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10 pages, 238 KB

Open AccessArticle

Smoothness of the Solution of a Boundary Value Problem for Degenerate Elliptic Equations

by Perizat Beisebay, Yerbulat Akzhigitov, Talgat Akhazhanov, Gulmira Kenzhebekova and Dauren Matin

Symmetry 2025, 17(9), 1562; https://doi.org/10.3390/sym17091562 - 18 Sep 2025

Viewed by 286

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of [...] Read more.

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of degeneracy. The analysis is grounded in several fundamental aspects of symmetry. Structural symmetry is reflected in the formulation of the differential operators; functional symmetry emerges in the properties of the associated weighted Sobolev spaces; and spectral symmetry plays a critical role in the behavior of the eigenvalues and eigenfunctions used to characterize solutions. By employing localization techniques, a priori estimates, and spectral theory, we establish new coefficient conditions ensuring smoothness in both semi-periodic and Dirichlet boundary settings. Moreover, we prove the boundedness and compactness of certain weighted operators, whose definitions and properties are tightly linked to underlying symmetries in the problem’s formulation. These results are not only of theoretical importance but also bear practical implications for numerical methods and models where symmetry principles influence solution regularity and operator behavior. Full article

(This article belongs to the Section Mathematics)

19 pages, 2036 KB

Open AccessArticle

Numerical Treatment of the Time Fractional Diffusion Wave Problem Using Chebyshev Polynomials

by S. S. Alzahrani, Abeer A. Alanazi and Ahmed Gamal Atta

Symmetry 2025, 17(9), 1451; https://doi.org/10.3390/sym17091451 - 4 Sep 2025

Viewed by 532

Abstract

This paper introduces an efficient numerical method based on applying the typical Petrov–Galerkin approach (

PGA

) to solve the time fractional diffusion wave equation (

TFDWE

). The method utilises asymmetric polynomials, namely, shifted second-kind Chebyshev polynomials (

SSKCPs

). New derivative [...] Read more.

This paper introduces an efficient numerical method based on applying the typical Petrov–Galerkin approach (

PGA

) to solve the time fractional diffusion wave equation (

TFDWE

). The method utilises asymmetric polynomials, namely, shifted second-kind Chebyshev polynomials (

SSKCPs

). New derivative formulas are derived and used for these polynomials to establish the operational matrices of their derivatives. The paper presents rigorous error bounds for the proposed method in Chebyshev-weighted Sobolev space and demonstrates its accuracy and efficiency through several illustrative numerical examples. The results reveal that the method achieves high accuracy with relatively low polynomial degrees. Full article

(This article belongs to the Section Mathematics)

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27 pages, 378 KB

Open AccessArticle

Weighted Fractional Sobolev Spaces on Timescales with Applications to Weighted Fractional p-Laplacian Systems

by Qibing Tan, Jianwen Zhou and Yanning Wang

Fractal Fract. 2025, 9(8), 500; https://doi.org/10.3390/fractalfract9080500 - 30 Jul 2025

Viewed by 541

Abstract

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on [...] Read more.

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

(This article belongs to the Special Issue Recent Advances in Nonlocal Problems Involving the Fractional Laplacian Operators)

10 pages, 344 KB

Open AccessArticle

On Estimates of Functions in Norms of Weighted Spaces in the Neighborhoods of Singularity Points

by Viktor A. Rukavishnikov and Elena I. Rukavishnikova

Mathematics 2025, 13(13), 2135; https://doi.org/10.3390/math13132135 - 30 Jun 2025

Viewed by 328

Abstract

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of [...] Read more.

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the

R_{ν}

-generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method. Full article

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25 pages, 310 KB

Open AccessArticle

Weighted Optimal Quadrature Formulas in Sobolev Space and Their Applications

by Kholmat Shadimetov and Khojiakbar Usmanov

Algorithms 2025, 18(7), 374; https://doi.org/10.3390/a18070374 - 20 Jun 2025

Viewed by 378

Abstract

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great [...] Read more.

The optimization of computational algorithms is one of the main problems of computational mathematics. This optimization is well demonstrated by the example of the theory of quadrature and cubature formulas. It is known that the numerical integration of definite integrals is of great importance in basic and applied sciences. In this paper we consider the optimization problem of weighted quadrature formulas with derivatives in Sobolev space. Using the extremal function, the square of the norm of the error functional of the considered quadrature formula is calculated. Then, minimizing this norm by coefficients, we obtain a system to find the optimal coefficients of this quadrature formula. The uniqueness of solutions of this system is proved, and an algorithm for solving this system is given. The proposed algorithm is used to obtain the optimal coefficients of the derivative weight quadrature formulas. It should be noted that the optimal weighted quadrature formulas constructed in this work are optimal for the approximate calculation of regular, singular, fractional and strongly oscillating integrals. The constructed optimal quadrature formulas are applied to the approximate solution of linear Fredholm integral equations of the second kind. Finally, the numerical results are compared with the known results of other authors. Full article

(This article belongs to the Section Analysis of Algorithms and Complexity Theory)

11 pages, 255 KB

Open AccessArticle

Three Bounded Solutions for a Degenerate Nonlinear Dirichlet Problem

by Elisabetta Tornatore

Axioms 2025, 14(4), 289; https://doi.org/10.3390/axioms14040289 - 11 Apr 2025

Viewed by 339

Abstract

The existence of at least three bounded weak solutions is established for a nonlinear elliptic equation with unbounded coefficients. The approach is based on variational methods and critical point theorems. Full article

(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)

22 pages, 266 KB

Open AccessArticle

Spectral Theory and Hardy Spaces for Bessel Operators in Non-Standard Geometries

by Saeed Hashemi Sababe

Mathematics 2025, 13(4), 565; https://doi.org/10.3390/math13040565 - 8 Feb 2025

Viewed by 713

Abstract

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov [...] Read more.

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. The spectral theory of families of boundary-interpolating operators is also expanded, offering precise eigenvalue estimates and functional calculus applications. Furthermore, we explore Bessel operators under non-standard measures, such as fractal and weighted geometries, uncovering new analytical phenomena. Key implications include advanced insights into singular integrals, heat kernel behavior, and the boundedness of Riesz transforms, with potential applications in fractal geometry, constrained wave propagation, and mathematical physics. Full article

(This article belongs to the Special Issue New Perspectives in Harmonic Analysis)

11 pages, 268 KB

Open AccessArticle

Inverse Problems of Recovering Lower-Order Coefficients from Boundary Integral Data

by Sergey Pyatkov and Oleg Soldatov

Axioms 2025, 14(2), 116; https://doi.org/10.3390/axioms14020116 - 1 Feb 2025

Cited by 1 | Viewed by 740

Abstract

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also considered. Overdetermination conditions are the integrals [...] Read more.

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also considered. Overdetermination conditions are the integrals over the boundary of a solution’s domain with weights. We focus on existence and uniqueness theorems and stability estimates for solutions to these inverse problems. An operator equation to which the problem is reduced is studied with the use of the contraction mapping principle. A solution belongs to some Sobolev space and has all generalized derivatives occurring into the equation summable to some power. The method of the proof is constructive, and it can be used for developing new numerical algorithms for solving the problem. Full article

(This article belongs to the Special Issue Axioms and Methods for Handling Differential Equations and Inverse Problems)

24 pages, 337 KB

Open AccessArticle

Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings

by Jiayi Qiu, Guanggui Chen, Yanyan Xu, Ying Luo and Hang Ren

Axioms 2025, 14(1), 42; https://doi.org/10.3390/axioms14010042 - 6 Jan 2025

Viewed by 645

Abstract

This article primarily examines the approximation properties of a weighted Sobolev space

W_{2, κ}^{r}

defined on a sphere

S^{d - 1}

equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The [...] Read more.

This article primarily examines the approximation properties of a weighted Sobolev space

W_{2, κ}^{r}

defined on a sphere

S^{d - 1}

equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The exact asymptotic orders of the Gel’fand n-width and the linear n-width of

W_{2, κ}^{r}

on

S^{d - 1}

are derived for these settings, providing a comprehensive understanding of their approximation characteristics. Full article

(This article belongs to the Section Mathematical Analysis)

22 pages, 580 KB

Open AccessArticle

Identification of Boundary Conditions in a Spherical Heat Conduction Transmission Problem

by Miglena N. Koleva and Lubin G. Vulkov

Symmetry 2024, 16(11), 1507; https://doi.org/10.3390/sym16111507 - 10 Nov 2024

Viewed by 1615

Abstract

Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat [...] Read more.

Although numerous analytical and numerical methods have been developed for inverse heat conduction problems in single-layer materials, few methods address such problems in composite materials. The following paper studies inverse interface problems with unknown boundary conditions by using interior point observations for heat equations with spherical symmetry. The zero degeneracy at the left interval

0 < r < R_{1}

leads to solution difficulties in the one-dimensional interface problem. So, we first investigate the well-posedness of the direct (forward) problem in special weighted Sobolev spaces. Then, we formulate three groups of unknown boundary conditions and inverse problems upon internal point measurements for the heat equation with spherical symmetry. Second-order finite difference scheme approaches for direct and inverse problems are developed. Computational test examples illustrate the theoretical statements proposed. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Partial Differential Equations)

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17 pages, 1741 KB

Open AccessArticle

On the Large-x Asymptotic of the Classical Solutions to the Non-Linear Benjamin Equation in Fractional Sobolev Spaces

by Nabendra Parumasur and Olabisi Aluko

Fractal Fract. 2024, 8(11), 635; https://doi.org/10.3390/fractalfract8110635 - 28 Oct 2024

Viewed by 1252

Abstract

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known

H^{s}

-well-posedness results to the case of [...] Read more.

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known

H^{s}

-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)

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13 pages, 309 KB

Open AccessArticle

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

by Said Mesloub, Eman Alhazzani and Hassan Eltayeb Gadain

Fractal Fract. 2024, 8(9), 526; https://doi.org/10.3390/fractalfract8090526 - 10 Sep 2024

Viewed by 1347

Abstract

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular [...] Read more.

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order

θ \in [0, 1]

. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

20 pages, 1772 KB

Open AccessArticle

Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems

by Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2024, 12(11), 1748; https://doi.org/10.3390/math12111748 - 4 Jun 2024

Viewed by 937

Abstract

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, [...] Read more.

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. Full article

(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems)

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13 pages, 294 KB

Open AccessArticle

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

by Eman Alhazzani, Said Mesloub and Hassan Eltayeb Gadain

Fractal Fract. 2024, 8(4), 189; https://doi.org/10.3390/fractalfract8040189 - 26 Mar 2024

Cited by 4 | Viewed by 1609

Abstract

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used [...] Read more.

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Advanced Results for Cases with Singularities)

22 pages, 338 KB

Open AccessArticle

Weighted Optimal Formulas for Approximate Integration

by Kholmat Shadimetov and Ikrom Jalolov

Mathematics 2024, 12(5), 738; https://doi.org/10.3390/math12050738 - 29 Feb 2024

Viewed by 907

Abstract

Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of [...] Read more.

Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of the differential operator

{[1 - \frac{1}{{(2 π)}^{2}} \frac{d^{2}}{d x^{2}}]}^{[m]}

in the Hilbert space

H_{2}^{μ} (R)

, called

D_{m} [β]

. We modify the Sobolev algorithm to construct optimal quadrature formulas using a discrete operator. We provide a weighted optimal quadrature formula, using this algorithm for the case where

m = 1

. Finally, we construct an optimal quadrature formula in the Hilbert space

H_{2}^{μ} (R)

for the weight functions

p (x) = 1

and

p (x) = e^{2 π i ω x}

when

m = 1

. Full article

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