Fundamental solution neural networks for solving inverse Cauchy problems for the Laplace and biharmonic equations

On the critical behavior for the semilinear biharmonic heat equation with forcing term in exterior domain

A mixed FEM for time-fractional biharmonic integro-differential equations

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in , highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics.

Existence and Convergence Results for Nonlinear Biharmonic Equations on Graphs

A new stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation on polygonal meshes

Many practical problems, such as a solar panel, a microplate inside an electronic device, a thin layer of metal exposed to a laser, or the bending of a reinforced concrete slab, are described by various boundary value problems for the biharmonic equation. Solving biharmonic equations using iterative methods is extremely inconvenient due to the requirement to perform a large number of arithmetic operations, in addition, the number of iterations in which often turns out to be very large. The consideration of biharmonic equations with Dirichlet and Neumann boundary conditions limits the use of difference methods for their numerical solution. Therefore, the development of highly accurate and efficient direct numerical methods for solving such equations is of particular scientific interest. For this purpose, in this article, for the numerical solution of boundary value problems for the biharmonic equation, it is proposed to use a direct numerical method - a discrete version of the preliminary integration method with high accuracy and efficiency.

We consider the fourth-order PDE uxxxx+2uxxyy+uyyyy−utt=h(u). The Lie and Noether symmetry generators are constructed, and we reduce the PDE to simpler ODEs. Furthermore, we use some well-known methods to compute the conserved vectors associated with the PDE. An analysis of reduced ordinary differential equations (ODEs), invariant solutions, and their physical interpretations is presented.

Abstract This work focuses on the stability and convergence of boundary spline-approximation methods for non-homogeneous biharmonic equations with non-homogeneous boundary conditions in simple piecewise smooth domains. The problem is translated into two boundary value problems (BVPs) for the homogeneous biharmonic equation with non-homogeneous boundary data. Each of these BVPs is reduced to a Sherman-Lauricella equation with a specific right-hand side. The corresponding integral equations are then solved by spline Galerkin methods and their solutions are used to determine numerical solutions of the biharmonic problem under consideration. As far as the errors of the methods are concerned, it is remarkable that for piecewise smooth domains, the approximate solutions do not inherit the convergence order of the approximate solutions of the Sherman-Lauricella equations, but halve it. This is in sharp contrast to the case of smooth boundaries. The approach used is quite universal and can be applied to biharmonic problems in both regular and irregular domains. Examples show a good numerical convergence of the method.

Fourier-feature induced physics informed randomized neural network method to solve the biharmonic equation

Asymptotics is constructed for eigenvalues and eigenfunctions of the biharmonic equation with the Neumann conditions perturbed by the spectral Winkler–Steklov conditions at small parts of the plate’s edge. The zero eigenvalue has multiplicity three and the corresponding eigenfunctions are linear. Asymptotic expansions of positive eigenvalues differ essentially in the mid- and high-frequency ranges of the spectrum. In particular, eigenfunctions at low and medium frequencies are distributed along the entire domain while eigenfunctions at high frequencies are concentrated in the vicinity of the edge perturbations.

Abstract In this paper, we consider the obstacle scattering problem for biharmonic wave equations with the Dirichlet boundary condition in both two and three dimensions. Firstly, some basic properties are derived for the scattered fields, which leads to a simple criterion for the uniqueness of the solution. Then a new definnition for the far-field pattern is introduced, where the correspondence between the far-field pattern and scattered field is established. With these preparations, we prove the existence of a unique solution in associated Sobolev spaces by the boundary integral equation method, which relies on a natural decomposition of the biharmonic operator and the theory of the pseudodifferential operator. Moreover, the inverse problem in determining the shape and location of the obstacle is studied. By establishing some novel reciprocity relations between the far-field pattern and the scattered field, we show that the obstacle can be uniquely recovered from the far-field measurements at two frequencies or near field measurements at a fixed frequency.

Abstract This paper concerns the direct and inverse source scattering problems for the stochastic biharmonic wave equation with a Lipschitz-type nonlinearity. The driven source is assumed to be a white noise. Given the rough random source, the direct problem is shown to be well-posed in the sense of distributions. For the inverse problem, it is shown that the strength of the random source can be uniquely determined by the expectation of the high frequency limit of the correlated far-field pattern. Moreover, we extend the results to the more general case where the driven source is a generalized Gaussian random function whose covariance operator is a classical pseudodifferential operator. For this case, we demonstrate by ergodicity that the strength of the random source can be uniquely determined by a single realization of the far-field pattern almost surely. This paper presents the first uniqueness result on inverse source scattering problems for nonlinear wave equations.

Sign-changing solutions for a perturbed biharmonic equation with critical exponent

Interpolated Galerkin finite elements on triangular meshes for the biharmonic equation

In this paper, we are concerned with the existence of ground states to the following biharmonic equation on the lattice graph $$ \Delta^2 u-\Delta u+V(x)u=f(x, u), \quad x \in \mathbb{Z}^N. $$ The analysis is performed if the potential $V$ and the reaction $f$ are $T$-periodic in $x$, and the mapping $u \mapsto {f(x, u)}/{\vert u\vert}$ is non-decreasing on $\mathbb{R}\setminus \{0\}$. By using the variational methods, we establish the existence of ground states for the above problem. Moreover, if the potential $V$ has a bounded potential well and $f(x, u)=f(u)$ with $u \mapsto {f(u)}/{\vert u\vert}$ non-decreasing on $\mathbb{R}\setminus \{0\}$, the ground states are also obtained for the above equation. Finally, we extend the main results on the lattice graph $\mathbb{Z}^N$ to quasi-transitive graphs. In our analysis, the mappings $u \mapsto {f(x, u)}/{\vert u\vert}$ or $u \mapsto {f(u)}/{\vert u\vert}$ are only non-decreasing on $\mathbb{R}\setminus \{0\}$, which allows to consider larger classes of nonlinearities in the reaction.

We study the two-dimensional steady-state creeping flow in a converging–diverging channel gap formed by two immobile rollers of identical radius. For this purpose, we analyse the Stokes equation in the streamfunction formulation, i.e. the biharmonic equation, which has homogeneous and particular solutions in the roll-adapted bipolar coordinate system. The analysis of existing works, investigating the particular solutions allowing arbitrary velocities at the two rollers, is extended by an investigation of homogeneous solutions. These can be reduced to an algebraic eigenvalue problem, whereby the associated discrete but infinite eigenvalue spectrum generates symmetric and asymmetric eigenfunctions with respect to the centre line between the rollers. These represent nested viscous vortex structures, which form a counter-rotating chain of vortices for the smallest unsymmetrical eigenvalue. With increasing eigenvalue, increasingly complex finger-like structures with more and more layered vortices are formed, which continuously form more free stagnation points. In the symmetrical case, all structures are mirror-symmetrical to the centre line and with increasing eigenvalues, finger-like nested vortex structures are also formed. As the gap height in the pressure gap decreases, the vortex density increases, i.e. the number of vortices per unit length increases, or the length scales of the vortices decrease. At the same time the rate of decay between subsequent vortices increases and reaches and asymptotic limit as the gap vanishes.

A fourth order mixed compact finite difference scheme for biharmonic equations

A novel class of Hessian recovery-based numerical methods for solving biharmonic equations and their applications in phase field modeling

Existence of positive solutions for the semipositone biharmonic equation with Navier boundary conditions