IJPAM: Volume 62, No. 4 (2010)

ASYMPTOTIC EXPRESSIONS FOR THE EIGENVALUES
AND EIGENVECTORS OF A SYSTEM OF SECOND
ORDER DIFFERENTIAL EQUATIONS WITH
A TURNING POINT

Debasish Sengupta
Department of Mathematics
Vivekananda College
269, Diamond Harbour Road, Thakurpukur, Kolkata, 700063, INDIA
e-mail: drdebasish.sengupta@gmail.com


Abstract.The present paper deals with a system of second order differential equations with turning points.

In particular we consider the second order differential system

\begin{displaymath} {y}''(x)+(\lambda ^2R(x)+Q(x))y(x)=0,\quad 0\le x \le \pi , \end{displaymath}

where $y(x) = (y_{1}(x), y_{2}(x))^{T}$,

\begin{displaymath} Q(x)=\left( {{\begin{array}{*{20}c} {p(x)} \hfill & {r(x)} ... ...fill \\ 0 \hfill & {t(x)} \hfill \\ \end{array} }} \right), \end{displaymath}

$s(x) = x^{m}s_{1}(x)$, $t(x) = x^{m}t_{1}(x)$, $s_{1}(x) > 0$, $t_{1}(x) > 0$ and $p(x)$, $q(x)$, $r(x)$, $s_{1}(x)$, $t_{1}(x)$ are real-valued functions having continuous second order derivatives at $x$, $0 \leq x \leq \pi$, $m$ being a positive constant and $\lambda$, a real parameter.

We determine the asymptotic solutions for such a system for large values of the parameter $\lambda$ and apply these to determine the asymptotic distributions of the eigenvalues and the asymptotic values of the normalizing constants in two cases when the boundary conditions are: (i) the Dirichlet and (ii) the Neumann.

Received: May 7, 2010

AMS Subject Classification: 35B40, 37K40

Key Words and Phrases: asymptotic solutions, turning points, Dirichlet boundary conditions, Neumann boundary conditions, normalizing constants

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 62
Issue: 4