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This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H 2 and standard L 2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the error and stability analysis of the method of fundamental solution (MFS) is explored for biharmonic equations. The bounds of errors are derived for the fundamental solutions r 2ln r in bounded simply-connected domains, and the polynomial convergence rates are obtained for certain smooth solutions. The bounds of condition number are also derived to show the exponential growth rates for disk domains. Numerical experiments are carried out to support the above analysis, which is the first time to provide the rigorous analysis of the MFS using r 2ln r for biharmonic equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the adaptive fashion of the Ciarlet–Raviart mixed method for biharmonic equation/eigenvalue problem with simply supported boundary condition in R d. We propose an a posteriori error indicator of the Ciarlet–Raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the Ciarlet–Raviart approximate eigenfuction, and prove the reliability and efficiency of the indicators. We also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability. We design an adaptive Ciarlet–Raviart mixed method with piecewise polynomials of degree less than or equal to m , and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order O (d o f − 2 m d ) (d = 2 , m = 2 , 3 ; d = 3 , m = 3). [ABSTRACT FROM AUTHOR]

• For the biharmonic eigenvalue problem, Wang et al. [Adv. Comput. Math. 45(2019)] first analyzed the mixed DG method (MDG method) under the condition that the eigenfunction u and Δ u. have the same regularity. In this paper, we establish an MDG discretization which is different with that in the existing work and we conduct our analysis under the condition that u and Δ u have different regularities. • The existing literature derived the error estimate of biharmonic eigenvalue by using the estimate of Laplacian eigenvalue problem. In this paper, we establish an identity (see Lemma 3.5) and use it to get the a priori error estimate of biharmonic eigenvalue. We prove that the convergence rate of approximate eigenvalues reaches 2p-2 in h and p − (2 s − 7) in when p ≥ 2 and the eigenfunction u ∈ H S (Ω) (s ≥ p + 1) which is consistent with the estimate in the existing work [Babuska and Osborn, Eigenvalue problems, 1991] for the classical mixed finite element method for biharmonic eigenvalue problem. • Using the identity (see Lemma 3.5) we also discuss the a posteriori error estimate for the biharmonic eigenvalue. The literature on the a posteriori error estimate for Ciarlet-Raviart mixed method is still quite a few. We give an a posteriori error estimator of the approximate eigenvalue and analyze its reliability. The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p ≥ 3 , we derive the a priori error estimates of the approximate eigenfunction in the broken H 1 norm and L 2 norm which are optimal in h and suboptimal in p. When p = 2 , the approximate eigenfunctions converge but with only suboptimal convergence rate. When p ≥ 2 and the eigenfunction u ∈ H s (Ω) (s ≥ p + 1), we prove that the convergence rate of approximate eigenvalues reaches 2 p − 2 in h and p − (2 s − 7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues. [ABSTRACT FROM AUTHOR]

Geometrically continuous ( G k ) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are C k also for non-tensor-product layout. This paper describes and analyzes one such concrete C 1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developed G 1 surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson’s equation on the disk using interior vertices of valence 3 and symmetric layout, we observe O ( h 3 ) convergence in the L ∞ norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis. [ABSTRACT FROM AUTHOR]

We employ spectral Legendre–Galerkin and mixed Legendre–Galerkin approximations to solve the first bi-harmonic equation in one dimension, respectively. By orthogonal properties of Legendre polynomials, we obtain an explicit a posteriori error indicator for spectral Legendre–Galerkin methods. Furthermore, in virtue of an auxiliary variable, we present spectral mixed Legendre–Galerkin methods and study the a priori estimate and a posteriori error indicator. Especially, these indicators only depend on the expansions of the right-hand item. Numerical examples are presented to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fourth-order elliptic equation of the form ▵ 2 u - ▵ u + V ( x ) u - κ 2 ▵ ( u 2 ) u = h ( x , u ) , x ∈ R N , where ▵ 2 ≔ ▵ ( ▵ ) is the biharmonic operator, κ ⩾ 0 , N ⩽ 6 , V ∈ C ( R N , R ) and h ∈ C ( R N × R , R ) . Under appropriate assumptions on V ( x ) and h ( x , u ) , some existence and multiplicity results for nontrivial solutions are obtained via variational methods. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new construction of a stable cubic spline-wavelet basis on the interval satisfying complementary boundary conditions of the second order. It means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the second order, while the dual wavelet basis preserves the full degree of polynomial exactness. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator. [Copyright &y& Elsevier]

Abstract: In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions. [Copyright &y& Elsevier]

Abstract: We provide a rigorous analysis of structure of a ray tube in geometrical shock dynamics. Our aim is to shed more lights on the cross-sectional area function of a ray tube. We have shown that for a given ray tube and a given initial value of cross-sectional area, then its cross-sectional area can be determined uniquely everywhere. We give a definition of cross-sectional area of a ray tube in precise mathematical terms and from that we derive a set of relations, each describing the cross-sectional area for an arbitrary ray tube in geometrical shock dynamics. We have shown that from our results one can deduce Whitham’s area function relation as a partial differential equation from our general formulations. Some applications are discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we discuss the properties of the Schwarzian derivative, integral means and the affine and linear invariant families of biharmonic mappings. First, we introduce the Schwarzian derivative for biharmonic mappings F =∣z∣2 G + H, and obtain several necessary and sufficient conditions for to be analytic. Second, we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings. Finally, we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the Jacobian of functions in these invariant families. [Copyright &y& Elsevier]

Abstract: In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable and its Laplacian are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach. [Copyright &y& Elsevier]

Abstract: Using slope enclosures may provide sharper bounds for the range of a function than using enclosures of the derivative. Hence, slope enclosures may be useful in verifying the assumptions for existence tests or in algorithms for global optimization. Previous papers by Kolev and Ratz show how to compute slope enclosures for convex and concave functions. In this paper, we generalize these formulas and show how to obtain slope enclosures for a function that has exactly one point of inflection or whose derivative has exactly one point of inflection. [Copyright &y& Elsevier]

Abstract: This paper considers the existence and uniqueness of the solution to the Cauchy problem for a class of Boussinesq equation, proves the global existence and finite time blowup of the solution to the problem by the potential well method. [Copyright &y& Elsevier]

Abstract: In this paper, we point out that the generalized iteration method presented in a recent paper by Ling and Hu [X.-F. Ling, X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379] belongs to the inexact Uzawa algorithm, and an improper statement in a theorem in that paper is corrected. [Copyright &y& Elsevier]

Abstract: In this paper, we study the uniform attractor for semilinear wave equation with mixed differential quotient terms and critical nonlinearity. We prove the existence of the uniform attractor in . [Copyright &y& Elsevier]

Abstract: A class of discrete-time orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations with initial and boundary conditions are considered. These schemes are constructed by using piecewise cubic Hermite interpolations in space combined with finite difference methods in time. It is proved that the schemes have the conservation laws of discrete energy, and possess second order accuracy in maximum norm and fourth order in -norm for time and space, respectively. The conservation laws and rate of convergence are verified in numerical experiments. Moreover, the propagations of solitary waves and collisions of two head-on solitary waves are also well simulated. [Copyright &y& Elsevier]

Abstract: In the present article, we implement the new iterative method (NIM) proposed by Daftardar-Gejji and Jafari [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753–763] to solve linear/nonlinear partial differential equations of integer and fractional order. The results obtained are compared with those obtained by other iterative methods such as adomian decomposition, homotopy perturbation and variational iteration methods. [Copyright &y& Elsevier]

Abstract: In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases. [Copyright &y& Elsevier]

Abstract: In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a numerical method for solving the Cauchy problem of the modified Helmholtz equation. By using Green’s formula, the Cauchy problem is transformed to a moment problem. Then we propose a Tikhonov type regularization algorithm for obtaining an approximate solution to the Neumann date on the unspecified boundary. Convergence analysis and error estimation are also discussed. For numerical verification, several numerical examples in the two-dimensional case are presented. [Copyright &y& Elsevier]

Abstract: In this paper we consider the approximate controllability of a class of degenerate systems. The equations may be weakly degenerate and strongly degenerate on the boundary. We prove that the control systems are approximate controllable by constructing the controls via the conjugate problems. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the problem of how to solve the system of linear operator equations in Hilbert spaces and applied the result to reproducing kernel spaces. Under the assumption that the operator equations has a unique solution, its formal solution is given. As a result, when are both reproducing kernel spaces, the formal solution turned into analytic solution. Truncating the series (where formal solution and the analytic solution are denoted by a series), the approximate solution of the operator equations is obtained. When increasing the number of the nodes, the error of the approximate solution is monotone decreasing in the sense of the norm of the Hilbert space. The final numerical experiment shows the efficiency of our method. [Copyright &y& Elsevier]

Abstract: In this paper, a general technique is proposed for solving the solution of a parabolic differential equation with integral boundary condition in the reproducing kernel space. The representation of solution for the parabolic differential equation with integral boundary condition is given. The solution is given by the form of series and its approximate solution is obtained by truncating the series. Numerical results show that the method employed in the paper is valid. It is worthy to note that the method used in the paper can be generalized to solving linear ordinary or partial differential equation with initial condition and boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the linearly viscoelastic equations for Koiter shells. The existence and uniqueness of the solution are proved by Galerkin method. [Copyright &y& Elsevier]

Abstract: We develop a nonstandard difference-integral method based on a nonstandard finite difference method coupled with a CE–SE scheme. We use the viscous Burgers’ equation with preestablished conditions as a benchmark for testing our method. Numerical results obtained show that this new method is more robust and efficient than the associated standard difference-integral method. [Copyright &y& Elsevier]

Abstract: While Darcy’s equations are adequate for studying a large class of flows through porous media, there are several situations wherein it would be inappropriate to use Darcy’s equations. One such example is a flow wherein the range of pressures involved is very large and high pressures and pressure gradients are at play. Here, after developing an approximation for the flow through a porous solid, that is a generalization of an equation developed by Brinkman, we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy Law”. We find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the Brinkman equations or Darcy’s equations (or the Navier–Stokes equation if one neglects the porosity) in that the solutions for the velocity as well as the vorticity lack symmetry and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower. We also find that for a certain range of values for the non-dimensional parameters describing the flow, boundary layers develop in that the vorticity is confined next to the boundary adjacent to which the viscosity is lower, such boundary layers being absent in the other classical cases. [Copyright &y& Elsevier]

Abstract: We discuss the numerical differentiation from the noisy data. We propose a method, i.e., adding a differential term, for the differentiation problem. An error estimate shows that the method allows for the approximate recovery of the derivative function. Some interesting numerical tests display the properties of the proposed method. [Copyright &y& Elsevier]

Abstract: We propose a new numerical method for solving stochastic boundary-value problems. The method uses the deterministic method of lines to treat the time, space and randomness separately. The emphasis in the present study is given to stochastic partial differential equations with forced additive noise. The spatial discretization is carried out using a second-order finite volume method, while the associated stochastic differential system is numerically solved using a class of stochastic Runge–Kutta methods. The performance of the proposed methods is tested for a stochastic advection–diffusion problem and a stochastic Burgers equation driven with white noise. Numerical results are presented in both one and two space dimensions. [Copyright &y& Elsevier]

Abstract: This paper deals with Cauchy problem for a degenerate equation with variable coefficient and absorption. We show that the solution to the Cauchy problem possesses the property that the interfaces globally exist or disappear in finite time depending on the behavior of the coefficient at infinity. [Copyright &y& Elsevier]

Abstract: In this article, a system of regularized long wave equations are studied. With the aid of the mathematic software Maple and using the direct method, some new exact solutions: compactons, solitons, solitary patterns and periodic solutions are obtained. [Copyright &y& Elsevier]

Abstract: This paper deals with the Cauchy problem for the equation u t =(u m ) xx + b(x, t)(u p ) x , with m >1, 0< p <1, and b(x, t) being positive. First, we establish the local existence and uniqueness results for the problem under appropriate hypotheses. Then we show that instantaneous shrinking of the support of the solution depends on the behavior of b(x, t). [Copyright &y& Elsevier]

Abstract: We use sextic spline function to develop numerical method for the solution of system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the approximate solutions obtained by the present method are better than those produced by other collocation, finite difference and spline methods. A numerical example is given to illustrate practical usefulness of our method. [Copyright &y& Elsevier]

Abstract: We consider the blow-up solutions of the Cauchy problem for critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Raphaël’s recent arguments as well as Carles’ transform, the sharp upper and lower bounds of the blow-up rate are obtained. [Copyright &y& Elsevier]

Abstract: This paper considers the Dirichlet problem for the biharmonic equation. Finite volume and multi-grid methods have been proposed for solving this equation but in this paper, a new idea will be investigated. This equation is reduced to a system of integral equations (SIEs). Collocation methods for solving some SIEs with weakly singular kernels are rather complicated so that a poor choice of a set of collocation points may lead to instability. Therefore, the best special grids highly concentrated near the possible singularity points for one of the kernels is investigated. Also, the SIEs is reduced to a convex functional, and a minimal solution of this functional gives the minimum condition number for the SIEs. A parameter of collocation points is defined such that it is related to minimal solution of the functional. On the other hand, the speed of convergence rate depends on selecting the parameter of collocation points and vice versa. Finally, we test our method using several numerical examples and demonstrate its efficiency. [Copyright &y& Elsevier]

Abstract: In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions. [Copyright &y& Elsevier]

Abstract: One of the less-known integration methods is the weighted Newton–Cotes of closed type quadrature rule, which is denoted by:where w(x) is a positive function and is a positive value. There are various cases for the weight function w(x) that one can use. Because of special importance of the weight function of Gauss–Chebyshev quadrature rules, i.e. in numerical analysis, we consider this function as the main weight. Hence, in this paper, we face with the following formula in fact:It is known that the precision degree of above formula is n +1 for even n′s and is n for odd n′s, however, if we consider its bounds as two additional variables we reach a nonlinear system that numerically improves the precision degree of above formula up to degree n +2. In this way, we give several examples which show the numerical superiority of our approach. [Copyright &y& Elsevier]

Abstract: We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is split into a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by second-order finite difference discretizations. Using a radial grid with shifting a half mesh away from the origin, we can handle the coordinate singularity easily without pole conditions. The Sherman–Morrison formula is then applied to solve the resultant linear system in a cost-efficient way. The computational complexity of the method consists of O(MNlog2N) arithmetic operations for M×N grid points. The numerical accuracy check and some applications to the incompressible Navier–Stokes flows inside a disk are conducted. [Copyright &y& Elsevier]

In this paper, we consider a finite element approximation of a nonlinear biharmonic equation which is related to the well-known two-dimensional Navier–Stokes equations. First, optimal energy and H1-norm estimates are obtained. Second, a two-level additive Schwarz method is presented for the discrete nonlinear algebraic system. It is shown that if the Reynolds number is sufficiently small, the two-level Schwarz method is optimal, i.e., the convergence rate of the Schwarz method is independent of the mesh size and the number of subdomains. [Copyright &y& Elsevier]

In this article, using three spatial-grid points we propose two new two level implicit finite difference approximations of O(k2+h2) and O(k2+h4) in a coupled manner to the one-space dimensional unsteady quasi-linear biharmonic equation A(x,t,u,uxx)uxxxx+ut=f(x,t,u,ux,uxx,uxxx), 0, t>0 subject to the initial and boundary conditions u(x,0)=φ(x), u(0,t)=p0(t), uxx(0,t)=q0(t), u(1,t)=p1(t), uxx(1,t)=q1(t) are prescribed, where h>0 and k>0 are mesh sizes in x- and t-directions, respectively. The numerical solution of uxx is obtained as a by-product of the method and we do not require to discretize the boundary conditions. The methods are successfully tested on the problems having singularities. Numerical results are provided to demonstrate the convergence of new methods. [Copyright &y& Elsevier]

This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the "global" boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data. [ABSTRACT FROM AUTHOR]