scholarly journals On the exponential convergence to a limit of solutions of perturbed linear Volterra equations

2005 ◽  
pp. 1-16
Author(s):  
John A. D. Appleby ◽  
Siobhan Devin ◽  
David W. Reynolds
Keyword(s):  
Exponential Convergence ◽  
Volterra Equations

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

scholarly journals Characterisation of Exponential Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-27
Author(s):  
John A. D. Appleby ◽  
Siobhán Devin ◽  
David W. Reynolds
Keyword(s):  
Volterra Equation ◽  
Exponential Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Volterra Equations ◽  
Necessary And Sufficient

This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure and pth mean sense is obtained.

Modeling the co-evolution of economic institutions and the economy of resource producing regions

2020 ◽  
Vol 18 (4) ◽  
pp. 122-131
Author(s):  
Vadim F. Islamutdinov ◽  
Sergey P. Semenov
Keyword(s):  
Economic Growth ◽  
Resource Availability ◽  
Regional Economy ◽  
Cumulative Effect ◽  
Volterra Equations ◽  
Economic Institutions ◽  
Predator Prey Model ◽  
Resource Endowment ◽  
Disincentive Effect ◽  
Good Institution

The purpose of the study is to develop a model for the co-evolution of the regional economy and economic institutions. The research methods used: abstract-logical for the study of theoretical aspects and the experience of modeling co-evolution; and economic-mathematical for the development of own model of coevolution. The results of the study: approaches to modeling the evolution of economic institutions, as well as the co-evolution of the regional economy and economic institutions are considered, strengths and weaknesses of existing approaches to modeling co-evolution are identified, on the basis of the logistic model and Lotka-Volterra equations, an own co-evolution model has been developed, which includes three entities: regional economy, “good” institution and “bad” institution. Three versions of the model have been developed: the co-evolution of the regional economy and the “good” institution, the co-evolution of the regional economy and the “bad institution,” and a variant of the co-evolution of all three entities simultaneously, in which the “good” and “bad” institutions interact according to the “predator-prey” model, and their the cumulative effect determines the development of the regional economy. Numerical experiments have been carried out in the MathLab, which have shown the capabilities of the model to reflect the results of the co-evolution of the economy of a resource-producing region and economic institutions. In the first variant, a “good” institution promotes economic growth in excess of the level determined by resource availability. In the second variant, the “bad” institution has a disincentive effect on the GRP, as a result of which the GRP falls below the level determined by the resource endowment. In the third variant, the interaction of “good” and “bad” institutions still contributes to economic growth above the level determined by resource availability, but causes cyclical fluctuations in the GRP.

Exponential convergence in entropy and Wasserstein for McKean–Vlasov SDEs

2021 ◽  
Vol 206 ◽  
pp. 112259
Author(s):  
Panpan Ren ◽  
Feng-Yu Wang
Keyword(s):  
Exponential Convergence

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

scholarly journals Distributed Solving Sylvester equations with an explicit exponential Convergence

2020 ◽  
Vol 53 (2) ◽  
pp. 3260-3265
Author(s):  
Songsong Cheng ◽  
Xianlin Zeng ◽  
Yiguang Hong
Keyword(s):  
Exponential Convergence

scholarly journals Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

2015 ◽  
Vol 258 (2) ◽  
pp. 535-554 ◽  
Author(s):  
Boris Baeumer ◽  
Matthias Geissert ◽  
Mihály Kovács
Keyword(s):  
Multiplicative Noise ◽  
Volterra Equations
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