A uniformly convergent diffrence scheme for the singular perturbation of a self adjoint elliptic partial differential equation

In this paper, we consider a self adjoint elliptic first boundary value problem with a small parameter affecting the highest derivative.In the paper, we set up a new scheme by the asymptotic analysis method, compare asymptotic behavior between the solution of the difference equation and the solution of the differential equation, and show uniform convergence of the new scheme.     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

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Numerical solution of the singularly perturbed problem for the hyperbolic equation with initial jump

In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.     

The Lie-group shooting method for multiple-solutions of Falkner-Skan equation under suction-injection conditions

For the Falkner-Skan equation, including the Blasius equation as a special case, we develop a new numerical technique, transforming the governing equation into a non-linear second-order boundary value problem by a new transformation technique, and then solve it by the Lie-group shooting method. The second-order ordinary differential equation is singular, which is, however, much saving computational cost than the original third-order equation defined in a semi-infinite range. In order to overcome the singularity we consider a perturbed equation. The newly developed Lie-group shooting method allows us to search a missing initial slope at the left-end in a compact space of t∈[0,1], and moreover, the initial slope can be expressed as a closed-form function of r∈(0,1), where the best r is determined by matching the right-end boundary condition. All that makes the new method much superior than the conventional shooting method used in the boundary layer equation under imposed boundary conditions. When the initial slope is available we can apply the fourth-order Runge-Kutta method to calculate the solution, which is highly accurate. The present method is very effective for searching the multiple-solutions under very complex boundary conditions of suction or injection, and also allowing the motion of plate.     

Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation

We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence of solutions.     

On singular perturbation boundary-value problem of coupling type system of convection-diffusion equations

In this paper we consider the singular perturbation boundary-value problem of thefollowing coupling type system of convection-diffusion equationsWe advance two methods:the first one is the initial value solving method,by which theoriginal boundary-value problem is changed into a series of unperturbed initial-valueproblems of the first order ordinary differential equation or system so that an asymptoticexpansion is obtained;the second one is the boundary-value solving method,by which theoriginal problem is changed into a few boundary-value problems having no phenomenon ofboundary-layer so that the exact solution can be obtained and any classical numericalmethods can be used to obtain the numerical solution of consismethods can be used to obtainthe numerical solution of consistant high accuracy with respect to the perturbationparameterε     

On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.     

An initial boundary value problem for modeling a piezoelectric dipolar body

This study deals with the first initial boundary value problem in elasticity of piezoelectric dipolar bodies. We consider the most general case of an anisotropic and inhomogeneous elastic body having a dipolar structure. For two different types of restrictions imposed on the problem data, we prove two results regarding the uniqueness of solution, by using a different but accessible method. Then, the mixed problem is transformed in a temporally evolutionary equation on a Hilbert space, conveniently constructed based on the problem data. With the help of a known result from the theory of semigroups of operators, the existence and uniqueness of the weak solution for this equation are proved.     

The stability on the solution of the initial value problem

In this paper,using the differentiability of the solution with respect to the initial value and the parameter,we present a method which,different from Liapunov’s direct method.will determine the stability of the non-stationary solution of the initial value problem when the non-stationary solution remains unknown.     

Pattern selection in a reaction-diffusion equation

This paper studies the pattern selection in a spatially-periodic problem of a simplified reaction-diffusion system described by an evolutionary Duffing equation. Numerical explorations by a pseudo-spectral method reveal the dependence of the pattern formation on the initial profile as well as the parameter. Only patterns with 2 ^m humps are found to be meta-stable in the corresponding parameter ranges. The project supported by Chinese Special Funds for Major State Basic Research Project “Nonlinear Science”, and NSFC (10002002 and 19990510)     

A class of variational difference schemes for a singular perturbation problem

In this paper, a singularly perturbed boundary value problem for second order self-adjoint ordinary differential equation is discussed. A class of variational difference schemes is constructed by the finite element method. Uniform convergence about small parameter is proved under a weaker smooth condition with respect to the coefficients of the equation. The schemes studied in refs. [1], [3], [4] and [5] belong to the class.     

A robust numerical method for flow through a pipe driven by an oscillating pressure gradient

The problem of periodic flow of an incompressible fluid through a pipe, which is driven by an oscillating pressure gradient (e.g. a reciprocating piston), is investigated in the case of a large Reynolds number. This process is described by a singularly perturbed parabolic equation with a periodic right‐hand side, where the singular perturbation parameter is the viscosity ν. The periodic solution of this problem is a solution of the Navier–Stokes equations with cylindrical symmetry. We are interested in constructing a parameter‐robust numerical method for this problem, i.e. a numerical method generating numerical approximations that converge uniformly with respect to the parameter ν and require a bounded time, independent of the value of ν, for their computation. Our method comprises a standard monotone discretization of the problem on non‐standard piecewise uniform meshes condensing in a neighbourhood of the boundary layer. The transition point between segments of the mesh with different step sizes is chosen in accordance with the behaviour of the analytic solution in the boundary layer region. In this paper we construct the numerical method and discuss the results of extensive numerical experiments, which show experimentally that the method is parameter‐robust. Copyright © 2006 John Wiley & Sons, Ltd.     

On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u _t+uu_x)_x=1/2u _x ² with the simplest initial data such that u _x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.     

Diffusive limits of nonlinear hyperbolic systems with variable coefficients

We consider the initial-boundary value problem for a 2-speed system of first-order nonhomogeneous semilinear hyperbolic equations whose leading terms have a small positive parameter. Using energy estimates and a compactness lemma, we show that the diffusion limit of the sum of the solutions of the hyperbolic system, as the parameter tends to zero, verifies the nonlinear parabolic equation of the p-Laplacian type.     

Small parameter method for determining motion of a viscous incompressible fluid in a thrust bearing

We consider the plane stationary motion of a viscous incompressible fluid between two surfaces. The fixed surface is given by the equation y=h[1+f(x/h)], where the functionf(x/h=h) characterizes the deviation of the fixed surface from the plane y=h(h and , are constants). The moving surface is a plane which moves with constant velocity along the x axis and remains parallel to the plane y=h. The small parameter method is used to solve the problem. The problem formulation is presented in the first section, the solvability of the linear equations obtained using the small parameter method is investigated in the second section, and the third section studies the convergence of the method and finds the radius of convergence of the constructed series.     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

On the Large Time Behavior of the Solutions of a Nonlocal Ordinary Differential Equation with Mass Conservation

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Local and Global Well-Posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity

In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with H ² initial data.     

Application of equation regulation method to the solution of inverse problems in multiphase materials

In this paper, we propose a new method of determining local material properties of multiphase composites given the experimentally measured displacements and known traction boundary conditions. In the proposed method, an “observation” term is added to the original differential equation, and the modified equation is solved in terms of a regulation parameter. We call this approach the equation regulation (ER) method. By appropriately adjusting the value of the regulation parameter based on the noise level in the input data, we get faster convergence and improved stability than prevailing methods of solving the inverse problem in elliptic ordinary differential equations. Several numerical examples to the solution of this non-linear problem with continuous and discontinuous coefficient functions are given to show the accuracy and reliability of the proposed method.