An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

In this article, we propose a tailored finite point method (TFPM) for the numerical solution of a type of fourth‐order singular perturbation problem in two dimensions based on the equation decomposition technique. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. Our numerical examples show that our method has second order convergence rate in energy norm as $\varepsilon\to0$ . © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Uniformly convergent C 0‐nonconforming triangular prism element for fourth‐order elliptic singular perturbation problem

In this article, we introduce a C ⁰‐nonconforming triangular prism element for the fourth‐order elliptic singular perturbation problem in three dimensions by using the bubble functions. The element is proved to be convergent in the energy norm uniformly with respect to the perturbation parameter. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1785–1796, 2014     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Some properties of solutions for a fourth‐order parabolic equation

This paper is concerned with a fourth‐order parabolic equation in one spatial dimension. On the basis of Leray–Schauder's fixed point theorem, we prove the existence and uniqueness of global weak solutions. Moreover, we also consider the regularity of solution and the existence of global attractor. Copyright © 2012 John Wiley & Sons, Ltd.     

On a second‐order expansion of the truncated singular subspace decomposition

We present a second‐order expansion for singular subspace decomposition in the context of real matrices. Furthermore, we show that, when some particular assumptions are considered, the obtained results reduce to existing ones. Some numerical examples are provided to confirm the theoretical developments of this study. Copyright © 2016 John Wiley & Sons, Ltd.     

New estimates of mixed finite element method for fourth‐order wave equation

The conforming bilinear mixed finite element method is established for a fourth‐order wave equation. By applying the interpolation and projection simultaneously, the superclose and superconvergence results of order O (h ²) for the original variable u and intermediate variable p  =? a ₁(X )Δu in H¹‐norm are achieved for the semi‐discrete and fully discrete schemes, respectively (where a ₁(X ) is the coefficient appeared in the equation). The distinct advantage of this method is that it can reduce the smoothness of solutions u ,u _t , and p compared with the existing literature for getting optimal estimates alone. At last, some numerical results are carried out to validate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition

In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equations where w is a constant, and . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well‐known (AR)‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.     

Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

In this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non‐ C⁰ rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C⁰ extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact soliton solution for the fourth‐order nonlinear Schrödinger equation with generalized cubic‐quintic nonlinearity

In this paper, we investigate the fourth‐order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic‐quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth‐order generalized cubic‐quintic nonlinear Schrödinger equation through modified F‐expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

An iterative method for a Cauchy problem for the heat equation

^** Email: tomjo{at}itn.liu.se An iterative method for reconstruction of the solution to aparabolic initial boundary value problem of second order fromCauchy data is presented. The data are given on a part of theboundary. At each iteration step, a series of well-posed mixedboundary value problems are solved for the parabolic operatorand its adjoint. The convergence proof of this method in a weightedL²-space is included.     

Domain decomposition for a parabolic convection‐diffusion problem

A two‐dimensional convection‐diffusion problem of parabolic type is considered. A multidomain decomposition algorithm with nonoverlapping subdomains based on a upwind scheme and on a piecewise equidistant mesh is investigated. Uniform in a perturbation parameter convergence properties of the algorithm are established. Numerical experiments complement the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Solving a fourth‐order fractional diffusion‐wave equation in a bounded domain by decomposition method

In this article, the Adomian decomposition method has been used to obtain solutions of fourth‐order fractional diffusion‐wave equation defined in a bounded space domain. The fractional derivative is described in the Caputo sense. Convergence of the method has been discussed with some illustrative examples. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A fourth‐order parabolic equation in two space dimensions

In this paper, we consider an initial‐boundary problem for a fourth‐order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytic rogue wave solutions for a generalized fourth‐order Boussinesq equation in fluid mechanics

A bilinear transformation method is proposed to find the rogue wave solutions for a generalized fourth‐order Boussinesq equation, which describes the wave motion in fluid mechanics. The one‐ and two‐order rogue wave solutions are explicitly constructed via choosing polynomial functions in the bilinear form of the equation. The existence conditions for these solutions are also derived. Furthermore, the system parameter controls on the rogue waves are discussed. The three parameters involved in the equation can strongly impact the wave shapes, amplitudes, and distances between the wave peaks. The results can be used to deeply understand the nonlinear dynamical behaviors of the rogue waves in fluid mechanics.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Superconvergence of conforming finite element for fourth‐order singularly perturbed problems of reaction diffusion type in 1D

We consider conforming finite element approximation of fourth‐order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C¹ finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of in a discrete energy norm is established. The error bound is uniformly valid with respect to the singular perturbation parameter ?. Numerical tests indicate that the error estimate is sharp. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 550–566, 2014     

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

We study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041     

A Neumann–Neumann algorithm for a mortar finite element discretization of fourth‐order elliptic problems in 2D

Here we present and analyze a Neumann–Neumann algorithm for the mortar finite element discretization of elliptic fourth‐order problems with discontinuous coefficients. The fully parallel algorithm is analyzed using the abstract Schwarz framework, proving a convergence which is independent of the parameters of the problem, and depends only logarithmically on the ratio between the subdomain size and the mesh size.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009