Boundary Layers in Two‐Point Boundary Value Problems for Doubly Singularly Perturbed Third‐Order Integro‐Differential Equations

This article investigates a two‐point problem for a doubly singularly perturbed third‐order integro‐differential equation. The problem formulation is presented, which is then reduced to a parameterized problem using the parameterization method. The corresponding initial‐value problem is solved, and an algebraic equation system with respect to the introduced parameters is determined. The conditions for the existence and uniqueness of the solution are also established. The corresponding modified unperturbed problem is considered. Using the parameterization method, it becomes a parameter problem. The solution is obtained via the resolvent of a Fredholm integral equation of the second kind, and sufficient conditions for the uniqueness of the solution to the modified unperturbed problem are provided. The interrelation between singularly perturbed and modified unperturbed problems, an appropriate estimate is established, and provides an example illustrating the results obtained in the article.