On General Properties of Eigenvalues and Eigenfunctions of a Sturm–Liouville Operator: Comments on “ISS With Respect to Boundary Disturbances for 1-D Parabolic PDEs”

2017 ◽  
Vol 62 (11) ◽  
pp. 5970-5973 ◽  
Author(s):  
Yury Orlov
Keyword(s):  
Liouville Operator ◽  
Parabolic Pdes ◽  
Eigenvalues And Eigenfunctions ◽  
General Properties ◽  
Sturm Liouville

scholarly journals Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Erdoğan Şen ◽  
Jong Jin Seo ◽  
Serkan Araci
Keyword(s):  
Boundary Value Problem ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Coefficients ◽  
Discontinuous Boundary ◽  
Sturm Liouville ◽  
Special Case

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

Sturm-Liouville Problems with Discontinuities at Two Interior Points

2018 ◽  
Vol 85 (1-2) ◽  
pp. 70
Author(s):  
Hongmei Han
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Basic Solutions ◽  
The Resolvent Operator ◽  
Interior Points

<p>In this paper, we study the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We establish a new operator <em>A</em> associated with the problem, prove the operator <em>A</em> is self-adjoint in an appropriate space <em>H</em>, construct the basic solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions, then obtain asymptotic formulas for the eigenvalues and eigenfunctions, its Green function and the resolvent operator are also involved.</p>

scholarly journals Decay estimates for 1-D parabolic PDES with boundary disturbances

2018 ◽  
Vol 24 (4) ◽  
pp. 1511-1540 ◽  
Author(s):  
Iasson Karafyllis ◽  
Miroslav Krstic
Keyword(s):  
Stability Analysis ◽  
Liouville Operator ◽  
Eigenfunction Expansion ◽  
Decay Estimates ◽  
Parabolic Pdes ◽  
Sturm Liouville ◽  
The Stability ◽  
Nonlocal Terms

In this work, decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the L2 and H1 norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm–Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.

scholarly journals A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Fundamental Solutions ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Interior Points

We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.

Ambarzumyan-type theorem for the impulsive Sturm–Liouville operator

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ran Zhang ◽  
Chuan-Fu Yang
Keyword(s):  
Liouville Operator ◽  
Type Theorem ◽  
Sturm Liouville ◽  
Neumann Laplacian ◽  
Neumann Eigenvalues

AbstractWe prove that if the Neumann eigenvalues of the impulsive Sturm–Liouville operator {-D^{2}+q} in {L^{2}(0,\pi)} coincide with those of the Neumann Laplacian, then {q=0}.

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