In mathematics, a well-posed problem is one for which the following properties hold:
The problem has a solution
The solution is unique
The solution's behavior changes continuously with the initial conditions
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems.Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data.Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from numerical instability when
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In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.The function is often thought of as
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source r
A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathem
In this paper we consider the Holm-Staley b-family of equations in the Sobolev spaces H-s (R) for s > 3/2. Using a geometric approach we show that, for any value of the parameter b, the corresponding solution map, u(0) bar right arrow u(T), is nowhere locally uniformly continuous.
Ce cours présente une introduction aux méthodes d'approximation utilisées pour la simulation numérique en mécanique des fluides.Les concepts fondamentaux sont présentés dans le cadre de la méthode des différences finies puis étendus à celles des éléments finis et spectraux.
The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions.
Le cours donne une introduction à la théorie des EDO, y compris existence de solutions locales/globales, comportement asymptotique, étude de la stabilité de points stationnaires et applications, en particulier aux systèmes dynamiques et en biologie.