on $[0,T]$. Through a detailed formalism involving projectors and a (central) canonical form, it derives leading-order and recursive higher-order equations, along with solvability conditions and boundary-coupling constants. It then proves the asymptotic validity and uniqueness of the true solution, showing that $x(t,\varepsilon)$ approximates the constructed $x_l(t,\varepsilon)$ with error $O(\varepsilon^{l+1})$ for small $\varepsilon$, via a contraction-mapping argument applied to a transformed singular perturbation system.

Abstract

This paper deals with the boundary value problems for the singularly perturbed differential-algebraic system of equations. The case of turning points has been studied. The sufficient conditions for existence and uniqueness of the solution of the boundary value problems for DAEs have been found. The technique of constructing the asymptotic solutions has been developed