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Open AccessArticle

Trace Formulae for Second-Order Differential Pencils with a Frozen Argument

by

Yi-Teng Hu

and

Murat Şat

Mathematics 2023, 11(18), 3996; https://doi.org/10.3390/math11183996 - 20 Sep 2023

Viewed by 174

Abstract

This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, [...] Read more.

This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

Open AccessArticle

Keller–Osserman Phenomena for Kardar–Parisi–Zhang-Type Inequalities

by

Andrey B. Muravnik

Mathematics 2023, 11(17), 3787; https://doi.org/10.3390/math11173787 - 04 Sep 2023

Viewed by 300

Abstract

For coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results [...] Read more.

For coercive quasilinear partial differential inequalities containing nonlinearities of the Kardar–Parisi–Zhang type, we find conditions guaranteeing the absence of global positive solutions. These conditions extend both the classical result of Keller and Osserman and its recent Kon’kov–Shishkov generalization. Additionally, they complement the results for the noncoercive case, which had been previously established by the same author. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

Open AccessFeature PaperArticle

An Approach to Solving Direct and Inverse Scattering Problems for Non-Selfadjoint Schrödinger Operators on a Half-Line

by

Vladislav V. Kravchenko

and

Lady Estefania Murcia-Lozano

Mathematics 2023, 11(16), 3544; https://doi.org/10.3390/math11163544 - 16 Aug 2023

Viewed by 337

Abstract

In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost [...] Read more.

In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost solution in a unit disk of a complex variable related to the spectral parameter by a Möbius transformation. This representation leads to an efficient method of solving the corresponding direct scattering problem for a given potential, while the solution to the inverse problem is reduced to the computation of the first coefficient of the power series from a system of linear algebraic equations. The approach to solving these direct and inverse scattering problems is illustrated by several explicit examples and numerical testing. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

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Open AccessArticle

Direct Method for Identification of Two Coefficients of Acoustic Equation

by

Nikita Novikov

and

Maxim Shishlenin

Mathematics 2023, 11(13), 3029; https://doi.org/10.3390/math11133029 - 07 Jul 2023

Viewed by 338

Abstract

We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein [...] Read more.

We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

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Open AccessArticle

Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions

by

Tzong-Mo Tsai

Mathematics 2023, 11(12), 2787; https://doi.org/10.3390/math11122787 - 20 Jun 2023

Viewed by 417

Abstract

In this paper, we consider the differential equation

y^{″} + ω^{2} ρ (x) y = 0

, where

ω

is a positive parameter. The principal concern here is to find conditions on the function [...] Read more.

In this paper, we consider the differential equation

y^{″} + ω^{2} ρ (x) y = 0

, where

ω

is a positive parameter. The principal concern here is to find conditions on the function

ρ^{- 1 / 2} (x)

which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for sufficiently large

ω

. In particular, if

c_{ν k} (α)

denotes the kth positive zero of the general Bessel (cylinder) function

C_{ν} (x; α) = J_{ν} (x) cos α - Y_{ν} (x) sin α

of order

ν

and if

| ν | < 1 / 2

, we prove that

{(- 1)}^{m} Δ^{m + 2} c_{ν k} (α) > 0 (m = 0, 1, 2, \dots; k = 1, 2, \dots),

where

Δ a_{k} = a_{k + 1} - a_{k} .

This type of inequalities was conjectured by Lorch and Szego in 1963. In addition, we show that the differences of the zeros of various orthogonal polynomials with higher degrees possess sign regularity. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

Open AccessArticle

A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives

by

Vladimir E. Fedorov

and

Nikolay V. Filin

Mathematics 2023, 11(11), 2472; https://doi.org/10.3390/math11112472 - 27 May 2023

Viewed by 474

Abstract

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the [...] Read more.

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

Open AccessArticle

Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem

by

,

,

and

Mathematics 2022, 10(22), 4204; https://doi.org/10.3390/math10224204 - 10 Nov 2022

Viewed by 700

Abstract

In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function [...] Read more.

In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function for the boundary value problem by the first kind Chebyshev wavelet method. Two numerical examples show that the first kind Chebyshev wavelet method for solving the inverse nodal problems for the boundary value problem is valid. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

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Open AccessArticle

The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph

by

Xian-Biao Wei

,

Yan-Hsiou Cheng

and

Yu-Ping Wang

Mathematics 2022, 10(21), 3971; https://doi.org/10.3390/math10213971 - 26 Oct 2022

Cited by 1 | Viewed by 560

Abstract

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from [...] Read more.

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

Open AccessArticle

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

by

Natalia P. Bondarenko

Mathematics 2022, 10(20), 3882; https://doi.org/10.3390/math10203882 - 19 Oct 2022

Cited by 5 | Viewed by 643

Abstract

This paper is concerned with inverse spectral problems for higher-order (

n > 2

) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our [...] Read more.

This paper is concerned with inverse spectral problems for higher-order (

n > 2

) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for the constructive solution of inverse spectral problems and for the investigation of their solvability and stability. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

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Review

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Open AccessReview

Partial Inverse Sturm-Liouville Problems

by

Natalia P. Bondarenko

Mathematics 2023, 11(10), 2408; https://doi.org/10.3390/math11102408 - 22 May 2023

Viewed by 614

Abstract

This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) [...] Read more.

This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) from spectral characteristics, while the coefficients in the remaining part of the domain are known a priori. Usually, partial inverse problems require less spectral data than complete inverse problems. In this review, we pay considerable attention to partial inverse problems on graphs and to the unified approach based on the reduction of partial inverse problems to Sturm-Liouville problems with entire analytic functions in a boundary condition. We not only describe the results of selected studies but also compare them with each other and establish interconnections. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

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