Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials (original) (raw)
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An algorithm for discontinuous inverse Sturm-Liouville problems with symmetric potentials
Computers & mathematics with applications, 1989
Almtriet-In this paper we consider two Sturm-Liouville problems with symmetric potentials and symmetric discontinuities satisfying symmetric boundary and jump conditions. In the first section we derive a simple expression for the difference of potentials when only a finite number of eigenvalues differ. In the second section this result is used to construct an algorithm for solving the discontinuous inverse Sturm-Liouville problem numerically.
Discontinuous Inverse Sturm-Liouville Problems with Symmetric Potentials
1988
In this paper we study the Inverse Sturm-Liouville problem on a finite interval with a symmetric potential function with two interior discontinuities. In the introductory chapter we survey previous results on the existence and uniqueness of solutions to inverse Sturm-Liouville problems and discuss earlier numerical methods. In chapter 1 we present a uniqueness proof for the inverse Sturm-Liouville problem on a finite interval with a symmetric potential having two interior jump discontinuities. In chapter 2 we show that any absolutely continuous function can be expanded in terms of the eigenfunctions of a Sturm-Liouville problem with two discontinuities. In chapter 3 we consider two Sturm-Liouville problems with different symmetric potentials with two discontinuities satisfying symmetric boundary conditions and symmetric jump conditions. We find that if only a finite number of eigenvalues differ then a simple expression for the difference of the potentials can be established. In addition, the locations of the discontinuities are uniquely determined. Finally, in chapter 4 we derive an algorithm for solving the discontinuous inverse Sturm-Liouville problem numerically and present the results of numerical experiments.
Boundary Value Problems, 2013
In this paper, we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than that of the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.
Discontinuous Sturm-Liouville Problems Involving an Abstract Linear Operator
Journal of Applied Analysis & Computation, 2020
In this paper we introduce to consideration a new type boundary value problems consisting of an "Sturm-Liouville" equation on two disjoint intervals as −p(x)y ′′ + q(x)y + By|x = µy, x ∈ [a, c) ∪ (c, b] together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction x = c, where B : L2(a, c) ⊕ L2(c, b) → L2(a, c)⊕L2(c, b) is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define "alternative" inner products in Sobolev spaces, "incorporating" with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator B.
Computational procedure for Sturm-Liouville problems
Journal of Computational Physics, 1983
have been incorporated to develop a procedure for the automatic computation of the eigenvalues and the eigenfunctions of one-dimensional linear Sturm-Liouville boundary value eigenproblems, both singular and nonsingular. The continuous coefftcients of a regular Sturm-Liouville problem have been approximated by a finite number of step functions. In each step the resulting boundary value problem has been integrated exactly and the solutions have then been matched to construct the continuously differentiable solution of the original problem and the corresponding eigencondition.
On a discontinuous Sturm-Liouville type problem with retarded argument
2016
In this work a Sturm-Liouville type problem with retarded argument which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.
A hierarchy of Sturm-Liouville problems
Mathematical Methods in the Applied Sciences, 2003
Sturm-Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classes of these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to those of the 'usual' Sturm-Liouville problem which has constant boundary conditions. In fact it will be shown how these classes can be converted into each other, and into the 'usual' Sturm-Liouville problem, by means of transformations preserving all but ÿnitely many eigenvalues. Copyright ? 2003 John Wiley & Sons, Ltd.
Inverse Sturm-Liouville problem with discontinuity conditions
This paper deals with the boundary value problem involving the di erential equation ly := -y''+qy = λy, subject to the standard boundary conditions along with the following discontinuity conditions at a point a ε(0,π) y(a + 0) = a1y(a - 0), y'(a + 0) = a1-1y'(a - 0) + a2y(a - 0), where q(x), a1,a2 are real, q ε L2(0,π ) and λ is a parameter independent of x. We develop the Hochestadt's result based on the transformation operator for inverse Sturm-Liouville problem when there are discontinuous conditions. Furthermore, we establish a formula for q(x) - q~(x) in the finite interval where q(x) and q~(x) are analogous functions.