Sturm-Liouville Research Articles

Wave equation for Sturm-Liouville operator with singular potentials

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.

Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

Uniqueness and reconstruction of the periodic Strum–Liouville operator with a finite number discontinuities

In this paper we study the inverse problem for Sturm–Liouville operator with periodic condition and a finite number discontinuities. We give the uniqueness theorem by two spectra and the sign sets, and the reconstruction algorithm.

EIGENVALUES OF STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER DEPENDENT BOUNDARY AND INTERFACE CONDITIONS

In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.

Using the Accelerated Convergence Method for Solving the Singular Sturm–Liouville Problem

This article is dedicated to the memory of L.D. Akulenko, with whom the author of the article worked for more than 40 years. Within the framework of the accelerated convergence method developed jointly, a number of classes of problems related to the Sturm–Liouville problems were solved. Based on the research results, several dozen articles and a generalizing monograph [1] were published. In this paper, we describe the adaptation of the method to solving singular Sturm–Liouville problems.

Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity

Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity

Lower bounds for self-adjoint Sturm–Liouville operators

In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm–Liouville differential expression 1 r ( − d d x p d d x + q ) \begin{equation*} \frac {1}{r} \left (-\frac {\mathrm d}{\mathrm dx} p \frac {\mathrm d}{\mathrm dx} + q\right ) \end{equation*} in the weighted L 2 L^2 -Hilbert space L 2 ( R ; r d x ) L^2(\mathbb R; rdx) .

On the polynomial integrability of the critical systems for optimal eigenvalue gaps

This exploration consists of two parts. First, we will deduce a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent p ∈ (1, ∞) of the Lebesgue spaces concerned. These systems can be used to obtain the optimal lower or upper bounds for eigenvalue gaps of Sturm–Liouville operators and are equivalent to non-convex Hamiltonian systems of two degrees of freedom. Second, with appropriate choices of exponents p, the critical systems are polynomial systems in four dimensions. These systems will be investigated from two aspects. The first one is that by applying the canonical transformation and the Darboux polynomial, we obtain the necessary and sufficient conditions for polynomial integrability of these polynomial critical systems. As a special example, we conclude that the system with p = 2 is polynomial completely integrable in the sense of Liouville. The second is that the linear stability of isolated singular points is characterized. By performing the Poincaré cross section technique, we observe that the systems have very rich dynamical behaviors, including periodic trajectories, quasi-periodic trajectories, and chaos.

Integration of the negative order Korteweg-de Vries equation by the inverse scattering method

In this paper, we consider the integration of the negative order Korteweg-de Vries equation by the inverse scattering method. The evolution of the spectral data of the Sturm-Liouville operator with a potential associated with the solution of the negative order Korteweg-de Vries equation is determined. The obtained results make it possible to apply the method of inverse scattering problem to solve the negative order Korteweg-de Vries equation in the class of rapidly decreasing functions.

OPTIMIZATION SPECTRAL PROBLEM FOR THE STURM-LIOUVILLE OPERATOR IN THE SPACE OF VECTOR FUNCTIONS

An inverse spectral optimization problem is considered: for a given matrix potential \({{Q}_{0}}(x)\) it is required to find the matrix function \(\hat {Q}(x)\) closest to it, such that the k-th eigenvalue of the Sturm–Liouville matrix operator with potential \(\hat {Q}(x)\) matched the given value \(\lambda {\kern 1pt} *\). The main result of the paper is the proof of existence and uniqueness theorems. Explicit formulas for the optimal potential are established through solutions to systems of nonlinear differential equations of the second order, known in mathematical physics as systems of nonlinear Schrödinger equations

The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si–SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below 4.2 K) is a challenging task, which mostly lies in the discontinuities in material properties (e.g. effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered finite-volume discretization of the Sturm–Liouville form of the effective-mass Schrödinger equation and generalized Poisson’s equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas formed at the Si–SiO2 interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as 50 mK) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron–electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters.

A Conformable Inverse Problem with Constant Delay

This paper aims to express the solution of an inverse Sturm-Liouville problem with constant delay using a conformable derivative operator under mixed boundary conditions. For the problem, we stated and proved the specification of the spectrum. The asymptotics of the eigenvalues of the problem was obtained and the solutions were extended to the Regge-type boundary value problem. As such, a new result, as an extension of the classical Sturm-Liouville problem to the fractional phenomenon, has been achieved. The uniqueness theorem for the solution of the inverse problem is proved in different cases within the interval (0,π). The results in the classical case of this problem can be obtained at α=1. 2000 Mathematics Subject Classification. 34L20,34B24,34L30.

Inverse nodal problems for Sturm–Liouville operators on quantum star graphs

In this work, we study the nodal inverse problem for Sturm–Liouville operators on quantum star graphs. We show that the zeros of eigenfunctions characterize the potential, and we present a simple algorithm that enables us to approximate it. The method does not rely on a priori information about the asymptotic behavior of eigenvalues. We manage to reduce the problem to a family of constant coefficient eigenvalue problems that can be solved explicitly and contain enough information to recover the potential. We also consider the multiplicity of eigenvalues and their vanishing properties, which only hold generically.

Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative.

Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

This paper focuses on the study of Sturm-Liouville eigenvalue problems. In the classical Chebyshev collocation method, the Sturm-Liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled Chebyshev points. We are concerned with the computation of high-order eigenvalues of Sturm-Liouville problems using an effective method of discretization based on the Chebfun software algorithms with domain truncation. We solve some numerical Sturm-Liouville eigenvalue problems and demonstrate the efficiency of computations.

Eigenvalues of the Laplace-Beltrami operator on a prolate spheroid

The Laplace-Beltrami operator on a prolate spheroid admits a sequence of eigenvalues. These eigenvalues are determined by a singular Sturm-Liouville problem. Properties of the eigenvalues are obtained using the minimum-maximum principle and the Prüfer angle. In particular, eigenvalues are approximated by those of generalized matrix eigenvalue problems, and their behavior is studied when the eccentricity of the spheroid approaches 0 or 1.

Friedrichs extensions of a class of discrete Hamiltonian systems with one singular endpoint

AbstractThis paper is concerned with Friedrichs extensions for a class of discrete Hamiltonian systems with one singular endpoint. First, Friedrichs extensions of symmetric Hamiltonian systems are characterized by imposing some constraints on each element of domains of the maximal relations H. Furthermore, it is proved that the Friedrichs extension of each of a class of non‐symmetric systems is also a restriction of the maximal relation H by using a closed sesquilinear form. Then, the corresponding Friedrichs extensions are characterized. In addition, ‐self‐adjoint Friedrichs extensions are studied, and two results are given for elements of , which make the expression of the Friedrichs extension simpler. All results are finally applied to Sturm–Liouville equations with matrix‐valued coefficients.

ON THE APPLICATION OF THE SOLUTION OF THE DEGENERATE NONLINEAR BURGERS EQUATION WITH A SMALL PARAMETER AND THE THEORY OF <i>p</i>-REGULARITY

The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form\(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,\) (1)where \(F:\Omega \to C([0,\pi ] \times [0,T])\), \(T 0\), \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}\) and \(u(0,t) = u(\pi ,t) = 0\), \(u(x,0) = \varphi (x)\), \(f(x,t) \in C([0,\pi ] \times [0,T])\), \(\varphi (x) \in C[0,\pi ]\). We will be interested in the most important in applications case of a small parameter ε with oscillating initial conditions of the form \(\varphi (x) = k\sin x\), where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), which corresponds to \(k = k{\kern 1pt} * = 0\) and at what initial Under certain conditions on the values of k, it is possible to construct an analytical approximation of this solution for small ε.We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form \(u(x,t) = v(t)u(x)\), where \(v(t) = c{{e}^{{ - t}}}\), \(u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])\). In this case, the problem under consideration is degenerate at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), since \({\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])\). This follows from the Sturm-Liouville theory. To achieve our goals, we apply the apparatus of p-regularity theory [6, 7, 15, 16] and show that the mapping \(F(u,\varepsilon )\) is 3-regular at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), т.е. p = 3.

Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem

Three-Point Difference Schemes of High Order of Accuracy for the Sturm–Liouville Problem