Investigation of boundary-value problems for the singular perturbed differential equation of high order

I. V. Amirkhanov ; E. P. Zhidkov ; D. Z. Muzafarov ; N. R. Sarker ; I. Sarhadov ; Z. A. Sharipov

Matematičeskoe modelirovanie, Tome 19 (2007) no. 11, pp. 65-79

Résumé

By the different methods the boundary-value problems for the differential equations of high order with the small parameter $\varepsilon$ at higher derivatives are investigated. A comparative analysis of the obtained results is given at diminution of $\varepsilon$. The existence of a boundary layer for a derivative from the solutions is established. It is shown, that at diminution of $\varepsilon$ the solutions of one boundary-value problem (when for the solution $\psi(r)$ of the given equation sets the next boundary conditions: $\psi(0)=0$, $\psi''(0)=0$, $\psi^{\mathrm{IV}}(0)=0$, $\cdots$; $\psi(\infty)=0$) converge to the solutions of a degenerate problem (Schrödinger equation), and for the other (when the boundary conditions are given by: $\psi(0)=0$, $\psi'(0)=0$, $\psi''(0)=0$, $\cdots$; $\psi(\infty)=0$) such convergence doesn't exist.