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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 k$$ k $$ 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. [ABSTRACT FROM AUTHOR]

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency. [ABSTRACT FROM AUTHOR]

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 (h k) convergence rate for the exact solution u in the H 1 norm and the O (h k + 1) convergence rate in the L 2 norm. Finally, numerical examples support the results reached by the theory. [ABSTRACT FROM AUTHOR]

This article focused on investigating the spatial behavior of the quasi-static biharmonic conduction equation within the framework of type Ⅲ of the second gradient in a two-dimensional cylindrical domain. The results of growth or decay estimates were established by using a second-order differential inequality. When the distance tends to infinity, the energy either grows exponentially or decays exponentially. The results showed that the Saint-Venant principle was also valid for the quasi-static biharmonic conduction equation. [ABSTRACT FROM AUTHOR]

We consider the finite element approximation of a coupled fluid‐structure interaction (FSI) system, which comprises a three‐dimensional (3D) Stokes flow and a two‐dimensional (2D) fourth‐order Euler–Bernoulli or Kirchhoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface which is assumed to be fixed. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, since it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to generate some numerical results concerning the approximate solutions to the FSI model. For this, we propose a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coefficient inverse problem related to partial reconstruction of the memory matrix of a viscoelastic medium from the results of probing the medium with a family of wave fields excited by point sources. A spatially non-overdetermined formulation is studied, for which the manifolds of point sources and detectors do not coincide and have a total dimension of 3. The requirements for these manifolds are established that ensure the unique solvability of the inverse problem. The results are achieved by reducing the problem to a chain of coupled systems of linear integral equations of the Lavrent'ev type. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type. [ABSTRACT FROM AUTHOR]

We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented. [ABSTRACT FROM AUTHOR]

This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multifrequency phaseless data. In the first stage, we introduce some artificial auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multifrequency data. The proposed method is easy to implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We consider viscous steady streaming induced by oscillatory flow past a cylinder between two plates, where the cylinder's axis is normal to the plates. While this phenomenon was first studied in the 1930s, it has received renewed interest recently for possible applications in particle manipulations and non-Newtonian flows. The flow is driven at the ends of the channel by the boundary condition which is a series solution of the oscillating flow problem in a rectangular channel in the absence of a cylinder. We use a combination of Fourier series and an asymptotic expansion to study the confinement effects for steady streaming. The Fourier series in time naturally simplifies to a finite series. In contrast, it is necessary to truncate the Fourier series in, which is in the direction of the axis of the cylinder, to solve numerically. The successive equations for the Fourier coefficients resulting from the asymptotic expansion are then solved numerically using finite element methods. We use our model to evaluate how steady streaming depends on the domain width and distance from the cylinder to the outer walls, including the possible breaking of the four-fold symmetry due to the domain shape. We utilize the tangential steady streaming velocity along the radial chord at an angle of π/4 to analyze our solutions over an extensive range of oscillating frequencies and multiple levels in the -direction. Finally, higher-order solutions are computed and an asymptotic correction to steady streaming is included. [ABSTRACT FROM AUTHOR]

The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

An inhomogeneous biharmonic equation is considered on a unit sphere of three-dimensional space. The solution of this equation belonging to the spherical Sobolev space is approximated by a sequence of solutions of the same equation, but with special right-hand sides, which are linear combinations of shifts of the Dirac delta function. It is proved that, for given nodes on the sphere that determine shifts, there exist special solutions of the equation: spherical biharmonic splines, and the weights corresponding to each of them are solutions of the accompanying nondegenerate system of linear algebraic equations. A relation is established between the quality of approximation of the solution of the differential problem by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → 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 (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the deviations of the functions of the Hölder class H1 from the biharmonic Poisson integrals defined in the upper half-plane were found. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic Choquard-type problem Δ 2 u - β Δ u = λ u + (I μ ∗ F (u)) f (u) , in R 4 , ∫ R 4 | u | 2 d x = c 2 > 0 , u ∈ H 2 (R 4) , where β ≥ 0 , λ ∈ R , I μ = 1 | x | μ with μ ∈ (0 , 4) , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem. [ABSTRACT FROM AUTHOR]

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen [SIAM J. Sci. Comput., 44 (2022), pp. A1205-A1226] recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polyg- onal domains. In this paper, we present the LARS algorithm (lightning-AAA rational Stokes) for computing two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply con- nected domains using lightning and AAA rational approximation [Y. Nakatsukasa, O. Sète, and L. N. Trefethen, SIAM J. Sci. Comput., 40 (2018), pp. A1494-A1522]. After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∊ V := H2 0 (\Omega) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces Vh(P) and a smoother allows rough source terms F ∊ V* =H-20(Ω). The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator J: Vh → V from the nonconforming virtual element space Vh. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on u ∊ V. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [ABSTRACT FROM AUTHOR]

In this paper, the following critical biharmonic elliptic problem { Δ 2 u = λ u + μ u ln ⁡ u 2 + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u = ∂ u ∂ ν = 0 , x ∈ ∂ Ω is considered, where Ω ⊂ R N is a bounded smooth domain with N ≥ 5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced. [ABSTRACT FROM AUTHOR]

In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing H^2-conforming virtual element, an H^1-nonconforming virtual element is obtained with the same degrees of freedom as the usual H^1-conforming virtual element, such that it locally has H^2-regularity on each polygon in meshes. To enforce the C^1 continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx + 2uxxyy + uyyyy = f(u). To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]

The main objective of this article is to consider a biharmonic problem with Navier boundary conditions. Among others, some criteria for the existence, multiplicity and nonexistence of positive solutions are established by employed fixed point theorems in a cone. In addition, we not only consider the sublinear case, but also we will study the case of appropriate combinations of superlinearity and sublinearity. [ABSTRACT FROM AUTHOR]

In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré--Korn-type inequalities for hybrid fields. [ABSTRACT FROM AUTHOR]

In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham paradigm, leading to an arbitrary-order construction that applies to meshes composed of general polygonal elements. The discrete plates complex is then used to derive a novel numerical scheme for Kirchhoff–Love plates, for which a full stability and convergence analysis are performed. Extensive numerical tests complete the exposition. [ABSTRACT FROM AUTHOR]

In this paper, we are inspired by Ngô, Nguyen and Phan's (2018 Nonlinearity 31 5484–99) study of the pointwise inequality for positive C 4-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality − Δ u u + α | ∇ u | 2 u 2 + β u − q + 1 2 ⩽ C R 2 for positive C 4-solutions of the biharmonic equations with negative exponent − Δ 2 u = u − q i n B R where B R denotes the ball centered at x 0 with radius R, n ⩾ 3, q > 1, and some constants α ⩾ 0, β ⩾ 0, C > 0. [ABSTRACT FROM AUTHOR]

New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves fast Fourier transforms and, in comparison to its one-step counterpart, it has the advantage of requiring the solution a symmetric positive definite Schur complement system rather than a nonsymmetric one. As a consequence, the corresponding step of the new method is performed using a preconditioned conjugate gradient method. The total cost of the method on a N × N partition of the unit square is O (N 2 log N) . To demonstrate the optimal accuracy of the method, the results of numerical experiments are provided. [ABSTRACT FROM AUTHOR]

Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n ≠ 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 . [ABSTRACT FROM AUTHOR]

A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This study delves into the inverse problem of identifying unknown sources within nonhomogeneous biharmonic equations confined to a bounded domain. The inherent ill-posed nature of this problem, as per Hadamard’s definition, necessitates a solution strategy. To address this challenge, we propose the Fractional Tikhonov Regularization, aimed at stabilizing the solution and providing convergence estimates. We establish both a priori and a posteriori regularization parameter selection rules to guide the regularization process. Mean Square Error estimation in Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} is conducted by employing the Fourier method and Sobolev embedding techniques with respect to the observed data. Additionally, convergence results are presented, specifically in scenarios where the observed data follows a Gaussian white noise distribution. [ABSTRACT FROM AUTHOR]

In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that, the Green's function of the Navier problem and the integral representation of the solution to the Navier problem are presented. To study the Neumann-2 problem, the Green's function of the Neumann problem for the Poisson equation is discussed and on its basis the Green's function of the Neumann-2 problem is constructed. To illustrate the results obtained, solutions of the three considered homogeneous problems for the polynomial right-hand side of the equation are found. [ABSTRACT FROM AUTHOR]

Gopal and Trefethen recently introduced "lightning solvers" for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner. [ABSTRACT FROM AUTHOR]

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green's functions. [ABSTRACT FROM AUTHOR]

A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra e A ~ 2 over the field ℂ is considered. All H-analytic functions are described in the explicit form. The obtained description is applied to the integral representation of these functions, and the mentioned functions are also applied when solving some PDEs. [ABSTRACT FROM AUTHOR]

New h-, p-, and hp-versions of the least-squares collocation method are proposed and implemented. They yield approximate solutions of boundary value problems for an inhomogeneous biharmonic equation in irregular and multiply-connected domains. Formulas for the extension operation in the transition from coarse to finer grids on a multigrid complex are given in the case of applying various spaces of polynomials. It is experimentally shown that numerical solutions of boundary value problems produced by the developed versions of the method have a higher order of convergence to analytical solutions with no singularities. The results are compared with those of other authors produced by applying finite difference, finite element, and other methods based on Chebyshev polynomials. Examples of problems with singularities are considered. The developed versions of the method are used to simulate the bending of an elastic isotropic plate of irregular shape subjected to transverse loading. [ABSTRACT FROM AUTHOR]

This paper presents a novel computational-geometry-based approach to generating random packing of complex-shaped 3D particles with quantitatively controlled sizes and shapes for discrete modeling of granular materials. The proposed method consists of the following five essential steps: (1) partitioning of the packing domain into a prescribed number of random polyhedrons with desired sizes and form-scale shapes using the constrained Voronoi tessellation; (2) extraction of key points from the edges and facets of each polyhedron; (3) construction of a freeform curve network in each polyhedron based on Bézier curve fitting; (4) generation of solid particles with smooth, convex surfaces using the biharmonic-based surface interpolation of the constructed network; and (5) creation of concavity by superimposing spherical harmonic-based random noise. To ensure that the obtained shape descriptors (e.g., the elongation, flatness, roundness and convexity ratio) match the hypothesized values, an inverse Monte Carlo algorithm is employed to iteratively fine-tune the control parameters during particle generation. The ability of the proposed approach to generate granular particles with the desired geometric properties and packing is demonstrated through several examples. This study paves a viable pathway for realistic modeling of granular media pertaining to various engineering and industrial processes. [ABSTRACT FROM AUTHOR]

The biharmonic equation arises in areas of continuum mechanics, including mechanics of elastic plates and the slow flow of viscous fluids. In this paper, we make an effort to establish the generalized versions of Lions-type theorem under various conditions and then apply them to study the existence of ground state solutions for the biharmonic equation. [ABSTRACT FROM AUTHOR]

A radialbasis function (RBF)–differentialquadrature (DQ) method is applied for the numerical solution of elliptic boundary value problems (BVPs) in three-dimensional axisymmetric domains. By appropriately selecting the collocation points, for any choice of RBF, the proposed discretization leads to linear systems in which the coefficient matrices possess block circulant structures. Matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) are employed for the efficient solution of these systems. Three types of BVPs are considered, namely ones governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy–Navier equations of elasticity. The high accuracy of the proposed technique as well as its ability to solve large-scale problems is demonstrated on several numerical examples. [ABSTRACT FROM AUTHOR]

The spatial properties of solutions for a class of thermoelastic plates with biharmonic operators were studied in a semi-infinite strip in R2. Firstly, an energy expression was constructed. Then, by means of the differential inequality technique, a differential inequality whose energy expression can be controlled with its 1st derivative was derived. Finally, the spatial decay estimates of the solution were obtained. The result can be regarded as an application of the Saint-Venant principle to hyperbolic parabolic coupled biharmonic equations. [ABSTRACT FROM AUTHOR]

A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]