Search results for "Equations"

Systematic derivation of partial differential equations for second order boundary value problems

2022

Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre-given list. This expands significantly end users possibilities to employ boundary value problems in modeling. To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems …

Non-standard Problems in an Ordinary Differential Equations Course

2018

International audience; We report first results from a teaching intervention in an ordinary differential equations (ODEs) course for engineering students. Our aim is to challenge traditional approaches to teaching of Existence and Uniqueness Theorems (EUTs) through the design of problems that students cannot solve by applying well-rehearsed techniques or familiar methods. We analyse how the use of nonstandard problems contributes to the development of students' conceptual understanding of EUTs and ODEs.

Effective charge from lattice QCD

2020

Using lattice configurations for quantum chromodynamics (QCD) generated with three domain-wall fermions at a physical pion mass, we obtain a parameter-free prediction of QCD's renormalisation-group-invariant process-independent effective charge, $\hat\alpha(k^2)$. Owing to the dynamical breaking of scale invariance, evident in the emergence of a gluon mass-scale, this coupling saturates at infrared momenta: $\hat\alpha(0)/\pi=0.97(4)$. Amongst other things: $\hat\alpha(k^2)$ is almost identical to the process-dependent (PD) effective charge defined via the Bjorken sum rule; and also that PD charge which, employed in the one-loop evolution equations, delivers agreement between pion parton di…

Pupil's ecological reasoning with help of modeling tool

2006

Ecological concepts, and in particular population dynamics, has been found to be among the most difficult topics in biology. Some researchers pointed to students' relatively weak mathematical background as the main source of learning difficulties. This paper reports on an investigation of pupil's (n=73) reasoning about the ecological phenomena by using an iconic modelling tool, WorldMaker. The simulations eliminated the need for understanding of mathematical equations, and made the ecological concepts much more accessible to some children. However, many of the pupils reasoned from an anthropocentric perspective that obstructed their ability to predict ecological phenomena which requires sys…

The measurement of economic, social and environmental performance of countries: a novel approach

2010

This paper presents a new analytical framework for assessing spatial disparities among countries. It takes for granted that the analysis of a country's performance cannot be limited solely to either economic or social factors. The aim of the paper is to combine relevant economic and 'non-economic' (mainly social) aspects of a country's performance in an integrated logical framework. Based on this idea, a structural simultaneous equation model will be presented and estimated in order to explore the direction of the causal relationship between economic and non-economic aspects of a country's performance. Furthermore, an exploration of the trajectory that each country has registered over time …

ON THE EXISTENCE OF LIMIT CYCLES IN OPINION FORMATION PROCESSES UNDER TIME PERIODIC INFLUENCE OF PERSUADERS

2008

This paper concerns a model of opinion formation in a population of interacting individuals under the influence of external leaders or persuaders, which act in a time periodic fashion. The model is formulated within a general framework inspired to a discrete generalized kinetic approach, which has been developed in Ref. 6. It is expressed by a system of non-autonomous nonlinear ordinary differential equations. The dynamics of such a system is investigated and the existence of a globally asymptotically stable periodic solution is analytically proved in three example cases, each one corresponding to a different quantitative choice of the actions of the persuaders. Equivalently, in three part…

Bounds on plastic strains for elastic plastic structures in plastic shakedown conditions

2007

The problem related to the computation of bounds on plastic deformations for structures in plastic shakedown condition (alternating plasticity) is studied. In particular, reference is made to structures discretized by finite elements constituted by elastic perfectly plastic material and subjected to a special combination of fixed and cyclic loads. The load history is known during the steady-state phase, but it is unknown during the previous transient phase; so, as a consequence, it is not possible to know the complete elastic plastic structural response. The interest is therefore focused on the computation of bounds on suitable measures of the plastic strain which characterizes just the fir…

Asymptotic Hölder regularity for the ellipsoid process

2020

We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

Exact solutions of the Zakharov equations

2009

Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth plane domains

1984

The authors examine a finite element method for the numerical approximation of the solution to a div-rot system with mixed boundary conditions in bounded plane domains with piecewise smooth boundary. The solvability of the system both in an infinite and finite dimensional formulation is proved. Piecewise linear element fields with pointwise boundary conditions are used and their approximation properties are studied. Numerical examples indicating the accuracy of the method are given. peerReviewed