scholarly journals Inverse problems for Sturm-Liouville difference equations

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1297-1304 ◽  
Author(s):  
Martin Bohner ◽  
Hikmet Koyunbakan
Keyword(s):  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville

We consider a discrete Sturm-Liouville problem with Dirichlet boundary conditions. We show that the specification of the eigenvalues and weight numbers uniquely determines the potential. Moreover, we also show that if the potential is symmetric, then it is uniquely determined by the specification of the eigenvalues. These are discrete versions of well-known results for corresponding differential equations.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2265
Author(s):  
Malgorzata Klimek
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivatives ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville ◽  
Left And Right ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Liouville Operators

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

scholarly journals A counterexample in Sturm–Liouville completeness theory

2004 ◽  
Vol 134 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Paul Binding ◽  
Branko Ćurgus
Keyword(s):  
Boundary Conditions ◽  
Riesz Basis ◽  
Periodic Boundary ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Indefinite Weight ◽  
Sturm Liouville

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

scholarly journals A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem

2021 ◽  
Vol 14 (3) ◽  
pp. 706-722
Author(s):  
Francis Ohene Boateng ◽  
Joseph Ackora-Prah ◽  
Benedict Barnes ◽  
John Amoah-Mensah
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Finite Difference Methods ◽  
Dirichlet Boundary Conditions ◽  
Fictitious Domain ◽  
Dirichlet Boundary Value Problem ◽  
Wavelet Method ◽  
Difference Methods

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic  partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

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