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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]

We consider a time-fractional biharmonic equation involving a Caputo derivative in time of fractional order α ∈ (0 , 1) and a locally Lipschitz continuous nonlinearity. Local and global existence of solutions is discussed and detailed regularity results are provided. A finite element method in space combined with a backward Euler convolution quadrature in time is analyzed. Our objective is to allow initial data of low regularity compared to the number of derivatives occurring in the governing equation. Using a semigroup type approach, error estimates of optimal order are derived for solutions with smooth and nonsmooth initial data. Numerical tests are presented to validate the theoretical results. [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 mathematical analysis method is employed to solve the bending problem of slip clamped shallow spherical shell on elastic foundation. Using the slip clamped boundary conditions, the differential equations of the problem are simplified to a biharmonic equation. Using the R -function, the fundamental solution and the boundary equation of the biharmonic equation, a function is established. This function satisfies the homogeneous boundary condition of the biharmonic equation. The biharmonic equation of the slip clamped shallow spherical shell bending problem on elastic foundation is transformed to Fredholm integral equation of the second kind by using Green's formula. The vector expression of the integral equation kernel is derived. Choosing a suitable form of the normalized boundary equation, the singularity of the integral equation kernel is overcome. To obtain the numerical results, the discretization of the integral equation of the bending problem is conducted. The treatment of singular term in the discretization equation is to use the integration by parts. Numerical results of rectangular, trapezoidal, pentagonal, L-shaped and concave shape shallow spherical shells show high accuracy of the proposed method. The numerical results show fine agreement with the ANSYS finite element method (FEM) solution, which shows the proposed method is an effective mathematical analysis method. [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]

We consider an ill-posed linear homogeneous fourth-order elliptic equation. We show that the problem is ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the given data. We propose a regularization method via nonlocal conditions and under some a priori bound assumptions different estimates for the regularized solution are obtained. Numerical examples for a rectangle domain show the effectiveness of the new method in providing highly accurate numerical solutions as the noise level tends to zero. [ABSTRACT FROM AUTHOR]

A family of Pk+22×2-PkH(divdiv)mixed finite elements on triangular grids is constructed for the biharmonic equation. For the H(divdiv)mixed finite element functions, both the stress tensor and its divergence have continuous normal components, σh⋅nand divσh⋅n, on each edge. The double divergence of the H(divdiv)mixed finite element space is exact the full discontinuous Pkpolynomial space on a triangular grid. The stability of the mixed method is established. The stress solution converges at the optimal order in L2and H(divdiv)norms. Four-order of superconvergence is proved in L2norm for the displacement solution. The Pkdiscrete solution is processed on each triangle to obtain an optimal order Pk+4solution there. Numerical results are presented verifying the theory. Additional computation shows that all existing H(divdiv)mixed finite elements fail to solve jump-coefficient biharmonic equations while the new mixed element converges independently of the interface-jump. This is because the H(divdiv)element here is the full H(divdiv)−Pk+2space while the existing mixed elements are subspaces of the H(divdiv)−Pk+2space, on a triangular grid.

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]

A recently published approach for modeling the cross flow in an extruder channel using a new solution to the biharmonic equation is utilized in a study of chaotic mixing in a free‐helix single‐screw extruder. This novel extruder was designed and constructed with the screw flight, also referred to as the helix, detached from the screw core. The flight‐helix had straight sides that more closely emulated rectangular channel theory than the nominal sloped sides of a conventional single screw channel. Each of the screw elements could be rotated independently to obtain chaotic motion in the screw channel. Using the new extruder, experimental evidence for the increased mixing of a dye, for both a Dirac and droplet input, with a chaotic flow field relative to the traditional residence time distribution is presented. These experimental results are compared using the new biharmonic equation‐based model. Comparing the experimental chaotic mixing with theoretical calculations was facilitated by a recently published technique for accurately placing the dye in the extruder channel. Because of the ability to periodically rotate only the flight/helix, the chaotic mixing results are minimally confounded by the existence of Moffatt eddies. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]

We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally C 4 -smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L 2 -norm as well as to equivalents of the H s -seminorms for 1 ≤ s ≤ 4. In the studied case of spline degree p = 9 , the numerical results indicate in case of the Greville points a convergence of order O (h p − 3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O (h p − 2) for all (semi)norms except for the equivalent of the H 4 -seminorm, where the order O (h p − 3) is anyway optimal. [ABSTRACT FROM AUTHOR]

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted P k + 4 polynomial from four P k polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [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]

Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation. [ABSTRACT FROM AUTHOR]

In this paper we consider a severely ill posed problem associated with a two dimensional homogeneous biharmonic equation. By perturbing the original problem and using a two parameters regularization method, we get a stable solution which converges to the solution of the considered problem. Under some priori bound assumptions, different errors estimates for the regularized solution are obtained. These last ones depend on the choice of the space of the exact solution. To show the effectiveness of the proposed regularization method some numerical results are given. [ABSTRACT FROM AUTHOR]

This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress σ = −∇2u is sought in the Sobolev space H(divdiv, Ω; S ) simultaneously with the displacement u in L2(Ω). By stemming from the structure of H(div, Ω; S ) conforming elements for the linear elasticity problems proposed by Hu and Zhang (2014), the H(divdiv, Ω; S ) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H (div, Ω; S ) conforming spaces of Pk symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise Pk−2 polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k ⩾ 3, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis. [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]

The auxiliary space preconditioning (ASP) technique is applied to the hybridizable discontinuous Galerkin (HDG) schemes for three different elliptic problems with a parameter dependent low order term, namely, a symmetric interior penalty HDG scheme for the scalar reaction-diffusion equation, a divergence-conforming HDG scheme for a vectorial reaction-diffusion equation, and a C0-continuous interior penalty HDG scheme for the generalized biharmonic equation with a low order term. Uniform preconditioners are obtained for each case and the general ASP theory by J. Xu [Computing, 56 (1996), pp. 215--235] is used to prove the optimality with respect to the mesh size and uniformity with respect to the low order parameter. [ABSTRACT FROM AUTHOR]

We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy. [ABSTRACT FROM AUTHOR]

Earlier, the authors developed a method for reducing boundary value problems in continuum mechanics for systems of differential equations to boundary value problems for individual equations, called scalar. This approach is based on the B. G. Galerkin transformation for systems of partial differential equations and the block element method. In this article, based on the well-known properties of the block element method, three approaches are developed that make it possible to implement the application of the block element method to fairly complex scalar boundary value problems. For the first time, by analogy with the exponential substitution used in ordinary differential equations with constant coefficients, a block substitution is used in this article, which makes it possible to solve boundary value problems for partial differential equations. As a result of the study, the sought solutions of scalar boundary value problems are represented as a sum using the simplest block elements, which greatly simplifies the study of the exact solutions obtained. [ABSTRACT FROM AUTHOR]

This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (P3) and possesses O (h 2) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with O (h 2) accuracy in the broken H2 norm. Besides, a discrete strengthened Miranda-Talenti estimate (∇h2 ·, ∇h2 ·) = (∆h ·, ∆h ·), which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence of solutions for the indefinite nonlinear fourth order elliptic problem with Navier boundary conditions: Δ²v - Δv + Vλ(x)v = f(x, v) in Ω, v = Δv = 0 on ∂Ω, where Ω is a bounded open domain in ℝN, N ≥ 2, Vλ(x) is sign-changing weight function. Also, the reaction source term f is not necessary positive. We will prove that for λ large enough, there exists a nontrivial solution. Our method is a variational one. The most delicate point is to choose the appropriate Sobolev space and the suitable norm. [ABSTRACT FROM AUTHOR]

The paper extends recently developed idea of stable evaluation of the Gaussian kernel. Owing to this, the Gaussian radial basis function that is sensitive to the shape parameter can be stably evaluated and applied to interpolation problems as well as to solve differential equations, giving highly accurate results. But it can be done only with grids being the Cartesian product of sets of points, what limits the use of this idea to rectangular domains. In the present paper, by the association of an appropriate transformation with the mentioned method, the latter is applied to solve biharmonic problems on quadrilateral irregular domains. As an example, in the present work this approach is applied to solve bending as well as free vibration problem of thin plates. In the paper some strategies for the implementation of the boundary conditions are also presented and examined due to the accuracy. The numerical tests show high accuracy and usefulness of the method.

In this paper, we consider the following sublinear biharmonic equations\begin{equation*} \Delta^2 u + V\left( x \right)u =K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*}where $N\geq5,~0

In this article, we study the multiplicity of positive solutions for the biharmonic equation of Kirchhoff type involving concave-convex nonlinearities, $$ \Delta^2u-\Big(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)\Delta u+V(x)u =\lambda f_1(x)|u|^{q-2}u+f_2(x)|u|^{p-2}u. $$ Using the Nehari manifold, Ekeland variational principle, and the theory of Lagrange multipliers, we prove that there are at least two positive solutions, one of which is a positive ground state solution.

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.

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 C1 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 24 in the L2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O(h3). 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(h4). An optimal reduction of 24 is observed when the functions on the C1 geometry are relaxed to be C0.

Coefficient inverse problems for second- and third-order equations with one or two unknown coefficients are investigated. The initial data are specified as the solution of an equation for a set of sounding sources averaged over time with power-law weights. It is shown that the original nonlinear inverse problems can be equivalently reduced to integral equations that are linear or nonlinear depending on the averaging method. It is proved that these equations have a unique solution determining the desired solution of the inverse problems. The results of a numerical experiment concerning the solution of a linear integral equation with a special kernel are presented. [ABSTRACT FROM AUTHOR]

We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities ƒ(x)|u|q−1u and h(x)|u|q−1u under certain conditions on ƒ(x), h(x), p and q. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of ƒ(x) and h(x) on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When h(x)+ ≠ 0, we prove that the equation has at least one nontrivial solution if ƒ(x)+ = 0 and that the equation has at least two nontrivial solutions if ƒΩ|ƒ+|r dx ∈ (0,Ar), where r and A are explicit numbers. These results are novel, which improve and extend the existing results in the literature. [ABSTRACT FROM AUTHOR]

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In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions. [ABSTRACT FROM AUTHOR]

Two‐dimensional biharmonic boundary‐value problems are considered by the linear barycentric rational method, the unknown function was approximated by the barycentric rational function. For the biharmonic equation, we change the biharmonic equation into the two Poisson equations by depression of order. The linear equations of discrete the biharmonic equation was changed into matrix form. For the basis of barycentric rational function, we present the convergence rate of linear barycentric rational method for biharmonic equation by depression of order. At last, several numerical examples are provided to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated. [ABSTRACT FROM AUTHOR]

In the present paper, we devote our effort to Cauchy boundary value problems for biharmonic equations. In general, the investigated problem is ill-posed. Therefore, we develop a filter method to defeat the ill-posedness of the problem. Explicit convergence rate is established under both a priori and a posteriori parameter choice rules. Finally, a numerical example is presented to illustrate the ill-posedness of the problem as well as the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]