MDPI - Publisher of Open Access Journals

14 pages, 543 KB

Open AccessFeature PaperArticle

Numerical Methods for Partial Inverse Spectral Problems with Frozen Arguments on Star-Shaped Graphs

by Chung-Tsun Shieh, Tzong-Mo Tsai and Jyh-Shyang Wu

Mathematics 2026, 14(1), 156; https://doi.org/10.3390/math14010156 - 31 Dec 2025

Viewed by 194

In this paper, the authors investigate a partial inverse spectral problem for Sturm–Liouville operators with frozen arguments on star-shaped graphs. The problem is to reconstruct the potential on one edge from the known potentials on the other edges together with two sequences of [...] Read more.

In this paper, the authors investigate a partial inverse spectral problem for Sturm–Liouville operators with frozen arguments on star-shaped graphs. The problem is to reconstruct the potential on one edge from the known potentials on the other edges together with two sequences of eigenvalues from a prescribed spectral set. The proposed approach is constructive. First, the characteristic function associated with the given spectral data is constructed, allowing the unknown potential contribution to be isolated. The potential is then recovered by expanding the resulting expressions in an appropriate Riesz basis and solving a corresponding system of linear equations. Based on established uniqueness results, this procedure yields a constructive numerical algorithm. Numerical examples demonstrate reliable reconstruction for both smooth and piecewise continuous potentials, providing a practical scheme for frozen-argument problems on star graphs. Full article

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

► Show Figures

Figure 1

26 pages, 382 KB

Open AccessArticle

Some Realisation of the Banach Space of All Continuous Linear Functionals on ℓ₁ Approximated by Weakly Symmetric Continuous Linear Functionals

by Mykhailo Varvariuk and Taras Vasylyshyn

Symmetry 2025, 17(11), 1896; https://doi.org/10.3390/sym17111896 - 6 Nov 2025

Viewed by 281

Abstract

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{1}

(i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation [...] Read more.

A general notion of a weakly symmetric continuous linear functional on a Banach space, in the case where the space is

ℓ_{1}

(i.e., the space of all absolutely summable sequences of complex numbers), reduces to a continuous linear functional whose Riesz representation is a periodic sequence. We consider the completion of the space of all such linear continuous functionals on

ℓ_{1}

with periods of Riesz representations equal to powers of 2. It is known that this completion is a Banach space with a Schauder basis. In this work, we construct a sequence Banach space with the standard Schauder basis

{e_{m} = (\underset{m - 1}{\underset{︸}{0, \dots, 0}}, 1, 0, \dots)}_{m = 1}^{\infty}

that is isometrically isomorphic to this completion. Results of the work can be used to describe spectra of topological algebras of analytic functions on

ℓ_{1}

that can be approximated by weakly symmetric functions. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis: 3rd Edition)

12 pages, 248 KB

Open AccessArticle

On Structral Properties of Some Banach Space-Valued Schröder Sequence Spaces

by Yılmaz Yılmaz, A. Nihal Tuncer and Seçkin Yalçın

Symmetry 2025, 17(7), 977; https://doi.org/10.3390/sym17070977 - 20 Jun 2025

Viewed by 513

Abstract

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties [...] Read more.

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties of vector-valued Schröder sequence spaces. Further, we show that these sequence spaces have a kind of Schauder basis. We also prove that

ℓ_{1} (S, V)

possesses the Dunford–Pettis property and demonstrate that

ℓ_{p} (S, V)

satisfies the approximation property for

1 \leq p < \infty

under certain conditions and

ℓ_{\infty} (S, V)

has the Hahn–Banach extension property. Finally, we show that

ℓ_{2} (S, V)

has the Radon–Riesz property whenever V has it. Full article

(This article belongs to the Section Mathematics)

14 pages, 312 KB

Open AccessArticle

Inverse Problem for a Fourth-Order Hyperbolic Equation with a Complex-Valued Coefficient

by Asselkhan Imanbetova, Abdissalam Sarsenbi and Bolat Seilbekov

Mathematics 2023, 11(15), 3432; https://doi.org/10.3390/math11153432 - 7 Aug 2023

Cited by 3 | Viewed by 1598

Abstract

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form [...] Read more.

This paper studies the existence and uniqueness of the classical solution of inverse problems for a fourth-order hyperbolic equation with a complex-valued coefficient with Dirichlet and Neumann boundary conditions. Using the method of separation of variables, formal solutions are obtained in the form of a Fourier series in terms of the eigenfunctions of a non-self-adjoint fourth-order ordinary differential operator. The proofs of the uniform convergence of the Fourier series are based on estimates of the norms of the derivatives of the eigenfunctions of a fourth-order ordinary differential operator and the uniform boundedness of the Riesz bases of the eigenfunctions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Functional Analysis and Their Applications)

8 pages, 265 KB

Open AccessArticle

Non-Parametric Regression and Riesz Estimators

by Christos Kountzakis and Vasileia Tsachouridou-Papadatou

Axioms 2023, 12(4), 375; https://doi.org/10.3390/axioms12040375 - 14 Apr 2023

Cited by 1 | Viewed by 1738

Abstract

In this paper, we consider a non-parametric regression model relying on Riesz estimators. This linear regression model is similar to the usual linear regression model since they both rely on projection operators. We indicate that Riesz estimator regression relies on the positive basis [...] Read more.

In this paper, we consider a non-parametric regression model relying on Riesz estimators. This linear regression model is similar to the usual linear regression model since they both rely on projection operators. We indicate that Riesz estimator regression relies on the positive basis elements of the finite-dimensional sub-lattice generated by the rows of some design matrix. A strong motivation for using the Riesz estimator model is that the data of explanatory variables may come from categorical variables. Calculations related to Riesz estimator regression are very easy since they arise from the measurability in finite-dimensional probability spaces. Moreover, we show that the fitted model of Riesz estimators is an ordinary least squares model. Any vector of some Euclidean space is supposed to be a rendom variable under the objective probability values, being used in expected utility theory and its applications. Finally, the reader may notice that goodness-of-fit measures are similar to those defined for the usual linear regression. Due to the fact that this model is non-parametric, it may include samples relevant to finance and actuarial science variables. Full article

(This article belongs to the Special Issue Mathematical and Computational Finance Analysis)

12 pages, 329 KB

Open AccessArticle

Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

by Mokhtar Kirane and Abdissalam A. Sarsenbi

Fractal Fract. 2023, 7(2), 131; https://doi.org/10.3390/fractalfract7020131 - 30 Jan 2023

Cited by 13 | Viewed by 2792

Abstract

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems [...] Read more.

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

17 pages, 873 KB

Open AccessArticle

Binary Operations in Metric Spaces Satisfying Side Inequalities

by María A. Navascués, Pasupathi Rajan and Arya Kumar Bedabrata Chand

Mathematics 2022, 10(1), 11; https://doi.org/10.3390/math10010011 - 21 Dec 2021

Cited by 3 | Viewed by 3188

Abstract

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was [...] Read more.

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space

E .

We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space. Full article

(This article belongs to the Special Issue Fractal and Computational Geometry)

► Show Figures

Figure 1

10 pages, 1425 KB

Open AccessArticle

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

by Gábor Maros and Ferenc Izsák

Fractal Fract. 2021, 5(3), 75; https://doi.org/10.3390/fractalfract5030075 - 21 Jul 2021

Viewed by 2204

Abstract

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation [...] Read more.

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment. Full article

(This article belongs to the Section Numerical and Computational Methods)

► Show Figures

Figure 1

14 pages, 338 KB

Open AccessEditor’s ChoiceArticle

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

by Emilia Bazhlekova

Fractal Fract. 2021, 5(3), 63; https://doi.org/10.3390/fractalfract5030063 - 30 Jun 2021

Cited by 6 | Viewed by 2669

Abstract

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction [...] Read more.

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established. Full article

(This article belongs to the Special Issue Fractional Dynamics: Theory and Applications)

14 pages, 669 KB

Open AccessArticle

The Dirichlet Problem for the Perturbed Elliptic Equation

by Ulyana Yarka, Solomiia Fedushko and Peter Veselý

Mathematics 2020, 8(12), 2108; https://doi.org/10.3390/math8122108 - 25 Nov 2020

Cited by 17 | Viewed by 2427

Abstract

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier [...] Read more.

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation. Full article

(This article belongs to the Special Issue Mathematical Methods and Analysis for the Industrial Management and Business)

► Show Figures

Figure 1

21 pages, 5393 KB

Open AccessArticle

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

by Dana Černá and Václav Finĕk

Axioms 2017, 6(1), 4; https://doi.org/10.3390/axioms6010004 - 22 Feb 2017

Cited by 15 | Viewed by 5483

Abstract

We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets [...] Read more.

We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefﬁcients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efﬁciency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility. Full article

(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)

► Show Figures

Figure 1

Search Results (11)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (11)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI