A Singular Singularly Perturbed Boundary Value Problem of the Second Order Quasilinear Systems

Abstract

The singular singularly perturbed boundary value problem ϵ d 2x/dt 2 = A(x, t)dx/dt + B(x, t), x(0, ϵ) = α(ϵ), ϵ[a dx(0, ϵ)/dt + b dx(1, ϵ)/dt] = β(ϵ) for an m-dimensional system of quasilinear differential equations is considered under the assumption that there is a vector-valued function F( x, t) such that ∇ x F( x, t) = A( x, t). The asymptotic solution is constructed by a modified Vasil′eva method where there are 3 boundary layer corrections, i.e., the left boundary layer contains 2 different stretched variables t/√ϵ and t/ϵ. The existence and uniqueness of the exact solution and the uniform validity of the formal asymptotic solution for the boundary value problem are proved by using the Banach-Picard fixed-point theorem.