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

Ferreira, Chelo; López, José L.; Pérez Sinusía, Ester

J. Integral Equations Appl. 27, No. 1, 27-45 (2015).

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.

Reviewer: Nicolae Lupa (Timisoara)

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

Keywords:

Olver’s method; second order differential equations; turning points; regular singular points; Volterra integral equations of the second kind; asymptotic expansions; Green functions; Airy functions; Bessel functions

Citations:

Zbl 0303.41035