A sturm-liouville problem with physical and spectral parameters in boundary conditions

1999 ◽  
Vol 66 (2) ◽  
pp. 127-134 ◽  
Author(s):  
J. Ben Amara ◽  
A. A. Shkalikov
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Spectral Parameters ◽  
Sturm Liouville

Free Acoustic Oscillations Inside a Triaxial Ellipsoid

1995 ◽  
Author(s):  
Tatyana V. Levitina
Keyword(s):  
Boundary Conditions ◽  
Sound Field ◽  
Liouville Problem ◽  
Neumann Condition ◽  
Triaxial Ellipsoid ◽  
Acoustic Oscillations ◽  
Spectral Parameters ◽  
Ellipsoidal Coordinates ◽  
Sturm Liouville ◽  
Eigen Frequencies

Abstract If a Dirichlet or Neumann condition is imposed on the surface of the ellipsoid, the variables are separated in the scalar wave equation in ellipsoidal coordinates, and the problem in hand is reduced to a system of three identical ordinary differential equations, each being defined on a separate interval and subject to its own boundary conditions. Thus, the three-parameter self-adjoint Sturm-Liouville problem arises: the equations are coupled by two separation constants and the eigen frequency of the ellipsoid, i. e., the spectral parameters, which must be so chosen that all the equations of the system have simultaneously nontrivial solutions, each satisfying the corresponding boundary conditions. The effective globally converging numerical algorithm is proposed for calculating eigen frequencies and separation constants. When the modes of an ellipsoid are found, the caustic surfaces can be easily determined. The merit of the method is illustrated on the example of several calculations of the sound field and caustic surfaces in an ellipsoid.

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.

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

2017 ◽  
Vol 53 (1) ◽  
pp. 47-55
Author(s):  
V. A. Sadovnichii ◽  
Ya. T. Sultanaev ◽  
A. M. Akhtyamov
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Sturm Liouville

scholarly journals A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jinming Cai ◽  
Zhaowen Zheng
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Fundamental Solutions ◽  
Asymptotic Formulas ◽  
Limit Circle ◽  
Operator Formulation ◽  
Sturm Liouville ◽  
Completeness Of Eigenfunctions

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

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