IJPAM: Volume 72, No. 4 (2011)

ASYMPTOTIC EXPRESSIONS FOR THE EIGENVALUES AND
EIGENVECTORS OF A SYSTEM OF SECOND ORDER
DIFFERENTIAL EQUATIONS WITH
A TURNING POINT (EXTENSION-1)

Debasish Sengupta
Department of Mathematics
Vivekananda College
Jadavpur University
269, Diamond Harbour Road, Thakurpukur, Kolkata, 700 063, INDIA


Abstract. Consider the system of second order differential equation

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


where $y(x) = (y_{1}(x), y_{2}(x))^{T}$, $Q(x)\!= \! \left(\begin{array}{ll} p(x) & r(x))\\ r(x)&q(x) \end{array}\right)$, $R(x)\! =\!\left( \begin{array}{ll} s(x)&0\\ 0&t(x) \end{array}\right)$, $s(x)= x^{m} s_{1}(x)$, $t(x)=x^{m}t_{1}(x)$, $s_{1}(x) > 0$, $t_{1}(x) > 0$ 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 real constant and $\lambda$, a real parameter.

In the present paper we consider m, a negative real number and determine the asymptotic solutions alongwith their derivatives for such a system for large values of the parameter $\lambda$ and apply these to determine the asymptotic expressions for the distribution of the eigenvalues and the normalized eigenvectors under the Dirichlet boundary conditions.

Received: June 1, 2011

AMS Subject Classification: 35B40, 37K40

Key Words and Phrases: asymptotic solutions, turning points, Dirichlet boundary conditions, normalized eigenvectors

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 72
Issue: 4