New series of asymptotics of eigenvalues of the Sturm-Liouville problem with rapidly oscillating coefficients

1992 ◽  
Vol 52 (5) ◽  
pp. 1134-1136
Author(s):  
S. A. Nazarov
Keyword(s):  
Liouville Problem ◽  
Asymptotics Of Eigenvalues ◽  
Oscillating Coefficients ◽  
Sturm Liouville

scholarly journals Reconstruction of potential function in inverse Sturm-Liouville problem via partial data

2021 ◽  
Vol 11 (2) ◽  
pp. 186-198
Author(s):  
Mehmet Açil ◽  
Ali Konuralp
Keyword(s):  
Boundary Conditions ◽  
Potential Function ◽  
Liouville Problem ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Asymptotics Of Eigenvalues ◽  
Time Saving ◽  
Partial Data ◽  
Sturm Liouville ◽  
Saving Algorithm

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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