Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III (Q2351441)

The classical asymptotic method for second order differential equations containing a large parameter \(\Lambda\) is Olver' s method (divided into three canonical cases). The authors design a method that approximates a solution of the nonlinear differential equations \[ y''-\Lambda^3 xy=f(x,y) \text{ and } y''-\Lambda^2 \frac{y}{x}=f(x,y) \text{ in } [-a,a], \] where \(f:[-a,a]\times\mathbb{C}\to\mathbb{C}\) is a continuous function in its two variables satisfying a Lipschitz condition in the second variable, adds initial conditions and considers the corresponding initial value problems. Using the Banach fixed point theorem and the Green's function of some auxiliary problems, the authors obtain uniformly convergent expansions of that solution (corresponding to the Olver's cases II and III). Then, they show that these expansions are asymptotic expansions of the unique solution of the initial value problems for large parameter \(\Lambda\). For a better comparison with Olver's method, the authors particularize their theory to linear equations.