scholarly journals Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
M. M. Tharwat ◽  
A. H. Bhrawy ◽  
A. S. Alofi
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Sampling Theorem ◽  
High Accuracy ◽  
Internal Point ◽  
Derivative Sampling ◽  
Sturm Liouville ◽  
Truncation And Amplitude Errors ◽  
Hermite Approximation ◽  
Hermite Interpolations

The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.

scholarly journals Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
M. M. Tharwat ◽  
S. M. Al-Harbi
Keyword(s):  
Boundary Conditions ◽  
Error Bounds ◽  
Eigenvalue Problems ◽  
Sampling Theorem ◽  
Eigenvalue Parameter ◽  
Derivative Sampling ◽  
Sturm Liouville ◽  
Truncation And Amplitude Errors ◽  
Computable Error Bounds ◽  
Hermite Interpolations

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper, we use the derivative sampling theorem “Hermite interpolations” to compute approximate values of the eigenvalues of Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

Fully Legendre Spectral Galerkin Algorithm for Solving Linear One-Dimensional Telegraph Type Equation

2019 ◽  
Vol 16 (08) ◽  
pp. 1850118 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Error Analysis ◽  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Type Equation ◽  
High Accuracy ◽  
Basis Functions ◽  
Structured Matrices ◽  
One Dimensional ◽  
Wide Applicability ◽  
Careful Investigation

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

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.

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

Sturm-Liouville problems with singular non-selfadjoint boundary conditions

2005 ◽  
Vol 278 (12-13) ◽  
pp. 1509-1523 ◽  
Author(s):  
Walter Eberhard ◽  
Gerhard Freiling ◽  
Anton Zettl
Keyword(s):  
Boundary Conditions ◽  
Sturm Liouville
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