Singular Perturbations Research Papers - Academia.edu (original) (raw)

A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given.... more

A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg — de Vries, nonlinear Schrodinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamil-tonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative eft'ects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered. CONTENTS

HDR thesis (French habilitation to lead researches) in Applied Mathematics

In this paper, we use a numerical method to solve boundary-value problems for a singularly-perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift.... more

In this paper, we use a numerical method to solve boundary-value problems for a singularly-perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Similar boundary-value problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The stability and convergence analysis of the method is given. The effect of a small shift on the boundary-layer solution is shown via numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method.

This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the... more

This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the Crank-Nicolson scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of... more

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.

We study the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems are associated with expected first-exit time problems of the membrane potential in models... more

We study the boundary-value problems for singularly perturbed differential-difference equations with small shifts. Similar boundary-value problems are associated with expected first-exit time problems of the membrane potential in models for activity of neurons (SIAM J. Appl. Math. 1994; 54: 249–283; 1982; 42: 502–531; 1985; 45: 687–734) and in variational problems in control theory. In this paper, we present a numerical method to solve boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e. which contains both type of terms having negative shifts as well as positive shifts, and consider the case in which the solution of the problem exhibits rapid oscillations. The stability and convergence analysis of the method is given. The effect of small shift on the oscillatory solution is shown by considering the numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method. Copyright © 2004 John Wiley & Sons, Ltd.

Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing... more

Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are ε-uniformly convergent.

The goal of the tutorial is to give an overview of the newest unified design methodology of continuous-time or discrete-time nonlinear control systems which guarantees desired transient performances in the presence of plant parameter... more

The goal of the tutorial is to give an overview of the newest unified design methodology of continuous-time or discrete-time nonlinear control systems which guarantees desired transient performances in the presence of plant parameter variations and unknown external disturbances. The tutorial presents the up-to-date coverage of fundamental issues and recent research developments in design of nonlinear control systems with the highest derivative in feedback. The discussed design methodology allows us to provide effective control of nonlinear systems on the assumption of uncertainty. The approach is based on an application of a dynamical control law with the highest derivative of the output signal in the feedback loop. A distinctive feature of the control systems thus designed is that two-time-scale motions are forced in the closed-loop system. Stability conditions imposed on the fast and slow modes, and a sufficiently large mode separation rate, can ensure that the full-order closed-loop system achieves desired properties: the output transient performances are as desired, and they are insensitive to parameter variations and external disturbances. A general design methodology for control systems with the highest derivative in feedback for continuous-time systems, as well as corresponding discrete-time counterpart, will be presented during this tutorial. The method of singular perturbation is used to analyze the closed-loop system properties throughout.

In this paper, we study boundary value problems for a class of linear singularly perturbed discrete systems. We give conditions ensuring the existence and uniqueness of the solution and a convergent iterative algorithm to compute uniform... more

In this paper, we study boundary value problems for a class of linear singularly perturbed discrete systems. We give conditions ensuring the existence and uniqueness of the solution and a convergent iterative algorithm to compute uniform asymptotic solutions. We use the natural perturbation method, the stiffness of the system is removed and the boundary value problems are switched to plain initial subsystems of reduced order. This method improves the singular perturbation method known for this kind of systems.

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of... more

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.

Fitted Numerical Methods for Delay Differential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape. This thesis deals with... more

Fitted Numerical Methods for Delay Differential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of fitted numerical methods for some delay differential models that arise in biology. Very often such differential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Inefficiencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions. This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay differential equations. Though the developed approaches are very simplistic in nature,...

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. Motivated by their work, we formally derive this equation from two-dimensional potential ¯ow... more

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. Motivated by their work, we formally derive this equation from two-dimensional potential ¯ow equations governing the small amplitude long capillary-gravity waves on the surface of shallow water for Bond number very close to but less than 1/3. On the basis of far-®eld analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-®eld. We review various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (®fth-order) KdV equation. Using these methods, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-®eld.

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and... more

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class

This present research paper describes the application of Differential Quadrature Method (DQM) for getting the computational solution of singularly perturbed two point boundary value problems with varied condition in this method the... more

This present research paper describes the application of Differential Quadrature Method (DQM) for getting the computational solution of singularly perturbed two point boundary value problems with varied condition in this method the concept based on the approximation of the derivatives of the unknown functions involved in the differential equations at the grid point of the solution domain. It is a significant discretization technique in solving initial and /or boundary value problems precisely using a considerably small number of mesh points. To test the applicability of the method we have solved several related model problems and presented the computational results. The computed results have been compared with the exact/approximate solution to exhibit the accuracy and efficiency of the developed technique.

An Ak slow–fast system is a particular type of singularly perturbed ODE. The corresponding slow manifold is defined by the critical points of a universal unfolding of an Ak singularity. In this note we propose a formal normal form of... more

An Ak
slow–fast system is a particular type of singularly perturbed ODE. The corresponding slow manifold is defined by the critical points of a universal unfolding of an Ak
singularity. In this note we propose a formal normal form of Ak
slow–fast systems.

Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the... more

Process systems with material and energy recycle are well-known to exhibit complex dynamics and to present significant control challenges, due to the feedback interactions induced by the recycle streams. In this paper, we address the dynamic analysis and control of such ...