Ordinary differential equations singular perturbation

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equat., 31, 53. 

ASYMPTOTIC EXPANSIONS FOR ORDINARY DIFFERENTIAL EQUATIONS, Wolfgang Wasow. Outstanding text covers asymptotic power series, Jordan s canonical form, turning point problems, singular perturbations, much more. Problems. 3B4pp. 5K x 8X. 65456-7 Pa. 8.95... 

A perturbation analysis about this singular point yields a second order linear ordinary differential equation whose characteristic equation has the roots Ai and A2 where... 

R. E. O Malley Jr. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. 

This system and other problems for singularly perturbed ordinary differential equations will be investigated in Sections II-V. Solutions with boundary and/or interior layers will be considered. Our main goal will be the construction of an approximation to the solution valid outside the boundary (interior) layer as well as within the boundary (interior) layer, that is, so-called uniform approximation in the entire t domain. This approximation will have an asymptotic character. The definition of an asymptotic approximation with respect to a small parameter will be introduced in Section LB. 

It is known that, in the case of singularly perturbed elliptic equations for which (as the parameter s equals zero) the equation does not contain any derivatives with respect to the space variable, the principal term in the singular part of the solution is described by an ordinary differential equation similar to Eq. (1.16a) (see, e.g., [3-6]). Thus, it can be expected that, when solving singularly perturbed elliptic and parabolic equations using classical difference schemes, one faces computational problems similar to the computational problems for the boundary value problem (1.16). 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

In this section, using an example of a boundary value problem for a singularly perturbed ordinary differential equation, we discuss some principles for constructing special finite difference schemes. In Section II.D, these principles will be applied to the construction of special schemes for singularly perturbed equations of the parabolic type. 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

The authors presented the error analysis of A(alpha)-stable parallel multistep hybrid methods (PHMs) for the initial value problems of ordinary differential equations in singular perturbation form (see above). From these results one can see that the convergence results of the present methods are similar to those of linear multistep methods and so no order reduction occurs. 

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