A singular perturbation analysis with applications to delay differential equations

We prove an approximation result for the solutions of a singularly perturbed, nonautonomous ordinary differential equation which has interesting applications to problems in higher dimensions. Here our result is applied to a singularly perturbed, delay differential equation with state dependent time-lags (i.e., aninfinite dimensional problem). We find a new dynamical system (also in infinite dimensions), which describes, in a certain sense, the dynamics of our delay equations for very small values of the singular parameter.     

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.     

Uniformly higher order accurate extrapolations to solutions of uniformly convergent discretization methods for singularly perturbed problems

We discuss the uniformly higher order accurate extrapolations, which are based on the uniform expansion for global error, to solutions of uniformly convergent discretization methods for singularly perturbed problems. By applying the approach to the Il'in-Allen-Southwell scheme for a non-self-adjoint problem, we obtain an extrapolation solution which is uniformly convergent with order two. We confirm the result by numerical calculations.Supported by the National Natural Science Foundation of China.     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Uniform difference scheme for a singularly perturbed linear 2nd order hyperbolic problem with zeroth order reduced equation

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.     

On singularly perturbed quasilinear systems

In this paper, the objective is to give sufficient conditions for the existence of solution of the nonlinear two-point boundary value problem(1.1). And we employ these results to consider the boundary layer phenomena of the quasilinear weakly coupled singularly perturbed system (DP)_q.     

Asymptotic models for the discrete optimal control of the deformation of an elastic membrane

This paper considers the singularly perturbed static problem of the optimal control of the deformation of an elastic membrane by means of external loads (control without constraints) applied to several small areas distant from each other. The objective functional is equal to the sum of the square of the root-mean-square approximation error and the square of the norm of the external load. Asymptotic models are constructed using the method of matched asymptotic expansions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 131–144, September–October, 2006.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

A second order uniform difference scheme for a singularly perturbed turning point problem

We construct a positive type difference scheme.for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ε, to the solution of the singularly perturbed B.V.P. Numerical examples are provided.     

Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

Construction of an asymptotic solution of the Cauchy problem for a degenerate linear system in the singular case

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of coefficients of derivatives in the case where the limit pencil of matrices is singular. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 271–288, April–June, 2008.     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

Application of Tikhonov’s theorem for automatic stabilization of controlled motion

A decomposition of a control algorithm is proposed in the form of a combination of control laws for the slow and fast variables of a controlled system. This is possible due to the fulfilment of Tikhonov’s theorem hypotheses for singularly perturbed systems.     

Step-like contrast structure for class of nonlinear singularly perturbed optimal control problem

The step-like contrast structure for a class of nonlinear singularly perturbed optimal control problems is considered. The existence of the step-like contrast structure for the singularly perturbed optimal control problem is proved by equivalence, which is based on the necessary conditions. The authors not only give the conditions under which there exists a step-like contrast structure, but also determine where the internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of the step-like contrast structure solution is constructed by the direct scheme method. Finally, an example is presented to show the result.     

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)

In this paper the existence of solutions of the singularly perturbed boundary valueproblems on infinite interval for the second order nonlinear equation containing a smallparameterε>0 :is examined,whereα_i,βare constants,and i=0,1 .Moreover,asymptoticestimates of the solutions for the above problems are given.     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

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Concentration fluctuations and dilution in two-dimensionally periodic heterogeneous porous media

Dilution of solute in two-dimensionally periodic heterogeneous porous media is assessed by numerically simulating advection-dispersion. The concentration fluctuations, created by advective heterogeneity, are destroyed by local dispersion, over a characteristic variance residence time (VRT). For an impulse introduction of solute, initially, plumes become increasingly irregular with time—the coefficient of variation (CV) of concentration grows with time. A priori, the spatial second moment and mean concentrations are insufficient measures of dilution, because concentration fluctuations can be large. At large times (t > VRT) the relative concentration fluctuations weaken—the concentration CV was observed to slowly decrease with time. At the center of mass the general trend of the decreasing CV follows VRT/t (predicted by Kapoor and Gelhar). The VRT is found to be an increasing function of the log hydraulic conductivity microscale. In employing effective dispersion coefficents to model the mean concentration, it needs to be recognized that excursions of concentrations around the mean are singularly determined by local dispersion.     

On the problem of stabilization of motion for a parametric family of nonlinear singularly perturbed systems

For nonlinear uncertain singularly perturbed systems, we construct a control guaranteeing their absolute parametric stability and estimate the set of values of the parameters for which this property of the system is preserved.