IJPAM: Volume 78, No. 1 (2012)

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

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


Abstract. Consider the system of second order differential equation

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


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

\begin{displaymath}Q(x)=
\left(\begin{array}{l}
p(x)~~~~r(x))\\ r(x)~~~~q(x)...
...\begin{array}{l}
s(x)~~~~0\\ 0~~~~t(x)
\end{array}\right)
,\end{displaymath}

and $p(x)$, $q(x)$, $r(x)$, $s(x)$, $t(x)$ are real-valued continuously differentiable functions of x on $[0, \pi]$.

In the present paper we assume that the elements of $R(x)$, i.e. $s(x) > 0$, $t(x) > 0$ for $x \in [0, \pi]$ and are at least twice continuously differentiable on $[0, \pi]$ and determine the asymptotic solutions along with their derivatives for the system for large values of the parameter $\lambda$ and finally apply these to determine the asymptotic expressions for the distribution of the eigenvalues and the normalized eigenvectors under the Dirichlet and Neumann boundary conditions.

Received: March 19, 2012

AMS Subject Classification: 35B40, 37K40

Key Words and Phrases: asymptotic solutions, turning points, Dirichlet and Neumann 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: 2012
Volume: 78
Issue: 1