5,869 research outputs found
An Initial Value Technique using Exponentially Fitted Non Standard Finite Difference Method for Singularly Perturbed Differential-Difference Equations
In this paper, an exponentially fitted non standard finite difference method is proposed to solve singularly perturbed differential-difference equations with boundary layer on left and right sides of the interval. In this method, the original second order differential difference equation is replaced by an asymptotically equivalent singularly perturbed problem and in turn the problem is replaced by an asymptotically equivalent first order problem. This initial value problem is solve by using exponential fitting with non standard finite differences. To validate the applicability of the method, several model examples have been solved by taking different values for the delay parameter δ , advanced parameter η and the perturbation parameter ε . Comparison of the results is shown to justify the method. The effect of the small shifts on the boundary layer solutions has been investigated and presented in figures. The convergence of the scheme has also been investigated
An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays
AbstractThis paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory
Fractal asymptotics
Recent advances in the periodic orbit theory of stochastically perturbed
systems have permitted a calculation of the escape rate of a noisy chaotic map
to order 64 in the noise strength. Comparison with the usual asymptotic
expansions obtained from integrals and with a previous calculation of the
electrostatic potential of exactly selfsimilar fractal charge distributions,
suggests a remarkably accurate form for the late terms in the expansion, with
parameters determined independently from the fractal repeller and the critical
point of the map. Two methods give a precise meaning to the asymptotic
expansion, Borel summation and Shafer approximants. These can then be compared
with the escape rate as computed by alternative methods.Comment: 15 pages, 5 postscript figures incorporated into the text; v2:
Quadratic Pade (Shafer) method added, also a few reference
Financial Applications of Random Matrix Theory: a short review
We discuss the applications of Random Matrix Theory in the context of
financial markets and econometric models, a topic about which a considerable
number of papers have been devoted to in the last decade. This mini-review is
intended to guide the reader through various theoretical results (the
Marcenko-Pastur spectrum and its various generalisations, random SVD, free
matrices, largest eigenvalue statistics, etc.) as well as some concrete
applications to portfolio optimisation and out-of-sample risk estimation.Comment: To appear in the "Handbook on Random Matrix Theory", Oxford
University Pres
Convergence of invariant densities in the small-noise limit
This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction
Fitted non-polynomial spline method for singularly perturbed differential difference equations with integral boundary condition
The aim of this paper is to present fitted non-polynomial spline method for singularly perturbed differential-difference equations with integral boundary condition. The stability and uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature
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