49,653 research outputs found
On some aspects of the geometry of differential equations in physics
In this review paper, we consider three kinds of systems of differential
equations, which are relevant in physics, control theory and other applications
in engineering and applied mathematics; namely: Hamilton equations, singular
differential equations, and partial differential equations in field theories.
The geometric structures underlying these systems are presented and commented.
The main results concerning these structures are stated and discussed, as well
as their influence on the study of the differential equations with which they
are related. Furthermore, research to be developed in these areas is also
commented.Comment: 21 page
Aspects of p-adic non-linear functional analysis
The article provides an introduction to infinite-dimensional differential
calculus over topological fields and surveys some of its applications, notably
in the areas of infinite-dimensional Lie groups and dynamical systems.Comment: 19 pages; LaTe
Strong convergence rates for backward EulerâMaruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients
In this work, we generalize the current theory of strong convergence rates for the backward EulerâMaruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and superlinear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models
Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations
In this work we study the problem of step size selection for numerical
schemes, which guarantees that the numerical solution presents the same
qualitative behavior as the original system of ordinary differential equations,
by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain
feedback stabilization methods are exploited and numerous illustrating
applications are presented for systems with a globally asymptotically stable
equilibrium point. The obtained results can be used for the control of the
global discretization error as well.Comment: 33 pages, 9 figures. Submitted for possible publication to BIT
Numerical Mathematic
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