Uniform difference method for parameterized singularly perturbed delay differential equations

This paper deals with the singularly perturbed initial value problem for quasilinear first-order delay differential equation depending on a parameter. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform meshes on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments illustrate in practice the result of convergence proved theoretically.     

On discontinuous Galerkin finite element method for singularly perturbed delay differential equations

A nonsymmetric discontinuous Galerkin FEM with interior penalties has been applied to one-dimensional singularly perturbed problem with a constant negative shift. Using higher order polynomials on Shishkin-type layer-adapted meshes, a robust convergence has been proved in the corresponding energy norm. Numerical experiments support theoretical findings.     

A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations

This study deals with the singularly perturbed initial value problem for a quasilinear first-order delay differential equation. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.     

Stability conditions for singularly perturbed delay differential equations

Singular perturbation problems containing a small positive parameter ε occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. A uniformly valid, reliable interpretable approximation of such problems is required. This paper provides sufficient conditions to ensure the exponential stability of the analytical and numerical solutions of the singularly perturbed delay differential equations with a bounded time-lag for suf.ciently small ε > 0. The Halanay inequality is used to prove the main results of the paper. A numerical example is provided to illustrate the methodology and clarify the need for a stiff solver for numerical solutions of these problems.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

Exponential stability of singularly perturbed impulsive delay differential equations

In this paper, the exponential stability of singularly perturbed impulsive delay differential equations (SPIDDEs) is concerned. We first establish a delay differential inequality, which is useful to deal with the stability of SPIDDEs, and then by the obtained inequality, a sufficient condition is provided to ensure that any solution of SPIDDEs is exponentially stable for sufficiently small ε>0. A numerical example and the simulation result show the effectiveness of our theoretical result.     

A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior

In this paper, we present a finite difference method for singularly perturbed linear second order differential-difference equations of convection–diffusion type with a small shift, i.e., where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. Here, the study focuses on the effect of shift on the boundary layer behavior or oscillatory behavior of the solution via finite difference approach. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameter on the boundary layer behavior and oscillatory behavior of the solution of the problem.     

Asymptotic expansion for the solution of singularly perturbed delay differential equations

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley-Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

Dynamics of a singularly perturbed system of two differential equations with delay

Theoretical and Mathematical Physics - We study a two-dimensional singularly perturbed system with delay, which is a simplification of models used in laser physics. We analyze several cases of a...     

Effective method for solving singularly perturbed systems of nonlinear differential equations

A boundary-value problem for a class of singularly perturbed systems of nonlinear ordinary differential equations is considered. An analytic-numerical method for solving this problem is proposed. The method combines the operational Newton method with the method of continuation by a parameter and construction of the initial approximation in an explicit form. The method is applied to the particular system arising when simulating the interaction of physical fields in a semiconductor diode. The Frechét derivative and the Green function for the corresponding differential equation are found analytically in this case. Numerical simulations demonstrate a high efficiency and superexponential rate of convergence of the method proposed. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.     

Uniformly convergent novel finite difference methods for singularly perturbed reaction–diffusion equations

In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.     

On the internal layer for a singularly perturbed system of second-order delay differential equations

We consider a nonlinear singularly perturbed boundary value problem with delay. By using the method of boundary functions and the theory of contrast structures, we prove the existence of a smooth solution with an internal transition layer and construct its uniform asymptotic expansion in a small parameter.     

An hp finite element method for singularly perturbed systems of reactiondiffusion equations

We consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We present results on a high order hp finite element scheme which includes elements of size O (εp) and O (μp) near the boundary, where ε, μ are the singular perturbation parameters and p is the degree of the approximating polynomials. Under the assumption of analytic input data, the method yields exponential rates of convergence as p → ∞, independently of ε and μ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fitted Galerkin spectral method to solve delay partial differential equations

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction‐diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential estimates for singularly perturbed linear fuctional differential equations

Exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts. The exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations. Differences between differential-difference equations and equations with distributed delays are emphasized.