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Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III. (English) Zbl 1337.34057
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.

MSC:
34E05 Asymptotic expansions of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B27 Green’s functions for ordinary differential equations
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
45D05 Volterra integral equations
47N20 Applications of operator theory to differential and integral equations
34A30 Linear ordinary differential equations and systems, general
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