Generalized Galerkin Approximations for Pseudoinverses and Operator Equations of the First Kind

The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows ：Let H be a Hilbert space, { Hn } a sequence of closed subspaces of H, Pn the orthogonal projection of H onto Hn, A∈B（H） and An∈B（Hn）. Suppose s-lim↑n→∞Hn=H, lim↑n→∞｜｜Pn°（A-An） ｜｜n=0,-↑R（An）=R（An）（n∈N）. Then the following four propositions are equivalent ： （a） sup↑n∈Nv∈An^-1 inf ｜｜υ｜｜〈∞ if un∈R（An） and lim↑n→∞un=0; （b） sup↑n∈N｜｜ An ｜｜ 〈∞; （c） if un∈R（An） and lim↑n→∞ un=u, then u∈R（A） and s-lim↑n→∞An^-1（un）=A^-1（u）; （d） if un∈R（An） and lim↑n→∞un=u.then u∈R（A） and lim↑n→∞Au^＋（un）=A^＋（u）. Furtherrnore, if any of the above propositions holds, we have thin N（A）=s-lim↑n→∞N（An ）,R（A） = s-lim↑n→∞R（An ）, -↑R（A） =R（A）.

Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time(CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman(GHJB) equations. The Galerkin approximation with Legendre polynomials(GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved.Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.

ON NONLINEAR GALERKIN APPROXIMATION

Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction diffusion equations. The convergence results are proved.

H2 and H∞-Feedback Control Design for Nonlinear Gene Networks via Successive Galerkin’s Approximation

This paper presents a design method of H2 and H∞-feedback control loop for nonlinear smooth gene networks that are in control affine form. Formulaic solution methodology for solving the nonlinear partial differential equations, namely the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations through successive Galerkin’s approximation is implemented and the results are compared. Throughout the implementation, there were several caveats that need to be further resolved for practical applications in general cases. Such issues and the clarification of causes are mathematically established and reviewed.

Crank-Nicolson ADI Galerkin Finite Element Methods for Two Classes of Riesz Space Fractional Partial Differential Equations

In this paper,two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered.These two models can be regarded as the generalization of the classical wave equation in two space dimensions.Combining with the Crank-Nicolson method in temporal direction,efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed,respectively.The corresponding stability and convergence analysis of the numerical methods are discussed.Numerical results are provided to verify the theoretical analysis.

The improved element-free Galerkin method forthree-dimensional wave equation

The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.

孔隙率对矩形沟槽上垂直多孔屏障的波浪散射效应影响

The effect of porosity on surface wave scattering by a vertical porous barrier over a rectangular trench is studied here under the assumption of linearized theory of water waves.The fluid region is divided into four subregions depending on the position of the barrier and the trench.Using the Havelock’s expansion of water wave potential in different regions along with suitable matching conditions at the interface of different regions,the problem is formulated in terms of three integral equations.Considering the edge conditions at the submerged end of the barrier and at the edges of the trench,these integral equations are solved using multi-term Galerkin approximation technique taking orthogonal Chebyshev’s polynomials and ultra-spherical Gegenbauer polynomial as its basis function and also simple polynomial as basis function.Using the solutions of the integral equations,the reflection coefficient,transmission coefficient,energy dissipation coefficient and horizontal wave force are determined and depicted graphically.It was observed that the rate of convergence of the Galerkin method in computing the reflection coefficient,considering special functions as basis function is more than the simple polynomial as basis function.The change of porous parameter of the barrier and variation of trench width and height significantly contribute to the change in the scattering coefficients and the hydrodynamic force.The present results are likely to play a crucial role in the analysis of surface wave propagation in oceans involving porous barrier over submarine trench.

LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS

This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.

Oblique Wave Scattering Problems Involving Vertical Porous Membranes

Oblique surface waves incident on a fixed vertical porous membrane of various geometric configurations is analyzed here.The mixed boundary value problem is modified into easily resolvable problems by using a connection.These problems are reduced to that of solving a couple of integral equations.These integral equations are solved by a one-term or a two-term Galerkin method.The method involves a basis functions consists of simple polynomials multiplied with a suitable weight functions induced by the barrier.Coefficient of reflection and total wave energy are numerically evaluated and analyzed against various wave parameters.Enhanced reflection is found for all the four barrier configurations.

Effect of Porosity on Oblique Wave Diffraction by Two Unequal Vertical Porous Barriers

The diffraction of obliquely incident wave by two unequal barriers with different porosity in infinitely deep water is investigated by using two-dimensional linearized potential theory.Reflection and transmission coefficients are computed numerically using appropriate Galerkin approximations for two partially immersed and two submerged barriers.The amount of energy dissipation due to the permeable barriers is derived using Green’s integral theorem.The coefficient of wave force is determined using the linear Bernoulli equation of dynamic pressure jump on the porous barriers.The numerical results of hydrodynamics quantities are illustrated graphically.

SUPERCONVERGENT APPROXIMATIONS FOR WIENER-HOPF EQUATIONS

In this paper, by means of Sloan's iteration technique, we present a kind of iterative correction method for Galerkin approximations of Wiener-Hopf equations,and show that this is notonly a high order method but also an adaptable one.

ASYMPTOTIC ERROR EXPANSION AND DEFECT CORRECTION FOR SOBOLEV AND VISCOELASTICITY TYPE EQUATIONS

In this paper we study the higher accuracy methods - the extrapolation and defect correction for the semidiscrete Galerkin approximations to the solutions of Sobolev and viscoelasticity type equations. The global extrapolation and the correction approximations of third order, rather than the pointwise extrapolation results are presented.

Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations

Based on a recent result on linking stochastic differential equations on R^d to （finite-dimensional） Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensionl stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity:Local Existence and Blow-up in Solutions

In this paper,we consider a class of Kirchhoff equation,in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms.Where the studied equation is given as followsutt-K(Nu(t))[Δ_(p(x))^(u)+Δ_(r(x))^(ut)]=F(x,t,u).Mere,K(Nu(t))is a Kirchhoff function,Δ_(r(x))^(ut)represent a Kelvin-Vbigt strong damp-ing term,and F(x,t,u)is a source term.According to an appropriate assumption,we obtain the local existence of the weak solutions by applying the Galerkin's approximation method.Furthermore,we prove a non-global existence result for certain solutions with negative/positive initial energy.More precisely,our aim is to find a sufficient conditions for p(x)/q(x)/r(x)/F(x/t/u)and the initial data for which the blow-up occurs.

FULL DISCRETE TWO-LEVEL CORRECTION SCHEME FOR NAVIER-STOKES EQUATIONS

In this paper, a full discrete two-level scheme for the unsteady Navier-Stokes equations based on a time dependent projection approach is proposed. In the sense of the new projection and its related space splitting, non-linearity is treated only on the coarse level subspace at each time step by solving exactly the standard Galerkin equation while a linear equation has to be solved on the fine level subspace to get the final approximation at this time step. Thus, it is a two-level based correction scheme for the standard Galerkin approximation. Stability and error estimate for this scheme are investigated in the paper.

On Poincare-Friedrichs Type Inequalities for the Broken Sobolev Space W^(2,1)

We are concerned with the derivation of Poincare-Friedrichs type inequalities in the broken Sobolev space W^(2,1)(Ω;T h)with respect to a geometrically conforming,simplicial triagulation T h of a bounded Lipschitz domainΩin R d,d∈N.Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as C^(0) Discontinuous Galerkin(C^(0)DG)approximations of minimization problems in the Sobolev space W^(2,1)(Ω),or more generally,in the Banach space BV^(2)(Ω)of functions of bounded second order total variation.As an application,we consider a C^(0) DG approximation of a minimization problem in BV^(2)(Ω)which is useful for texture analysis and management in image restoration.

A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic.The global expansions are contaminated by the presence of a local discontinuity,and the result is that the partial sums are oscillatory and feature non-uniform convergence.This characteristic behavior is called the Gibbs phenomenon.However,David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

Propagation of oblique water waves by an asymmetric trench in the presence of surface tension

An analysis is presented for the propagation of oblique water waves passing through an asymmetric submarine trench in presence of surface tension at the free surface.Reflection and transmission coefficients are evaluated applying appropriate multi-term Galerkin approximation technique in which the basis functions are chosen in terms of Gegenbauer polynomial of order 1/6 with suitable weights.The energy identity relation is derived by employing Green’s integral theorem in the fluid region of the problem.Reflection and transmission coefficients are represented graphically against wave numbers in many figures by varying several parameters.The correctness of the present method is confirmed by comparing the results available in the literature.The effect of surface tension on water wave scattering is studied by analyzing the reflection and transmission coefficients for a set of parameters.It can be observed that surface tension plays a qualitatively relevant role in the present study.

Scattering of water waves by thick rectangular barriers in presence of ice cover

Assuming linear theory,the two dimensional problem of water wave scattering past thick rectangular barrier in presence of thin ice cover,is investigated here.Mainly four types of thick barriers are considered here and also the ice cover is taken as a thin elastic plate.May be the barrier is partially immersed or bottom standing or fully submerged in water or in the form of thick rectangular wall with a submerged gap presence in water.The problem is formulated in terms of a first kind integral equation by considering the symmetric and antisymmetric parts of velocity potential function.The integral equation is solved by using multi term Galerkin approximation method involving ultraspherical Gegenbauer polynomials as its basis function.The numerical solutions of reflection and transmission coefficients are obtained for different parametric values and these are seen to satisfy the energy identity.These coefficients are depicted graphically against the wave number in a number of figures.Some figures available in the literature drawn by using different mathematical methods as well as laboratory experiments are also recovered following the present analysis without the presence of ice cover,thereby confirming the correctness of the results presented here.It is also observed that the reflection and transmission coefficients depend significantly on the width of the barriers.