High accuracy nonconforming finite elements for fourth order problems

The approach of nonconforming finite element method admits users to solve the partial differential equations with lower complexity,but the accuracy is usually low.In this paper,we present a family of highaccuracy nonconforming finite element methods for fourth order problems in arbitrary dimensions.The finite element methods are given in a unified way with respect to the dimension.This is an effort to reveal the balance between the accuracy and the complexity of finite element methods.     

Multiple Periodic Solutions of Differential Delay Equations with 2k-1 Lags

In this paper,we study the periodic solutions to a type of differential delay equations with 2 k-1 lags.The 4 k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system.This is a new type of differential delay equations compared with all the previous researches.And this paper provides a theoretical basis for the study of differential delay equations.An example is given to demonstrate our main results.     

A DIRECT METHOD OF OPERATIONAL CALCULUS(Ⅰ)——Research for the solution of second-order linear ordinary differential equations with variable coefficients

According to Mikusinski's fundamental view, we reconstruct a direct method of operation acalbulus which may be adapted to search for the solution of differential equations with variablecoefficients.The present paper is devoted to solutions of two types——indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations: The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.     

A MIXED FINITE ELEMENT METHOD FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device is described by a system of three quasilinear partial differential equations. One is elliptic in form for the electric potential and the other two are parabolic in form for the conservation of electron and hole concentrations. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by a Galerkin method that applies a variant of the method of characteristics to the transport terms. Optimal order convergence analysis in L2 is given for the proposed method.     

SURFACE FINITE ELEMENTS FOR

In this article we define a surface finite element method （SFEM） for the numerical solution of parabolic partial differential equations on hypersurfaces F in R^n＋1. The key idea is based on the approximation of F by a polyhedral surface Гh consisting of a union of simplices （triangles for n = 2, intervals for n = 1） with vertices on Г. A finite element space of functions is then defined by taking the continuous functions on Гh which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on Г. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demorrstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.     

Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs

A type of infinite horizon forward-backward doubly stochastic differential equations is studied.Under some monotonicity assumptions,the existence and uniqueness results for measurable solutions are established by means of homotopy method.A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given.A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.     

A DIRECT METHOD OF OPERATIONAL CALCULUS (Ⅰ)——Research for the solution of second-order linear ordinary differentiaI equations with Variable coefficients

According to Mikusinski's fundamental view,we reconstruct a direct method of operation a calbulus which may be adapted to search for the solution of differential equations with variable coefficients.The present paper is devoted to solutions of two types——indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations, The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.     

MIXED MONOTONE ITERATIVE TECHNIQUES FOR SEMILINEAR EVOLUTION EQUATIONS IN BANACH SPACES

This paper is concerned with initial value problems for semilinear evolution equations in Banach spaces. The abstract iterative schemes are constructed by combining the theory of semigroups of linear operators and the method of mixed monotone iterations. Some existence results on minimal and maximal (quasi)solutions are established for abstract semilinear evolution equations with mixed monotone or mixed quasimonotone nonlinear terms. To illustrate the main results, applications to ordinary differential equations and partial differential equations are also given.     

MODIFIED LEAST SQUARE METHOD ON COMPUTING DIRICHLET PROBLEMS

The singularity theory of dynamical systems is linked to the numerical computation of boundary value problems of differential equations. It turns out to be a modified least square method for a calculation of variational problem defined on Ck(Ω), in which the base functions are polynomials and the computation of problems is transferred to compute the coefficients of the base functions. The theoretical treatment and some simple examples are provided for understanding the modification procedure of the methods. A modified least square method on difference scheme is introduced with a general matrix form of dynamical systems. We emphasize the simplicity of the algorithm and only use Euler algorithm to compute initial value problems of ODEs. A better algorithm is needed to reduce the stiffness of ODEs.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

Nonclassical symmetry reductions of the Boussinesq equation

In this paper we discuss symmetry reductions and exact solutions of the Boussinesq equation using the classical Lie method of infinitesimals, the direct method due to Clarkson and Kruskal and the nonclassical method due to Bluman and Cole. In particular, we compare and contrast the application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and differential Gröbner bases as a technique for solving the overdetermined systems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exact solutions of partial differential equations are also mentioned. To conclude we describe some of the important open problems in the field of symmetry analysis of differential equations.     

Similarity Reductions and Exact Solutions for the Two-Dimensional Incompressible Navier–Stokes Equations

We study similarity reductions and exact solutions of the (2+1)-dimensional incompressible Navier–Stokes equations using the direct method originally developed by Clarkson and Kruskal [37]. The Navier–Stokes equations are reduced to their conventional stream function form, and it is shown that there exist essentially five reductions into lower-order partial differential equations. Furthermore, we study the possibilities for reducing each of these five forms to ordinary differential equations, some of which can be solved analytically, while others necessitate numerical treatment. In particular we exhibit several new reductions that are not obtained using the classical Lie group method of infinitesimal transformations, and thus we generate new exact solutions of the governing equations. Some of our solutions admit physical interpretations, and many of them contain well-known Navier–Stokes solutions as special examples.     

Symmetry Reductions of the Combined KdV—mKdV Equation

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Unified approach to numerical methods,part II: Finite elements,boundary methods,and their coupling

This is the second in a series of three papers devoted to the presentation of a direct procedure of analysis of numerical methods for partial differential equations. The procedure consists of applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions. Here, the general Green's formulas for operators defined in discontinuous fields developed in the first article, are applied to formulate the method of weighted residuals for arbitrary linear operators. Finite elements, boundary methods, and general procedures for coupling finite elements and boundary methods are discussed.     

Shallow Water Wave Systems

In this article we study various systems that represent the shallow water wave equation
v_xxt + αvv_t − βv_x∂_x^-1 ( v_t ) − v_t − v_x = 0,
where (∂ _x ⁻¹ f )( x )=∫ _x ^∞ f ( y ) d y , and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [ J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [ J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

Classical Symmetry Reductions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions

Classical reductions of a (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation are classified. These reductions to systems of partial differential equations in 1+1 dimensions admit symmetries that lead to further reductions, i.e., to systems of ordinary differential equations. All these systems have been reduced to second-order ordinary differential equations. We present some particular solutions involving two arbitrary functions.     

Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so-called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.     

源于Fermi-Pasta-Ulam问题的非线性发展方程的相似约化

利用Clarkson和Kruskal引入的直接法和Lou的改进的直接法，给出了源于Fermi-Pasta-Ulam问题的非线性方程的4种类型的相似约化。     

An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations

In this paper, based on differential characteristic set theory and the associated algorithm (also called Wu?s method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P.A. Clarkson and E.L. Mansfield in [21] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations.