Asymptotics of a second-order differential equation with a small parameter in the case when the reduced equation has two solutions. (Russian, English) Zbl 1201.34088

Dolbeeva, S. F.; Chizh, E. A.

Comput. Math. Math. Phys. 48, No. 1, 30-42 (2008); translation from Zh. Vychisl. Mat. Mat. Fiz. 48, No. 1, 33-45 (2008).

Summary: The boundary value problem for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative is examined. It is assumed that the reduced equation has two solutions with intersecting graphs. Near the intersection point, the asymptotic behavior of the solution to the original problem is fairly complex. A uniform asymptotic approximation to the solution that is accurate up to any prescribed power of the small parameter is constructed and justified.

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
34E10	Perturbations, asymptotics of solutions to ordinary differential equations

Keywords:

second-order nonlinear ordinary differential equation; small parameter; intersection point; asymptotic approximation