Laplacian Research Articles

On the existence of positive solutions of some nonlocal elliptic problems involving fractional p–Laplacian operator

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Characterization of Tubular Surfaces in Terms of Finite III-type

In this paper, we first define relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E 3 . Then, we will characterize the tubular surfaces in terms of their coordinate finite type.

On a class of fractional Γ(.)-Kirchhoff-Schrödinger system type

This paper focuses on the investigation of a Kirchhoff-Schrödinger type elliptic system involving a fractional \(\gamma(.)\)-Laplacian operator. The primary objective is to establish the existence of weak solutions for this system within the framework of fractional Orlicz-Sobolev Spaces. To achieve this, we employ the critical point approach and direct variational principle, which allow us to demonstrate the existence of such solutions. The utilization of fractional Orlicz-Sobolev spaces is essential for handling the nonlinearity of the problem, making it a powerful tool for the analysis. The results presented herein contribute to a deeper understanding of the behavior of this type of elliptic system and provide a foundation for further research in related areas.

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

Spectral Graph Convolutional Neural Networks in the Context of Regularization Theory.

Graph convolutional neural networks (GCNN) have been widely used in graph learning and related applications. It has been identified that the filters in the state-of-the-art spectral graph convolutional networks (SGCNs) are essentially low-pass filters that enforce smoothness across the graph and use the functions of graph Laplacian as a tool that injects graph structure into the learning algorithm. There had been research findings that connect the smoothness functional in graphs, graph Laplacian, and regularization operators. We review the existing SGCNs in this context and propose a framework where the state-of-the-art filter designs can be deduced as special cases. We designed new filters that are associated with well-defined low-pass behavior and tested their performance on semisupervised node classification tasks. Their performance was found to be superior to that of the other state-of-the-art techniques. We further investigate the representation capability of low-pass features and make useful observations. In this context, we discuss a few points to further optimize the network, new strategies for designing SGCNs, their challenges, and some latest related developments. Based on our framework, we also deduce the connection between support vector kernels and SGCN filters.

Heat Kernel of Networks with Long-Range Interactions

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of “information” across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with the k-path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization.

An Explicit Form of Harnack’s Inequality for Poisson’s Dirichlet Problem

The explicit form of Harnack’s inequality for non-negative harmonic functions in the open ball plays an important role in harmonic analysis and elliptic partial differential equations. In this paper, we establish the explicit form of Harnack’s inequality for Poisson’s Dirichlet problem in the open ball with non-negative boundary data. We involve the Green’s function for Laplacian operator to find it out. Harnack’s inequality for positive harmonic functions can be followed from Harnack’s inequality for Poisson’s Dirichlet problem when the source function is set to be zero.

The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth

Abstract In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.

Boundedness of Solutions of Nonautonomous Degenerate Logistic Equations

AbstractIn this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, K(t), inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in K(t) but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.

SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

Multiplicity of solutions for variable-order fractional Kirchhoff problem with singular term

In this paper, we consider a class of singular variable-order fractional Kirchhoff problem of the form: where is a bounded domain, is the variable-order fractional Laplacian operator, [u] s(·) is the Gagliardo seminorm and is a continuous and symmetric function. We assume that λ is a non-negative parameter, with and . We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

Computing compact finite difference formulas under radial basis functions with enhanced applicability

In this work, we use the combination of radial basis functions (RBFs) and polynomials to derive compact finite difference (FD) formulas. We use RBFs and integrated RBFs (IRBFs), to obtain two sets of formulas: RBF-Hermite FD (RBF-HFD) and IRBF weight formulas. The analytical coefficients and truncation errors for these formulations are computed for the first derivative, second derivative, and 2D Laplacian operator. Our presented approach expands the usability of the weights by introducing a general framework for deriving weighting formulas. Finally, the computational results underscore the effectiveness of the introduced approach.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.

Analysis and application of MLPG7 for diffusion equations with nonlinear reaction terms

This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied.

On the Boyarsky-Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

Numerical analysis of finite element method for a stochastic active fluids model

In this paper, we first investigate the well-posedness and regularity of mild solution to a stochastic active fluids model driven by the additive noise. A fully-discrete scheme is proposed for solving the given model, which is based on the finite element method for spatial discretization and the backward Euler method for temporal discretization. By overcoming the difficulty of error analysis caused by the discrete Laplacian operator, we obtain the convergence results of the developed scheme. Finally, some numerical examples are provided to validate the theoretical results and we also simulate the motion states of the active fluids.

The direct RBF-based partition of unity method for solving nonlinear fractional parabolic equations

This paper aims to analyze a novel localized radial basis function method known as the ’direct RBF-based partition of unity method’ for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy.

Optimal control problem with nonlinear fractional system constraint applied to image restoration

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on ‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model.