Biharmonic Equation Research Articles

A stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation

In this paper, we present and study a stabilizer-free weak Galerkin (SFWG) finite element method for the Ciarlet-Raviart mixed form of the biharmonic equation on general polygonal meshes. We utilize the SFWG solutions of the second order elliptic problem to define projection operators and build error equations. Further, using weak functions formed by discontinuous k-th order polynomials, we derive the O(hk) convergence rate for the exact solution u in the H1 norm and the O(hk+1) convergence rate in the L2 norm. Finally, numerical examples support the results reached by the theory.

Sobolev regularity of bivariate isogeometric finite element spaces in case of a geometry map with degenerate corner

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation.

Fractional Tikhonov regularization and error estimation in inverse source problems for Biharmonic equations: a priori and a posteriori analysis under deterministic and random perturbations

Fractional Tikhonov regularization and error estimation in inverse source problems for Biharmonic equations: a priori and a posteriori analysis under deterministic and random perturbations

Analysis and computation of an inverse source problem for the biharmonic wave equation

Abstract In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on the Fourier series expansion by multi-wavenumber boundary measurements. The stability of the proposed method is analyzed and numerical experiments are presented to verify the accuracy and efficiency.

On a biharmonic choquard equation involving critical exponential growth

The purpose of this work is to study the solvability of the following biharmonic Choquard equation: (1) Δ 2 u − Δu + V ( x ) u = [ | x | − μ ∗ ( K ( x ) F ( x , u ) ) ] K ( x ) f ( x , u ) , x ∈ R 4 , where the functions V and K may decay to zero at infinity like ( 1 + | x | α ) − 1 , with α ∈ ( 0 , 4 ) , and ( 1 + | x | β ) − 1 , with (8-\\mu )\\alpha /8 $ ]]> β > ( 8 − μ ) α / 8 , respectively, μ ∈ ( 0 , 4 ) , f is a continuous function that behaves like e γ 0 s 2 at infinity, for some 0 $ ]]> γ 0 > 0 and F is the primitive of f. By establishing a weighted version of the Adams inequality involving V and K, we investigate the existence of nontrivial solution for problem (1). Furthermore, we establish that the nontrivial solution is a bound state when α ∈ ( 0 , 2 ) .

A new Adams’ inequality involving the $$ (\frac{N}{2},p)-$$Bilaplacian operators and applications to some biharmonic nonlocal equation

A new Adams’ inequality involving the $$ (\frac{N}{2},p)-$$Bilaplacian operators and applications to some biharmonic nonlocal equation

P(x)-biharmonic equations involving (h,r(x))-Hardy singular coefficients with no-flux boundary conditions

In this article, we investigate $p(x)$-biharmonic equations involving two kinds of different Hardy potentials, in which the $r(x)$-Hardy potentials are seldom mentioned in previous papers. New criteria for the existence of generalized solutions are reestablished when the nonlinear terms satisfying appropriate assumptions. The results are based on variational methods and the theory of variable exponent Sobolev spaces.

Fading regularization method for an inverse boundary value problem associated with the biharmonic equation

In this paper, we propose a numerical algorithm that combines the fading regularization method with the method of fundamental solutions (MFS) to solve a Cauchy problem associated with the biharmonic equation. We introduce a new stopping criterion for the iterative process and compare its performance with previous criteria. Numerical simulations using MFS validate the accuracy of this stopping criterion for both compatible and noisy data and demonstrate the convergence, stability, and efficiency of the proposed algorithm, as well as its ability to deblur noisy data.

Multiplicity results for biharmonic equations with critical growth

Multiplicity results for biharmonic equations with critical growth

Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

Continuous finite elements satisfying a strong discrete Miranda–Talenti identity

Abstract This article introduces continuous $H^{2}$-nonconforming finite elements in two and three space dimensions that satisfy a strong discrete Miranda–Talenti inequality in the sense that the global $L^{2}$ norm of the piecewise Hessian is bounded by the $L^{2}$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^{1}$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in nondivergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Complete topological asymptotic expansion for L2 and H1 tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an L2 and H1 tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the H1 cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the L2 tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.

Spherical Spline Solutions of an Inhomogeneous Biharmonic Equation

Spherical Spline Solutions of an Inhomogeneous Biharmonic Equation

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

On the solvability of a space-time fractional nonlinear Schrödinger system

This paper is devoted to the theoretical analysis of a coupled nonlinear system of fractional Schrödinger equations in Rn×R+,n≥1, considering time fractional derivative in the Caputo sense, and a fractional spatial dispersion defined in terms of the Fourier transform. We prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, considering power-type nonlinearities and initial data in the framework of weak-Lp spaces, which contain singular functions with infinite energy. As consequence of the embedding of weak-Lp spaces in Lloc2, for p>2, the obtained solutions have finite local L2-mass. In addition, we discuss the scenario in which it is possible to obtain the existence of self-similar solutions, which is a symmetric property that reproduces the structure of physical phenomena in different spatio-temporal scales. Our results are applicable, in the fractional setting, to the nonlinear Schrödinger and Biharmonic equations, as well as in a large class of dispersive systems appearing in nonlinear optics.

Linearized inverse problem for biharmonic operators at high frequencies

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate.

Global existence and blow‐up of solutions for a class of p$$ p $$‐biharmonic wave equations with damping terms and power sources

Global existence and blow‐up of solutions for a class of p$$ p $$‐biharmonic wave equations with damping terms and power sources

Existence of Weak Solutions for a Class of $(p,q)$-biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity

Existence of Weak Solutions for a Class of $(p,q)$-biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity

A new approach to problems of thermoelasticity in stresses

It is known that the plane problems of the theory of elasticity and thermoelasticity are usually reduced to solving the biharmonic equation with respect to the Airy stress function. Solutions to spatial problems can also be defined using the Maxwell and Morera stress functions, etc. In this work, for solving a spatial thermoelastic problem, in the framework of the Beltrami-Michell equations, Poisson-type differential equations are found for each component of the stress tensor. In this case, the right-hand sides of these equations depend on temperature and elastic constants. The boundary conditions consist of the usual surface forces and “additional” boundary conditions derived from the equilibrium equations. The differential equations and boundary conditions of the plane problem of thermoelasticity in stresses are discussed in more detail. Grid equations for plane and spatial problems of thermoelasticity were compiled by the finite-difference method and solved by the iterative method. The problems of a thermoelastic rectangle and a parallelepiped located in a given temperature field are solved numerically. The validity of the formulated boundary value problems and the reliability of the results obtained are substantiated by comparing the numerical results with the solutions of the problems of a thermoelastic plate and a parallelepiped known in the literature.