Search results for " Differential equations"
showing 10 items of 146 documents
Une quête d'exactitude : machines, algèbre et géométrie pour la construction traditionnelle des équations différentielles
2015
In La Géométrie, Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one—geometry became a mean of visualization, no longer of construction. Descartes’s…
A fully adaptive wavelet algorithm for parabolic partial differential equations
2001
We present a fully adaptive numerical scheme for the resolution of parabolic equations. It is based on wavelet approximations of functions and operators. Following the numerical analysis in the case of linear equations, we derive a numerical algorithm essentially based on convolution operators that can be efficiently implemented as soon as a natural condition on the space of approximation is satisfied. The algorithm is extended to semi-linear equations with time dependent (adapted) spaces of approximation. Numerical experiments deal with the heat equation as well as the Burgers equation.
On a Retarded Nonlocal Ordinary Differential System with Discrete Diffusion Modeling Life Tables
2021
In this paper, we consider a system of ordinary differential equations with non-local discrete diffusion and finite delay and with either a finite or an infinite number of equations. We prove several properties of solutions such as comparison, stability and symmetry. We create a numerical simulation showing that this model can be appropriate to model dynamical life tables in actuarial or demographic sciences. In this way, some indicators of goodness and smoothness are improved when comparing with classical techniques.
Simplifying differential equations for multi-scale Feynman integrals beyond multiple polylogarithms
2017
In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to $\varepsilon$-form.
Simple differential equations for Feynman integrals associated to elliptic curves
2019
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is therefore of current interest, if these methods extend beyond the case of multiple polylogarithms. In this talk I discuss Feynman integrals, which are associated to elliptic curves and their differential equations. I show for non-trivial examples how the system of differential equations can be brought into an $\varepsilon$-form. Single-scale and multi-scale cases are discussed.
A new result on impulsive differential equations involving non-absolutely convergent integrals
2009
AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations.
Changing the General Factor of Personality and the c-fos Gene Expression with Methylphenidate and Self-Regulation Therapy
2012
[EN] A deepening in the biological nature of the general factor of personality (GFP) is suggested: the activation level of the stress system is here represented by the gene expression of c-fos. The results of a single case experimental design are reported. A model of four coupled differential equations that explains the human personality dynamics as a consequence of a single stimulant drug intake has been fitted to psychological and biological experimental data. The stimulant-drug conditioning and its adaptation to the considered mathematical model is also studied for both kinds of measures. The dynamics of the cfos expression presents a similar pattern to the dynamics of the psychological …
Spatio-temporal behaviour of five picophytoplankton populations in Tyrrhe- nian Sea: Model and data
2014
Recent works presented detailed analyses of spatio-temporal dynamics in marine ecosystems, reproducing real vertical distributions of phytoplankton biomass. These study however do not take into account the changes in environmental variables. On the contrary, seasonal variations can influence considerably the primary production, i.e. phytoplankton biomass, in marine ecosystems, determining significative consequences in the whole food chain, in particular fish species, whose growth is mainly explained by seasonal changes in the chlorophyll concentration, a marker of phytoplankton species. Here we present a one-dimensional reaction-diffusion-taxis model to describe the spatio-temporal dynamics…
Reaction-diffusion-taxis model for spatio-temporal dynamics of five picophytoplankton populations
2014
Recently new models were devised to study spatio-temporal dynamics of phytoplankton populations in view of obtaining more precise predictions of the vertical biomass distributions in marine ecosystems. These studies can be crucial from the point of view of shery. Indeed the abundance of fi sh species is strictly connected with primary production, i.e. phytoplankton biomass, responsible for chlorophyll concentration. In this work a one-dimensional deterministic reaction-di ffusion-taxis model is used to reproduce the spatio-temporal dynamics, along a water column, of five picophytoplankton populations sampled in a real ecosystem. In our analysis, to better reproduce the spatio-temporal behav…
Modelling of Pe C alloys solidification using the artificial heat source method
1997
Abstract In the paper the numerical solutions concerning the cast iron and also the carbon steel solidification are presented. In order to take into account the non-linearities appearing in differential equations describing the boundary-initial problem considered — a certain algorithm called the artificial heat source method has been used. The examples illustrating the possibilities of proposed method applications have been solved by means of the boundary element method, but the others numerical methods can be also utilized.