We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions a few reference biharmonic problems elasticity elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that challenging solve using classical numerical methods have not been addressed PINNs. Our...

The coefficients for a nine-point high-order accuracy discretization scheme for a biharmonic equation∇4u = f(x, y) (∇2 is the two-dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂u/∂n or (2) u and ∂u/∂n (where ∂/∂n is the normal to the boundary derivative) are specified at the boundary. For b...

A Laguerre minimal surface is an immersed surface in R being an extremal of the functional ∫ (H/K− 1)dA. In the present paper, we prove that any ruled Laguerre minimal surface distinct from a plane is up to motion a convolution of the helicoid x = y tan z, the cycloid r(t) = (t− sin t, 1−cos t, 0) and the Plücker conoid (ax+ by) = z(x+y) for some a, b ∈ R. To achieve invariance under Laguerre t...

Abstract. We consider multigrid algorithms for the biharmonic problem discretized by conforming 1 finite elements. Most finite elements for the biharmonic equation are nonnested in the sense that the coarse finite element space is not a subspace of the space of similar elements defined on a refined mesh. To define multigrid methods, certain intergrid transfer operators have to be constructed. W...

For the first biharmonic problem a mixed variational formulation is introduced which is equivalent to a standard primal variational formulation on arbitrary polygonal domains. Based on a Helmholtz decomposition for an involved nonstandard Sobolev space it is shown that the biharmonic problem is equivalent to three (consecutively to solve) second-order elliptic problems. Two of them are Poisson ...

In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the biharmonic equation. The technical approach is mainly base on a three critical points theorem of B. Ricceri. AMS Subject Classifications: 34B15.

We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.

This paper shows the existence of at least three solutions for Navier problem involving the p(x)-biharmonic operator. Our technical approach is based on a theorem obtained by B. Ricceri.

We consider additive Schwarz methods for the biharmonic Dirichlet problem and show that the algorithms have optimal convergence properties for some conforming nite elements. Some multilevel methods are also discussed.