Singular Weight Function Research Articles

A unified weighted inequality for fourth-order partial differential operators and applications

In this paper, we establish a fundamental inequality for fourth order partial differential operator P=△α∂s+β∂ss+Δ2 (▪) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator P with some suitable boundary conditions in the case of β<0 or α≠0,β=0. As application, we obtain a resolvent estimate for P, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.

Interpolating Meshless Methods for 3D Elastic Problems

Interpolating meshless methods can directly impose boundary conditions because of the interpolation property which shows advantages in dealing with problems with boundary conditions. The interpolating element-free Galerkin method (IEFGM), the improved interpolating element-free Galerkin method (IIEFGM), and the radial point interpolation method (RPIM) are applied in this paper to solve the two-dimensional and three-dimensional elastic problems. IEFGM and IIEFGM are two different ways to change the status that the traditional element-free Galerkin method (EFG) does not have the interpolation property. IEFGM uses an improved interpolating moving least-squares (IMLS) method that employed singular weight functions while IIEFGM takes the improved interpolating moving least-squares method based on non-singular weight function. RPIM, one of the most widely used interpolating meshless methods, is compared with IEFGM and IIEFGM in this paper. The numerical results of two-dimensional and three-dimensional elastic problems show that the three types of interpolating meshless methods obtain high precision displacement solutions and stress solutions.

Positive solutions of singular [formula omitted]-Hessian systems

The main objective of this paper is to consider a class of ki-Hessian systems with singular weight functions. Among others, we obtain new results on the existence and multiplicity of positive solutions. Moreover, many positive solutions are also considered.

Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data

There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ &gt; 0, when our control domain touches the degeneracy line {x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.

Flocking and collision avoidance problem of a singular Cucker–Smale model with external perturbations

In order to study the influence of external force or external perturbation on flocking behavior, a Cucker–Smale flocking model with singular communication weight and perturbation functions is proposed. By imposing appropriate restrictions on the perturbation functions, we first prove the system does not allow any collisions between agents under certain initial conditions. Moreover, under the same conditions, we obtain the system has an asymptotic flocking. Last, we give specific examples of perturbation functions and verify the correctness of the results through numerical simulations.

An improved interpolating dimension splitting element-free Galerkin method for 3D wave equations

In this paper, the improved interpolating dimension splitting element-free Galerkin (IIDSEFG) method based on nonsingular weight functions is proposed to solve 3D wave equations. By utilizing the dimension splitting method (DSM), a 3D wave equation is divided into a series of 2D problems. For 2D wave equations, the improved interpolating moving least-squares (IIMLS) method based on nonsingular weight functions is applied to construct the shape function. The finite difference method (FDM) is used both in time domain and splitting direction for obtaining the solutions. By adopting the IIDSEFG method, the essential boundary conditions can be imposed directly, and the truncation error caused by singular weight functions can be avoided. In this paper, the method of changing the diagonal element to 1 is selected as the boundary condition treatment method. Three numerical examples are chosen to verify the superiority of the IIDSEFG method. Compared with the improved element-free Galerkin (IEFG) method and the dimension splitting element-free Galerkin (DSEFG) method, the IIDSEFG method has greater computing speed and precision.

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

AbstractWe present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We also prove self‐similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on ℝ or for Hölder‐continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self‐similar singularity into the problem of establishing the nonlinear stability of an approximate self‐similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self‐similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well‐posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations. © 2021 Wiley Periodicals LLC.

On the semilinear fractional elliptic equations with singular weight functions

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $\end{document} where \begin{document}$ \alpha\in (0,1] $\end{document} , \begin{document}$ 1 2\alpha\right), $\end{document} the potential \begin{document}$ V_{\lambda }(x) = \lambda a(x)-b(x) $\end{document} and the parameter \begin{document}$ \lambda >0. $\end{document} Under some suitable assumptions on \begin{document}$ a,b $\end{document} and the weight functions \begin{document}$ f,g $\end{document} , we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$ \lambda $\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$ f, g $\end{document} are integrable or bounded on whole space \begin{document}$ \mathbb{R}^{N}. $\end{document}

Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem

In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for the forward stochastic degenerate parabolic equation with regular weighted function, based on which we obtain the Lipschitz stability for an inverse problem of determining a random source depending only on time in the forward stochastic degenerate parabolic equation.

Existence and multiplicity of nontrivial solutions for biharmonic equations with singular weight functions

In the paper, we study a class of biharmonic equations with singular weight functions as follows: Δ2u−βΔpu+Vλ(x)u=fxuq−2uinRN,u∈H2(RN),where N≥3,Δ2u=Δ(Δu),Δpu=div(|∇u|p−2∇u),β≥0 is a parameter, 2<p,q<2NN−2 and Vλ(x)=λa(x)−b(x) with λ>0. Under some suitable assumptions on a,b and f, we obtain the existence and multiplicity of nontrivial solutions for λ large enough. An interesting phenomenon is that we do not need the condition that weight function f is integrable or bounded on whole space RN, which can be regarded as an improvement work of Sun et al. (2017).

The Improved Interpolating Complex Variable Element Free Galerkin Method for Two-Dimensional Potential Problems

In this paper, the improved interpolating complex variable moving least squares (IICVMLS) method is applied, in which the complete basis function is introduced and combined with the singular weight function to achieve the orthometric basis function. Then, the interpolating shape function is achieved to construct the interpolating trial function. Incorporating the IICVMLS method and the Galerkin integral weak form, an improved interpolating complex variable element free Galerkin (IICVEFG) method is proposed to solve the 2D potential problem. Because the essential boundary conditions can be straightaway imposed in the above method, the expressions of final dispersed matrices are more concise in contrast to the non-interpolating complex variable meshless methods. Through analyzing four specific potential problems, the IICVEFG method is validated with greater computing precision and efficiency.

The interpolating element-free Galerkin method for solving Korteweg–de Vries–Rosenau-regularized long-wave equation with error analysis

The main target of this investigation is to develop a new numerical method for solving a class of wave models, i.e., the Korteweg–de Vries–Rosenau-regularized long-wave equation with application in plasma physics. The developed technique is concerning the interpolating element-free Galerkin method. The test and trial functions for the interpolating element-free Galerkin technique have been chosen from the interpolating moving least squares approximation. The interpolating moving least squares approximation shape functions unlike the moving least squares shape functions have the delta Kronecker property due to the use of a singular weight function. The stability and convergence of the new scheme are analyzed. Furthermore, the existence and uniqueness of solution have been proved for the full-discrete scheme. Finally, several examples in one- and two-dimensional cases have been studied to confirm the influence of the new scheme.

An improved local boundary integral equation method implemented by the transformed MLS approximation with the delta property

The meshless local boundary integral equation (LBIE) method is a promising method for solving problems of elasticity with nonhomogeneous material properties. However, the shape functions of the LBIE method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property. To impose boundary conditions, numerical integrations of LBIE method are carried out under consideration of boundary conditions and the fictitious nodal values would be solved out. In this paper, the transformed MLS shape functions can be obtained, satisfying the Kronecker delta property and requiring no singular weight function. And then the potential and flux in 2D potential problems would be regarded as independent variables each other. The numerical integrations of LBIEs can be processed without regard to boundary conditions and the coefficient matrix of LBIE method would be irrelated to boundary conditions. Finally boundary conditions would be directly imposed and the unknown nodal potentials or the unknown fluxes on the boundary nodes would be solved out. Three numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the proposed method with the traditional LBIE method shows the feasibility of the proposed method.

The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is applied to obtain the approximation function. The penalty method is applied to impose the displacement boundary condition, and Galerkin weak form of elastic large deformation problems based on total Lagrange formulation is used to form the final equations which is solved with the Newton–Raphson iteration method, then the improved element-free Galerkin (IEFG) method based on a nonsingular weight function for elastic large deformation problems is presented. The IMLS method can overcome the disadvantage of singular weight functions in the traditional MLS method, then the IEFG method in this paper has high computational accuracy and efficiency, which are shown by numerical examples of elastic large deformation problems. And the influences of the weight functions, scale parameter of influence domain, step number and penalty factor on the numerical results are discussed.

An Improved Interpolating Complex Variable Meshless Method for Bending Problem of Kirchhoff Plates

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

An interpolating element-free Galerkin scaled boundary method for the elasticity problem

As a newly-developed semi-analytical method, the scaled boundary finite element method is very powerful to deal with singular and unbounded problems. By combining the element-free Galerkin (EFG) method with the scaled boundary method in the frame of improved interpolating moving least-squares (IIMLS) method, an interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is firstly proposed to solve elasticity problems in this paper. In the IEFG-SBM, the solution in the radial direction is obtained analytically and only nodes are required to discretize the boundaries of the computational domain. In addition, higher accuracy and faster convergence are obtained due to the higher continuity of the shape functions in the circumferential direction. The IEFG-SBM does not need the fundamental solution and thus no singular integrations are involved. The shape function of the IIMLS method satisfies the Kronecker delta function property and thus the essential boundary conditions can be imposed directly as in the traditional finite element method. In comparison with the interpolating moving least-squares (IMLS) method proposed by Lancaster and Salkauskas, the key advantage of the IIMLS method is that it does not require singular weight function and thus any weight function used in the MLS approximation can also be applied in the IIMLS method. In addition, there are less unknown coefficients in the IIMLS method than in the conventional moving least-squares (MLS) approximation. Thus fewer nodes are required in the local influence domain and a higher computational accuracy can be reached in the IIMLS-based meshless method. At last, several numerical examples are presented to verify the effectiveness and accuracy of the developed method for the elasticity problem.

A fixed point operator for systems of vector p-Laplacian with singular weights

In this paper, after establishing a fixed point operator for a strongly coupled vector p-Laplacian with a singular and sign-changing weight function, which may not be integrable, we investigate the existence for the Dirichlet boundary value problems of strongly coupled vector p-Laplacian systems with a nonlinear term consisting of Hadamard product. The proofs are mainly based on topological degree arguments and the global continuation theorem.

Global interpolating MLS shape functions for structural problems with discrete nodal essential boundary conditions

The moving least squares (MLS)-based element-free Galerkin method (EFGM) has been widely studied, yielding valuable results and becoming a robust mesh-free technique for structural analysis. Due to the non-interpolating nature of MLS shape functions, the EFGM procedure leads to stiffness and mass matrices based on nodal parameters instead of true nodal displacements. The interpolating moving least squares (IMLS) methods with singular weight functions or the weighted nodal least squares (WNLS) methods were proposed in order to arrange the formulation in terms of actual displacements at the nodes. In this work, a later transformation matrix to retrieve the classic stiffness and mass matrices is discussed and numerically tested in problems with discrete prescribed displacements at boundaries, for which a straightforward finite element method (FEM)-type imposition of boundary conditions is possible.

Optimal estimates on free boundary propagation in thin‐film flow

AbstractWe present a method for the derivation of lower bounds on free boundary propagation for the thin‐film equation, one of the most prominent examples of a higher‐order degenerate parabolic equation. In particular, we obtain sufficient conditions for instantaneous forward motion of the free boundary, upper bounds on waiting times, as well as lower bounds on asymptotic propagation rates. Our estimates coincide (up to a constant factor) with the previously known reverse bounds and are therefore optimal. To the best of our knowledge, these results constitute the first lower bounds on free boundary propagation for any higher‐order degenerate parabolic equation. Our technique is based on new monotonicity formulas for solutions to the thin‐film equation which involve weighted entropies with singular weight functions. It turns out that our method is not restricted to the thin‐film equation, but also applicable to other higher‐order parabolic equations like quantum drift‐diffusion equations. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)