On two nonlinear singularly perturbed boundary value problems

Abstract

Boundary layer theory is a collection of perturbation methods which are frequently used to derive asymptotic solutions to differential equations with a small parameter ". However, it is often very difficult to prove rigorously that the approximations obtained are truly representations of exact solutions. Sometimes, boundary layer theory even leads to spurious solutions. In this thesis, based on shooting method, we study two nonlinear singularly perturbed boundary value problems rigorously, which the method of matched asymptotics cannot be applied directly. First, we are concerned with the positive solutions of the boundary-value problem 8>< >: "u00 ¡ ¾(u) = ¡°; u(0) = u(1) = 0; where ° is a positive constant. The nonlinear term ¾(u) behaves like a cubic, and it vanishes only at u = 0, where ¾0(0) > 0 and ¾00(0) < 0. This problem arises in a study of phase transitions in a slender circular cylinder composed of an incompressible phase-transforming material. Here, we determine the number of solutions to the problem for any given °, derive asymptotic formulas for these solutions, and show that the error terms associated with these formulas are exponentially small, except for one critical value of °. Then, we investigate the Carrier’s problem 8>< >: "2y00 = 1 ¡ y2 ¡ 2(1 ¡ x2)y; y(¡1) = y(1) = 0; where " is again a small positive parameter. Let N" denote the maximum number of spikes that a solution to Carrier’s problem can have. We show that N" is asymptotically equal to [K="], where K = 0:4725 ¢ ¢ ¢ , and the square brackets represent the greatest integer less than or equal to the quantity inside. If n(") stands for the number of solutions to this problem, then it is also shown that 4N" ¡ 3 · n(") · 4N". Our approach is based on the shooting method and on the construction of an envelope function for the minimum values of the solutions as " approaches zero.