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27 pages, 378 KiB

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 175

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)

29 pages, 2344 KiB

Open AccessArticle

A Discrete Model to Solve a Bifractional Dissipative Sine-Gordon Equation: Theoretical Analysis and Simulations

by Dagoberto Mares-Rincón, Siegfried Macías, Jorge E. Macías-Díaz, José A. Guerrero-Díaz-de-León and Tassos Bountis

Fractal Fract. 2025, 9(8), 498; https://doi.org/10.3390/fractalfract9080498 - 30 Jul 2025

Viewed by 281

Abstract

In this work, we consider a generalized form of the classical

(2 + 1)

-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two [...] Read more.

In this work, we consider a generalized form of the classical

(2 + 1)

-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

(This article belongs to the Special Issue Fractional Nonlinear Dynamics in Science and Engineering)

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13 pages, 549 KiB

Open AccessFeature PaperArticle

The Resistance Distance Is a Diffusion Distance on a Graph

by Ernesto Estrada

Mathematics 2025, 13(15), 2380; https://doi.org/10.3390/math13152380 - 24 Jul 2025

Viewed by 141

Abstract

The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing [...] Read more.

The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove that the resistance distance is given by a difference of “mass concentrations” obtained at the vertices of a graph by a diffusive process. The nature of this diffusive process is characterized here by means of an operator corresponding to the matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian. We prove also that this operator is indeed the Laplacian matrix of a signed version of the original graph, in which nonnearest neighbors’ “interactions” are also considered. In this way, the resistance distance is part of a family of squared Euclidean distances emerging from diffusive dynamics on graphs. Full article

(This article belongs to the Special Issue Advances in Combinatorics, Discrete Mathematics and Graph Theory)

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21 pages, 10783 KiB

Open AccessArticle

An ALoGI PU Algorithm for Simulating Kelvin Wake on Sea Surface Based on Airborne Ku SAR

by Limin Zhai, Yifan Gong and Xiangkun Zhang

Sensors 2025, 25(14), 4508; https://doi.org/10.3390/s25144508 - 21 Jul 2025

Viewed by 339

Abstract

The airborne Synthetic Aperture Radar (SAR) has the advantages of high-precision real-time observation of wave height variations and portability in the high frequency band, such as the Ku band. In view of the Four Fast Fourier Transform (4-FFT) algorithm, combined with a Gaussian [...] Read more.

The airborne Synthetic Aperture Radar (SAR) has the advantages of high-precision real-time observation of wave height variations and portability in the high frequency band, such as the Ku band. In view of the Four Fast Fourier Transform (4-FFT) algorithm, combined with a Gaussian operator, a Laplacian of Gaussian (LoG) Phase Unwrapping (PU) expression was derived. Then, an Adaptive LoG (ALoG) algorithm was proposed based on adaptive variance, further optimizing the algorithm through iteration. Building the models of Kelvin wake on the sea surface and height to phase, the interferometric phase of wave height can be simulated. These PU algorithms were qualitatively and quantitatively evaluated. The Principal Component Analysis (PCA) scores of the ALoG iteration (ALoGI) algorithm are the best under the tested noise levels of the simulation. Through a simulation experiment, it has been proven that the superiority of the ALoGI algorithm in high spatial resolution inversion for the sea-ship surface height of the Kelvin wake, with good stability and noise resistance. Full article

(This article belongs to the Section Radar Sensors)

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22 pages, 386 KiB

Open AccessArticle

Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains

by Abdelkader Moumen, Sabri T. M. Thabet, Hussien Albala, Khaled Aldwoah, Hicham Saber, Eltigani I. Hassan and Alawia Adam

Fractal Fract. 2025, 9(7), 442; https://doi.org/10.3390/fractalfract9070442 - 3 Jul 2025

Viewed by 383

Abstract

This article concerns a novel coupled implicit differential system under

φ

–Riemann–Liouville (

RL

) fractional derivatives with

p

-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains [...] Read more.

This article concerns a novel coupled implicit differential system under

φ

–Riemann–Liouville (

RL

) fractional derivatives with

p

-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains

[c, \infty)

. The explicit iterative solution’s existence and uniqueness (

E a U

) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias (

UHR

) stabilities are investigated. Ultimately, we provide a numerical application of a coupled

φ

-

RL

fractional turbulent flow model to illustrate and test the effectiveness of our outcomes. Full article

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

22 pages, 501 KiB

Open AccessArticle

Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations

by Yushan Li, Yuxuan Yang and Nanbo Chen

Mathematics 2025, 13(13), 2123; https://doi.org/10.3390/math13132123 - 28 Jun 2025

Viewed by 250

Abstract

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit [...] Read more.

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology. Full article

(This article belongs to the Special Issue Mathematical Methods and Models in Information Security with Applications)

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18 pages, 349 KiB

Open AccessArticle

A Brézis–Oswald-Type Result for the Fractional (r, q)-Laplacian Problems and Its Application

by Yun-Ho Kim and In Hyoun Kim

Fractal Fract. 2025, 9(7), 412; https://doi.org/10.3390/fractalfract9070412 - 25 Jun 2025

Viewed by 364

Abstract

This study derives the uniqueness of positive solutions to Brézis–Oswald-type problems involving the fractional

(r, q)

-Laplacian operator and discontinuous Kirchhoff-type coefficients. The Brézis–Oswald-type result and Ricceri’s abstract global minimum principle are critical tools in identifying this uniqueness. We consider [...] Read more.

This study derives the uniqueness of positive solutions to Brézis–Oswald-type problems involving the fractional

(r, q)

-Laplacian operator and discontinuous Kirchhoff-type coefficients. The Brézis–Oswald-type result and Ricceri’s abstract global minimum principle are critical tools in identifying this uniqueness. We consider an eigenvalue problem associated with fractional

(r, q)

-Laplacian problems to confirm the existence of a positive solution for our problem without the Kirchhoff coefficient. Moreover, we establish the uniqueness result of the Brézis–Oswald type by exploiting a generalization of the discrete Picone inequality. Full article

(This article belongs to the Section General Mathematics, Analysis)

19 pages, 388 KiB

Open AccessArticle

The Maximal Regularity of Nonlocal Parabolic Monge–Ampère Equations and Its Monotonicity in the Whole Space

by Xingyu Liu

Axioms 2025, 14(7), 491; https://doi.org/10.3390/axioms14070491 - 24 Jun 2025

Cited by 1 | Viewed by 265

Abstract

The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to [...] Read more.

The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to seek the maximal regularity of the parabolic Monge–Ampère equations by following an approach similar to that used for finding the maximal regularity of the parabolic fractional Laplacian operator. The maximal regularity of nonlocal parabolic Monge–Ampère equations guarantees the existence of solutions in the whole space. Based on these conditions, a modified sliding method, an enhancement of the moving planes method, is employed to establish the monotonicity property of the solutions for the nonlocal parabolic Monge–Ampère equations in the whole space. Full article

(This article belongs to the Special Issue Current Perspectives in Fractional Calculus: Theory, Methods, and Applications)

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19 pages, 8386 KiB

Open AccessArticle

An Ultra-Precision Smoothing Polishing Model for Optical Surface Fabrication with Morphology Gradient Awareness

by Guohao Liu, Yonghong Deng and Zhibin Li

Micromachines 2025, 16(7), 734; https://doi.org/10.3390/mi16070734 - 23 Jun 2025

Viewed by 398

Abstract

To improve the surface morphology quality of ultra-precision optical components, particularly in the suppression of mid-spatial frequency (MSF) errors, this paper proposes a morphology gradient-aware spatiotemporal coupled smoothing model based on convolutional material removal. By introducing the Laplacian curvature into the surface evolution [...] Read more.

To improve the surface morphology quality of ultra-precision optical components, particularly in the suppression of mid-spatial frequency (MSF) errors, this paper proposes a morphology gradient-aware spatiotemporal coupled smoothing model based on convolutional material removal. By introducing the Laplacian curvature into the surface evolution framework, a curvature-sensitive “peak-priority” mechanism is established to dynamically guide the local dwell time. A nonlinear spatiotemporal coupling equation is constructed, in which the dwell time is adaptively modulated by surface gradient magnitude, local curvature, and periodic fluctuation terms. The material removal process is modeled as the convolution of a spatially invariant removal function with a locally varying dwell time distribution. Moreover, analytical evolution expressions of PV, RMS, and PSD metrics are derived, enabling a quantitative assessment of smoothing performance. Simulation results and experimental validations demonstrate that the proposed model can significantly improve smoothing performance and enhance MSF error suppression. Full article

(This article belongs to the Section A1: Optical MEMS and Photonic Microsystems)

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13 pages, 289 KiB

Open AccessArticle

Finite Difference/Fractional Pertrov–Galerkin Spectral Method for Linear Time-Space Fractional Reaction–Diffusion Equation

by Mahmoud A. Zaky

Mathematics 2025, 13(11), 1864; https://doi.org/10.3390/math13111864 - 3 Jun 2025

Cited by 3 | Viewed by 528

Abstract

Achieving high-order accuracy in finite difference/spectral methods for space-time fractional differential equations often relies on very restrictive and usually unrealistic smoothness assumptions in the spatial and/or temporal domains. For spatial discretization, spectral methods using smooth basis functions are commonly employed. However, spatial–fractional derivatives [...] Read more.

Achieving high-order accuracy in finite difference/spectral methods for space-time fractional differential equations often relies on very restrictive and usually unrealistic smoothness assumptions in the spatial and/or temporal domains. For spatial discretization, spectral methods using smooth basis functions are commonly employed. However, spatial–fractional derivatives pose challenges, as they often lack guaranteed spatial smoothness, requiring non-smooth basis functions. In the temporal domain, finite difference schemes on uniformly graded meshes are commonly employed; however, achieving accuracy remains challenging for non-smooth solutions. In this paper, an efficient algorithm is adopted to improve the accuracy of finite difference/Pertrov–Galerkin spectral schemes for a time-space fractional reaction–diffusion equation, with a hyper-singular integral fractional Laplacian and non-smooth solutions in both time and space domains. The Pertrov–Galerkin spectral method is adapted using non-smooth generalized basis functions to discretize the spatial variable, and the L1 scheme on a non-uniform graded mesh is used to approximate the Caputo fractional derivative. The unconditional stability and convergence are established. The rate of convergence is

O (N^{μ - γ} + K^{- min {ρ β, 2 - β}}),

achieved without requiring additional regularity assumptions on the solution. Finally, numerical results are provided to validate our theoretical findings. Full article

(This article belongs to the Special Issue Advanced Research in Numerical Analysis of Partial Differential Equations)

12 pages, 250 KiB

Open AccessArticle

The Symmetric Derivative and Related Operators for Symmetric Forms with Polynomial Coefficients in

R^{n}

by Anna Kimaczyńska

Symmetry 2025, 17(6), 860; https://doi.org/10.3390/sym17060860 - 1 Jun 2025

Viewed by 303

Abstract

The symmetric derivative and related operators for symmetric forms with polynomial coefficients in

R^{n}

are investigated. The adjoints of these operators, with respect to the scalar product for symmetric forms with polynomial coefficients, are found and proved to be of a pure [...] Read more.

The symmetric derivative and related operators for symmetric forms with polynomial coefficients in

R^{n}

are investigated. The adjoints of these operators, with respect to the scalar product for symmetric forms with polynomial coefficients, are found and proved to be of a pure algebraic character. Full article

(This article belongs to the Section Mathematics)

11 pages, 2243 KiB

Open AccessArticle

Substrate Activation Efficiency in Active Sites of Hydrolases Determined by QM/MM Molecular Dynamics and Neural Networks

by Igor V. Polyakov, Yulia I. Meteleshko, Tatiana I. Mulashkina, Mikhail I. Varentsov, Mikhail A. Krinitskiy and Maria G. Khrenova

Int. J. Mol. Sci. 2025, 26(11), 5097; https://doi.org/10.3390/ijms26115097 - 26 May 2025

Viewed by 357

Abstract

The active sites of enzymes are able to activate substrates and perform chemical reactions that cannot occur in solutions. We focus on the hydrolysis reactions catalyzed by enzymes and initiated by the nucleophilic attack of the substrate’s carbonyl carbon atom. From an electronic [...] Read more.

The active sites of enzymes are able to activate substrates and perform chemical reactions that cannot occur in solutions. We focus on the hydrolysis reactions catalyzed by enzymes and initiated by the nucleophilic attack of the substrate’s carbonyl carbon atom. From an electronic structure standpoint, substrate activation can be characterized in terms of the Laplacian of the electron density. This is a simple and easily visible imaging technique that allows one to “visualize” the electrophilic site on the carbonyl carbon atom, which occurs only in the activated species. The efficiency of substrate activation by the enzymes can be quantified from the ratio of reactive and nonreactive states derived from the molecular dynamics trajectories executed with quantum mechanics/molecular mechanics potentials. We propose a neural network that assigns the species to reactive and nonreactive ones using the Laplacian of electron density maps. The neural network is trained on the cysteine protease enzyme-substrate complexes, and successfully validated on the zinc-containing hydrolase, thus showing a wide range of applications using the proposed approach. Full article

(This article belongs to the Collection State-of-the-Art Macromolecules in Russia)

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18 pages, 312 KiB

Open AccessArticle

Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group

by Huiying Wang, Chengwei Yu, Zhiqiang Zhang and Yue Zeng

Symmetry 2025, 17(5), 799; https://doi.org/10.3390/sym17050799 - 21 May 2025

Viewed by 341

Abstract

In the Heisenberg group

H^{n}

, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form [...] Read more.

In the Heisenberg group

H^{n}

, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form

\partial_{t} u - \sum_{i = 1}^{2 n} X_{i} A_{i} (X u) = K (x, t, u, X u)

, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For

2 \leq p \leq 4

, we establish the local Lipschitz regularity (

u \in C_{loc}^{0, 1}

), with the horizontal gradient satisfying

X u \in L_{loc}^{\infty}

; (ii) For

2 \leq p < 3

, we establish the local second-order horizontal Sobolev regularity (

u \in H W_{loc}^{2, 2}

), with the second-order horizontal derivative satisfying

X X u \in L_{loc}^{2}

. These results solve an open problem proposed by Capogna et al. Full article

(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)

15 pages, 280 KiB

Open AccessArticle

Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

by Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Mathematics 2025, 13(10), 1672; https://doi.org/10.3390/math13101672 - 20 May 2025

Viewed by 402

Abstract

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined [...] Read more.

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space

R^{n + 1}

, Euclidean sphere

S^{n + 1}

, and de Sitter space

S_{1}^{n + 1}

. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

19 pages, 289 KiB

Open AccessArticle

The Radial Symmetry and Monotonicity of Solutions of Fractional Parabolic Equations in the Unit Ball

by Xingyu Liu

Symmetry 2025, 17(5), 781; https://doi.org/10.3390/sym17050781 - 19 May 2025

Cited by 2 | Viewed by 349

Abstract

We use the method of moving planes to prove the radial symmetry and monotonicity of solutions of fractional parabolic equations in the unit ball. Since the fractional Laplacian operator is a linear operator, we investigate the maximal regularity of nonlocal parabolic fractional Laplacian [...] Read more.

We use the method of moving planes to prove the radial symmetry and monotonicity of solutions of fractional parabolic equations in the unit ball. Since the fractional Laplacian operator is a linear operator, we investigate the maximal regularity of nonlocal parabolic fractional Laplacian equations in the unit ball. The maximal regularity of nonlocal parabolic fractional Laplacian equations guarantees the existence of solutions in the unit ball. Based on these conditions, we first establish a maximum principle in a parabolic cylinder, and the principles provide a starting position to apply the method of moving planes. Then, we consider the fractional parabolic equations and derive the radial symmetry and monotonicity of solutions in the unit ball. Full article

(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)

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