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The Journal of the Indian Mathematical Society

The paper deals with the asymptotic expressions for the solutions along with their first derivati... more The paper deals with the asymptotic expressions for the solutions along with their first derivatives, the distribution of the eigenvalues and the normalized eigenvector for large eigenvalues corresponding to a system of second order differential equations having turning points at both ends of the interval, under certain suitable boundary conditions.

International journal of pure and applied mathematics, 2011

Consider the system of second order differential equation y '' (x) + (� 2 R(x) + Q(x))y(x... more Consider the system of second order differential equation y '' (x) + (� 2 R(x) + Q(x))y(x) = 0, 0 ≤ x ≤ �, where y(x) = (y1(x),y2(x)) T , Q(x)= � p(x) r(x)) s(x) = x m s1(x), t(x) = x m t1(x), s1(x) > 0, t1(x) > 0 and p(x), q(x), r(x), s1(x), t1(x) are real-valued functions having continuous second order derivatives at x, 0 ≤ x ≤ �, m being a real constant and �, 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 parameterand apply these to determine the asymptotic expres- sions for the distribution of the eigenvalues and the normalized eigenvectors under the Dirichlet boundary conditions.

The paper deals with the stability of solutions along with their derivatives of a certain system ... more The paper deals with the stability of solutions along with their derivatives of a certain system of second-order differential equation with respect to certain perturbation. We consider the system dy(x)/dx + A(x)y(x) = 0, (i) where y(x) = 1 11 12 2 12 22 ( ) ( ) ( ) , ( ) ( ( )) ( ) ( ) ( ) ij Y x a x a x A x a x Y x a x a x     = =         and aij(x), i, j = 1, 2 is real-valued continuous function of x, x ∈ [0, ∞). Let B(x) = (bij(x)), i, j = 1, 2 (bij(x)s being real-valued continuous function of x ∈ [0, ∞)) be a set of perturbations which changes (1) to dy(x)/dx + (A(x) + B(x))y(x) = 0. (ii) In this paper, certain results on the stability of solutions of the system (i) along with their derivatives, which are either bounded or tend to zero as the independent variable x tends to infinity with respect to the perturbation B(x) satisfying some conditions, are achieved.

Journal of Mathematical and Computational Science, Feb 3, 2013

Consider the system of second order differential equation y"(x)+(λ 2 R(x) +Q(x))y(x) = 0, 0 ≤ x ≤... more Consider the system of second order differential equation y"(x)+(λ 2 R(x) +Q(x))y(x) = 0, 0 ≤ x ≤ where y(x) = (y 1 (x), y 2 (x)) T ,Q(x)= , R(x) = , p(x),q(x),r(x),s(x),t(x) being real-valued continuously differentiable functions of x on [0, ]. In the present paper we determine the expressions for the first eigenvalues for the system in different cases for s(x), t(x) satisfying on[0,], the conditions a) s(x) = xs 1 (x), t(x) = xt 1 (x), s 1 (x) > 0, t 1 (x) > 0, or, b) s(x) = s 1 (x) / x, t(x) = t 1 (x) / x, s 1 (x) > 0, t 1 (x) > 0, or, c) s(x) > 0, t(x) > 0 by using the asymptotic expressions for the nth eigenvalue (λ n) and those of the corresponding normalized eigenvector (x, λ n) = ( 1 (x, λ n),  2 (x, λ n)) T under the Dirichlet and Neumann boundary conditions. Further, we determine the expressions for the regularized trace matrix for the system with s(x) = t(x) = 1, 0 ≤ x ≤ under the Neumann and general boundary conditions by employing the corresponding asymptotic expressions for the nth eigenvalue (λ n).

The Journal of the Indian Mathematical Society

The paper deals with the asymptotic expressions for the solutions along with their first derivati... more The paper deals with the asymptotic expressions for the solutions along with their first derivatives, the distribution of the eigenvalues and the normalized eigenvector for large eigenvalues corresponding to a system of second order differential equations having turning points at both ends of the interval, under certain suitable boundary conditions.

International journal of pure and applied mathematics, 2011

Consider the system of second order differential equation y '' (x) + (� 2 R(x) + Q(x))y(x... more Consider the system of second order differential equation y '' (x) + (� 2 R(x) + Q(x))y(x) = 0, 0 ≤ x ≤ �, where y(x) = (y1(x),y2(x)) T , Q(x)= � p(x) r(x)) s(x) = x m s1(x), t(x) = x m t1(x), s1(x) > 0, t1(x) > 0 and p(x), q(x), r(x), s1(x), t1(x) are real-valued functions having continuous second order derivatives at x, 0 ≤ x ≤ �, m being a real constant and �, 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 parameterand apply these to determine the asymptotic expres- sions for the distribution of the eigenvalues and the normalized eigenvectors under the Dirichlet boundary conditions.

The paper deals with the stability of solutions along with their derivatives of a certain system ... more The paper deals with the stability of solutions along with their derivatives of a certain system of second-order differential equation with respect to certain perturbation. We consider the system dy(x)/dx + A(x)y(x) = 0, (i) where y(x) = 1 11 12 2 12 22 ( ) ( ) ( ) , ( ) ( ( )) ( ) ( ) ( ) ij Y x a x a x A x a x Y x a x a x     = =         and aij(x), i, j = 1, 2 is real-valued continuous function of x, x ∈ [0, ∞). Let B(x) = (bij(x)), i, j = 1, 2 (bij(x)s being real-valued continuous function of x ∈ [0, ∞)) be a set of perturbations which changes (1) to dy(x)/dx + (A(x) + B(x))y(x) = 0. (ii) In this paper, certain results on the stability of solutions of the system (i) along with their derivatives, which are either bounded or tend to zero as the independent variable x tends to infinity with respect to the perturbation B(x) satisfying some conditions, are achieved.

Journal of Mathematical and Computational Science, Feb 3, 2013

Consider the system of second order differential equation y"(x)+(λ 2 R(x) +Q(x))y(x) = 0, 0 ≤ x ≤... more Consider the system of second order differential equation y"(x)+(λ 2 R(x) +Q(x))y(x) = 0, 0 ≤ x ≤ where y(x) = (y 1 (x), y 2 (x)) T ,Q(x)= , R(x) = , p(x),q(x),r(x),s(x),t(x) being real-valued continuously differentiable functions of x on [0, ]. In the present paper we determine the expressions for the first eigenvalues for the system in different cases for s(x), t(x) satisfying on[0,], the conditions a) s(x) = xs 1 (x), t(x) = xt 1 (x), s 1 (x) > 0, t 1 (x) > 0, or, b) s(x) = s 1 (x) / x, t(x) = t 1 (x) / x, s 1 (x) > 0, t 1 (x) > 0, or, c) s(x) > 0, t(x) > 0 by using the asymptotic expressions for the nth eigenvalue (λ n) and those of the corresponding normalized eigenvector (x, λ n) = ( 1 (x, λ n),  2 (x, λ n)) T under the Dirichlet and Neumann boundary conditions. Further, we determine the expressions for the regularized trace matrix for the system with s(x) = t(x) = 1, 0 ≤ x ≤ under the Neumann and general boundary conditions by employing the corresponding asymptotic expressions for the nth eigenvalue (λ n).