A difference scheme for solving the Timoshenko beam equations with tip body

In this article, a Timoshenko beam with tip body and boundary damping is considered. A linearized three-level difference scheme of the Timoshenko beam equations on uniform meshes is derived by the method of reduction of order. The unique solvability, unconditional stability and convergence of the difference scheme are proved. The convergence order in maximum norm is of order two in both space and time. A numerical example is presented to demonstrate the theoretical results.     

A finite difference scheme for solving the Timoshenko beam equations with boundary feedback

In this study, we derive a finite difference for a Timoshenko beam with boundary feedback by the method of reduction of order on uniform meshes. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second order convergent in _L∞$L_{\infty }$ norm. Numerical results demonstrate the theoretical results.     

Boundary feedback stabilization of the undamped Timoshenko beam with both ends free

In this paper, we study a boundary feedback system of a class of nonuniform undamped Timoshenko beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.     

Numerical Approximation and Error Analysis for the Timoshenko Beam Equations with Boundary Feedback

In this paper,the numerical approximation of a Timoshenko beam with bound- ary feedback is considered.We derived a linearized three-level difference scheme on uniform meshes by the method of reduction of order for a Timoshenko beam with boundary feedback.It is proved that the scheme is uniquely solvable,unconditionally stable and second order convergent in L_∞norm by using the discrete energy method. A numerical example is presented to verify the theoretical results.     

A finite difference scheme for solving the heat transport equation at the microscale

Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank‐Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999     

A new wavelet preconditioner for finite difference operators

This paper is devoted to the construction of a new multilevel preconditioner for operators discretized using finite differences. It uses the basic ingredients of a multiscale construction of the inverse of a variable coefficient elliptic differential operator derived by Tchamitchian [19]. It can be implemented fast and can therefore be easily incorporated in finite difference solvers for elliptic PDEs. Theoretical results, as well as numerical tests and implementation technical details are presented. This work has been partially supported by TMR Research Network Contract FMRX-CT98-0184.AMS subject classification 00A69, 65T60, 65Y99, 15A12     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.     

A linearized compact difference scheme for a class of nonlinear delay partial differential equations

A linearized compact difference scheme is presented for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. The unique solvability, unconditional convergence and stability of the scheme are proved. The convergence order is O(τ²+h⁴)$O({\tau }^2+h^4)$ in _L∞$L_{\infty }$ norm. Finally, a numerical example is given to support the theoretical results.     

A locking-free scheme for the LQR control of a Timoshenko beam

In this paper we analyze a locking-free numerical scheme for the LQR control of a Timoshenko beam. We consider a non-conforming finite element discretization of the system dynamics and a control law constant in the spatial dimension. To solve the LQR problem we seek a feedback control which depends on the solution of an algebraic Riccati equation. An optimal error estimate for the feedback operator is proved in the framework of the approximation theory for control of infinite dimensional systems. This estimate is valid with constants that do not depend on the thickness of the beam, which leads to the conclusion that the method is locking-free. In order to assess the performance of the method, numerical tests are reported and discussed.     

Exact finite difference scheme for linear differential equation with constant coefficients

An exact finite difference equation for the n-th order linear differential equation with real, constant coefficients is constructed. The exact finite difference scheme is expressed differently but equivalent to that given by Potts [3].     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

一类偏积分微分方程二阶差分全离散格式

本给出了数值求解一类偏积分微分方程的二阶全离散差分格式．采用了Crank-Nicolson格式；积分项的离散利用了Lubieh的二阶卷积积分公式；给出了稳定性的证明，误差估计及收敛性的结果．     

A MODIFIED UPWIND DIFFERENCE SCHEME FOR NONLINEAR PARABOLIC EQUATIONS IN A VARIABLE DOMAIN

The nonlinear parabolic equations in a variable domain are considered. A modified up-wind difference scheme is given in the variable domain. Stability in l norm and error estimate in norm are obtained.     

A finite difference scheme for traveling wave solutions to Burgers equation

We construct a finite difference scheme for the ordinary differential equation describing the traveling wave solutions to the Burgers equation. This difference equation has the property that its solution can be calculated. Our procedure for determining this solution follows closely the analysis used to obtain the traveling wave solutions to the original ordinary differential equation. The finite difference scheme follows directly from application of the nonstandard rules proposed by Mickens. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 815–820, 1998     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.     

A compact finite difference scheme for solving a three‐dimensional heat transport equation in a thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation differs from the traditional heat diffusion equation in having a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time. In this study, we develop a high‐order compact finite difference scheme for the heat transport equation at the microscale. It is shown by the discrete Fourier analysis method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 441–458, 2000     

A nonstandard finite difference scheme for contaminant transport with kinetic langmuir sorption

This study focuses on a contaminant transport model with Langmuir sorption under nonequilibrium conditions. The numerical instabilities of the standard finite difference schemes including the upwind method are investigated. By using the nonstandard finite difference method, a better finite difference model is constructed. The numerical simulation on a specific system configuration proves the advantages of the new finite difference model. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 767–785, 2011