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A new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math. 80, 1902–1931) where new transform techniques were developed for boundary value problems for Laplace's equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry. [ABSTRACT FROM AUTHOR]

In this paper, a meshless method based on finite point set is presented for solving the biharmonic equation with simply supported boundary condition. The biharmonic equation is split into a coupled system of two Poisson equations by introducing an intermediate function. The system of two Poisson equations is then solved by finite pointset method. This method is a local iterative method based on the weighted least square approximation. The advantage of this method is that two resultant of sizes only 6 × 6 matrices are solved at each particle for the original and intermediate solution. Numerical results indicate a good accuracy of the finite pointset method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a quasi boundary value method for an homogeneous biharmonic equation in a rectangular domain. It is known that the problem for the Biharmonic equations is severely ill-posed in the sense of Hadamard [12], i.e., the solution does not depend continuously on the given data. Convergence estimates for the regularized solutions are obtained under a priori and a posteriori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the mixed element method for the boundary value problem of the biharmonic equation ▵2u = f in two dimensions. We start from a u ∼ ▿ ∼ 2u ∼ div 2u formulation that is discussed in [4] and construct its stability on H1 0(Ω) × H1 0(Ω) × L2 sym(Ω) × H-1(div). Then we utilise the Helmholtz decomposition of H-1(div,Ω) and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove a multiplicity result for some biharmonic elliptic equation of Kirchhoff type and involving non-linearities with critical exponential growth at infinity. Using some variational arguments and exploiting the symmetries of the problem, we establish a multiplicity result giving two nontrivial solutions. [ABSTRACT FROM AUTHOR]

By variational methods we consider a p-biharmonic equation with nonlocal term on unbounded domain. We give sufficient conditions for the existence of solutions when some certain assumptions are fulfilled. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation [ABSTRACT FROM AUTHOR]

We consider the entire solution of the semilinear biharmonic equation Δ 2 u = e u , in R N , N ≥ 1 . For N = 3 , we obtain the asymptotic of the entire nonradial solution which extends the results of Lai and Ye (in press). Besides, a new singular solution of the above equation is constructed by derivative estimates for N ≥ 13 . [ABSTRACT FROM AUTHOR]

The article focuses on estimation of solution for biharmonic equation with unbounded domains of Neumann problem. Topics discussed include establishment of method for solution retrieval by operating with Hardy-type inequalities, determination of independent solutions by theorem for biharmonic equations, utilization of Liouville's theorem for solutions assessment of equations and establishment of Bernstein inequality by theory of Douglis-Nirenberg for elliptic systems.

In this paper, we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval [0, L], generated by the shifted Chebyshev polynomials. Moreover, we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation. The main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem. Convergence of the shifted Chebyshev polynomials expansion in two-dimensions is investigated. Also the power of this manageable method is illustrated. [ABSTRACT FROM AUTHOR]

In this paper we give sufficient conditions for the existence of solutions of a biharmonic equation of the form Δu² + V(x)u = f(u) in ℝ4 where V is a continuous positive potential bounded away from zero and the nonlinearity f(s) behaves like eα°s² at infinity for some α0 > 0. In order to overcome the lack of compactness due to the unboundedness of the domain ℝ4, we require some additional assumptions on V. In the case when the potential V is large at infinity we obtain the existence of a nontrivial solution, while requiring the potential V to be spherically symmetric we obtain the existence of a nontrivial radial solution. In both cases, the main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term f. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated. [Copyright &y& Elsevier]

In this paper, we research the standard 13-point difference scheme for solving the biharmonic equation. Existence and convergence of finite difference methods(FDM) solutions are obtained by estimating the lower bounds of the minimum eigenvalues of the discrete matrix and making use of the Taylor series expansions, respectively. The accuracy are proved to be O(h²). Moreover, the extrapolation techniques are used to improve the high accuracy of the solutions. For the given examples, numerical results show that the errors O(h4) is obtained from the first level extrapolation. The very accurate solutions with the errors O(10-8) are obtained by the third level extrapolation techniques under the homogeneous essential boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the critical growth biharmonic problem with a parameter and establish uniform lower bounds for , which is the supremum of the set of , related to the existence of positive solutions of the biharmonic problem. [Copyright &y& Elsevier]

The paper presents a new stability analysis for the finite difference method (FDM) for biharmonic equations, based on the effective condition number. Upper bounds of the effective condition number and the simplest effective condition number are derived. In general cases, the bound of the effective condition number is O(h-3.5 ), which are smaller than Cond. = O(h-4), where h is the minimal meshspace of the difference grids. Surprisingly, the bounds of the effective condition number are only O(1) for homogeneous boundary conditions. Namely, these new stability analysis are more valid than previous stability analysis. Numerical experiments are provided to verify the stability analysis. [ABSTRACT FROM AUTHOR]

Abstract: By a variant version of Fountain Theorem due to Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms. [Copyright &y& Elsevier]

Abstract: We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted -spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin. The solutions are obtained from the versions of mountain pass lemma and linking theorem. [Copyright &y& Elsevier]

Abstract: Given Ω a smooth bounded domain of , , we consider functions that are weak solutions to the equation where , and . In this article, we prove the maximal regularity of solutions to the above equation, depending on the value of and the relative position of Ω with respect to the origin. In particular, the solutions are in when . [Copyright &y& Elsevier]

Abstract: We describe the construction of Total Differential (TD) three-phase data for the implementation of the exact global pressure formulation for the modeling of three-phase compressible flow in porous media. This global formulation is preferred since it reduces the coupling between the pressure and saturation equations, compared to phase or weighted formulations. It simplifies the numerical analysis of the problem and boosts its computational efficiency. However, this global pressure approach exists only for three-phase data (relative permeabilities, capillary pressures) which satisfy a TD condition. Such TD three-phase data are determined by the choice of a global capillary pressure function and a global mobility function, which take both saturations and global pressure level as argument. Boundary conditions for global capillary pressure and global mobility are given such that the corresponding three-phase data are consistent with a given set of three two-phase data. The numerical construction of global capillary pressure and global mobility functions by and finite element is then performed using bi-Laplacian and Laplacian interpolation. Examples of the corresponding TD three-phase data are given for a compressible and an incompressible case. [Copyright &y& Elsevier]

Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the biharmonic equation with has positive solutions on if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball. [Copyright &y& Elsevier]

The article focuses on solution estimation for biharmonic equation of Dirichlet problem. Topics discussed include utilization of methods for smoothness analysis of solutions for biharmonic equation, solution for independent variables with infinity and boundary points, evaluation of energy involved Saint-Venant's principle for elasticity, consideration of infinity point for generalized solution of Navier-Stokes equation for the problem and establishment of solution by theorem for the problem.

We study the asymptotic behavior of least energy solutions to the equation Δ2u=c0K(x)upℇ with the Navier boundary condition as ℇ→+0, where Ω is a smooth bounded domain in RN (N≥5), c0=(N-4)(N-2)N(N+2) and pℇ=(N+4)/(N-4)-ℇ, ℇ>0. Under some assumptions on the coefficient function K, we obtain fairly precise asymptotics of the L∞-norm of least energy solutions. [ABSTRACT FROM AUTHOR]