Search Results (24)

The Navier equations are reformulated to be third-order partial differential equations. New anti-Cauchy-Riemann equations can express a general solution in 2D space for incompressible materials. Based on the third-order solutions in 3D space and the Boussinesq–Galerkin method, a third-order method of fundamental solutions (MFS) is developed. For the 3D Navier equation in linear elasticity, we present three new general solutions, which have appeared in the literature for the first time, to signify the theoretical contributions of the present paper. The first one is in terms of a biharmonic function and a harmonic function. The completeness of the proposed general solution is proven by using the solvability conditions of the equations obtained by equating the proposed general solution to the Boussinesq–Galerkin solution. The second general solution is expressed in terms of a harmonic vector, which is simpler than the Slobodianskii general solution, and the traditional MFS. The main achievement is that the general solution is complete, and the number of harmonic functions, three, is minimal. The third general solution is presented by a harmonic vector and a biharmonic vector, which are subjected to a constraint equation. We derive a specific solution by setting the two vectors in the third general solution as the vectorizations of a single harmonic potential. Hence, we have a simple approach to the Slobodianskii general solution. The applications of the new solutions are demonstrated. Owing to the minimality of the harmonic functions, the resulting bases generated from the new general solution are complete and linearly independent. Numerical instability can be avoided by using the new bases. To explore the efficiency and accuracy of the proposed MFS variant methods, some examples are tested. Full article

This paper mainly studies the existence of sign-changing solutions for the following $p\left(x\right)$-biharmonic Kirchhoff-type equations: $-\left(a+b{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{1}{p\left(x\right)}}{|\Delta u|}^{p\left(x\right)}\mathrm{d}x\right){\Delta }_{p\left(x\right)}^{2}u+V\left(x\right){|u|}^{p\left(x\right)-2}u$ = $K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N$, where ${\Delta }_{p\left(x\right)}^{2}u=\Delta \left({|\Delta u|}^{p\left(x\right)-2}\Delta u\right)$ is the $p\left(x\right)$ biharmonic operator, $a,b>0$ are constants, $N\ge 2$, $V\left(x\right),K\left(x\right)$ are positive continuous functions which vanish at infinity, and the nonlinearity f has subcritical growth. Using the Nehari manifold method, deformation lemma, and other techniques of analysis, it is demonstrated that there are precisely two nodal domains in the problem’s least energy sign-changing solution $u_b$. In addition, the convergence property of $u_b$ as $b\to 0$ is also established. Full article

The objective of this work is to investigate the global existence, general decay and blow-up results for a class of p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions. The global existence of solutions has been obtained by potential well theory and the general decay result of energy has been established, in which the exponential decay and polynomial decay are only special cases, by using the multiplier techniques combined with a nonlinear integral inequality given by Komornik. Finally, the blow-up of solutions is established with positive initial energy. To our knowledge, the global existence, general decay, and blow-up result of solutions to p-Biharmonic-type hyperbolic equations with delay and acoustic boundary conditions has not been studied. Full article

In this paper, we consider whether the zero extension of a solution to the Dirichlet problem for the biharmonic equation in a smaller domain remains a solution to the corresponding extended problem in a larger domain. We analyze classical and strong solutions, and present a necessary and sufficient condition under each framework, respectively. Full article

In this paper, based on a new representation of the Green’s function of the Navier problem for the 3-harmonic equation in the unit ball, an integral representation of the solution of the corresponding Navier problem is found. Then, for the Navier problem for a homogeneous 3-harmonic equation, the obtained representation of the solution is reduced to a form that does not explicitly contain the Green’s function. Full article

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., ${\Delta }^{2}x=0$. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book. Full article

In this article, we consider the singular $p$-biharmonic problem involving Hardy potential and critical Hardy–Sobolev exponent. Firstly, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold. Then, we investigate the least-energy sign-changing solutions by considering the Nehari nodal set. In both cases, the critical Sobolev exponent is of great importance as the solutions exists only if we are below the critical Sobolev exponent. Full article

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ${▵}^{2}u=f(u,▵u)$ on an annular domain $\Omega$ in ${\mathbb{R}}^N$ with the Navier boundary conditions ${u|}_{\partial \Omega }=0$ and ${▵u|}_{\partial \Omega }=0$, where $f:{\mathbb{R}}^{+}\times {\mathbb{R}}^{-}\to {\mathbb{R}}^{+}$ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow $f(\xi ,\eta )$ to have superlinear or sublinear growth on $\xi ,\eta$ as $\left|\right(\xi ,\eta \left)\right|\to 0$ and ∞. Our discussion is mainly based on the fixed-point index theory in cones. Full article

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ${▵}^{2}u=f\left(\right|x|,u,|\nabla u|,▵u)$ in an annular domain $\Omega =\{x\in {\mathbb{R}}^N:r_1<|x|<r_2\}$($N\ge 2$) with the boundary conditions ${u|}_{\partial \Omega }=0$ and ${▵u|}_{\partial \Omega }=0$, where $f:[r_1,r_2]\times \mathbb{R}\times {\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R}$ is continuous. Under certain inequality conditions on f involving the principal eigenvalue ${\lambda }_1$ of the Laplace operator $-▵$ with boundary condition ${u|}_{\partial \Omega }=0$, an existence result and a uniqueness result are obtained. The inequality conditions allow for $f(r,\xi ,\zeta ,\eta )$ to be a superlinear growth on $\xi ,\zeta ,\eta$ as $\left|\right(\xi ,\zeta ,\eta \left)\right|\to \infty$. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates. Full article

To describe stationary physical systems, well-known boundary problems for shielded Poisson and Sophie Germain equations are used. The obtained shielded harmonic and biharmonic systems are approximated using the finite element method and fictitiously continued. The resulting problems are solved using the developed method of iterative extensions. To expedite the convergence of this method, the relationships between physical quantities on the extension of systems and additional parameters of the iterative method are employed. The formulations of sufficient convergence conditions for the iterative process utilize interdisciplinary connections with functional analysis, applying discrete analogs of the principles of function extensions while preserving norm and class. In the algorithmic implementation of the iterative extensions method, automation is applied to control the selection of the optimal iterative parameter value during information processing. In accordance with the fictitious domain methodology, solvable problems from domains with a complex geometry are reduced to problems in a rectangle in the two-dimensional case and in a rectangular parallelepiped in the three-dimensional case. But now, in the problems being solved, the minimization of the error of the iterative processes is carried out with a norm stronger than the energy norm. Then, all relative errors are estimated from above in the used norms by terms of infinitely decreasing geometric progressions. A generalization of the developed methodology to boundary value problems for polyharmonic equations is possible. Full article

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^{k}M$ to the cotangent bundle $T^{*}{T}^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. Full article

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Full article

Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where $n\ne 4$ boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic $C^1$ splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson’s equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by $2^4$ in the $L^2$ norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as $O(h^3$). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O($h^4$). An optimal reduction of $2^4$ is observed when the functions on the $C^1$ geometry are relaxed to be $C^0$. Full article

Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson’s problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases. Full article

In this study, we describe new advances in the multiscale methodology to allow a more realistic interpretation of volcanic deformation fields by investigating geometrically irregular bodies and multi-source scenarios. We propose an integrated approach to be applied to InSAR measurements, employing the Multiridge and ScalFun methods and the Total Horizontal Derivative (THD) technique: this strategy provides unconstrained information on the source geometrical parameters, such as the depth, position, shape, and horizontal extent. To do this, we start from conditions where the biharmonic deformation field satisfies Laplace’s equation and homogeneity law. We test the use of the multiscale procedures to model single and multisource scenarios with irregular geometries by retrieving satisfactory results for a set of simulated sources. Finally, we employ the proposed approach to the 2004–2009 uplift episode at the Yellowstone Caldera (U.S.) measured by ENVISAT InSAR to provide information about the volcanic plumbing system. Our results indicate a single $~50\times 20$ km² extended source lying beneath the caldera at around $10$ km b.s.l. (depth to the center), which is shallower below both the resurgent domes (6–7 km b.s.l. depth to the top). Full article