Geometrical approaches to differential equations
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The geometric approach to differential equations: direction fields
Volume 24 , Issue 6. The full text of this article hosted at iucr.
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In his Erlanger Programm Klein proposed that Euclidean and non-Euclidean geometry be regarded as special cases of projective geometry. For example, in Euclidean plane geometry the space is the familiar plane, and the transformations are rotations, reflections, translations, and their composites, none of which change either length or angle, the basic properties of figures in Euclidean geometry. Different geometries would have different spaces and different groups, and the figures would have different basic properties.
Klein produced an account that unified a large class of geometries—roughly speaking, all those that were homogeneous in the sense that every piece of the space looked like every other piece of the space. This excluded, for example, geometries on surfaces of variable curvature , but it produced an attractive package for the rest and gratified the intuition of those who felt that somehow projective geometry was basic. Mathematicians could now ask why they had believed Euclidean geometry to be the only one when, in fact, many different geometries existed.
The first to take up this question successfully was the German mathematician Moritz Pasch , who argued in that the mistake had been to rely too heavily on physical intuition. In his view an argument in mathematics should depend for its validity not on the physical interpretation of the terms involved but upon purely formal criteria. Indeed, the principle of duality did violence to the sense of geometry as a formalization of what one believed about physical points and lines; one did not believe that these terms were interchangeable.
In wondering why it was that mathematics—and in particular geometry—produced correct results, he came to feel increasingly that it was not because of the lucidity of its definitions. Rather, mathematics worked because its elementary terms were meaningless. What kept it heading in the right direction was its rules of inference.
Proofs were valid because they were constructed through the application of the rules of inference , according to which new assertions could be declared to be true simply because they could be derived, by means of these rules, from the axioms or previously proven theorems.
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The theorems and axioms were viewed as formal statements that expressed the relationships between these terms. The rules governing the use of mathematical terms were arbitrary, Hilbert argued, and each mathematician could choose them at will, provided only that the choices made were self-consistent. A mathematician produced abstract systems unconstrained by the needs of science , and if scientists found an abstract system that fit one of their concerns, they could apply the system secure in the knowledge that it was logically consistent.
For the first time there was a way of discussing geometry that lay beyond even the very general terms proposed by Riemann. Not all of these geometries have continued to be of interest, but the general moral that Hilbert first drew for geometry he was shortly to draw for the whole of mathematics. By the late 19th century the debates about the foundations of geometry had become the focus for a running debate about the nature of the branches of mathematics.
The algebraic theory of numbers and the transformed theory of equations had focused attention on abstract structures in mathematics. Questions that had been raised about numbers since Babylonian times turned out to be best cast theoretically in terms of entirely modern creations whose independence from the physical world was beyond dispute.
Finally, geometry, far from being a kind of abstract physics, was now seen as dealing with meaningless terms obeying arbitrary systems of rules.
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Although there had been no conscious plan leading in that direction, the stage was set for a consideration of questions about the fundamental nature of mathematics. Similar currents were at work in the study of logic , which had also enjoyed a revival during the 19th century. The work of the English mathematician George Boole and the American Charles Sanders Peirce had contributed to the development of a symbolism adequate to explore all elementary logical deductions. In Germany the logician Gottlob Frege had directed keen attention to such fundamental questions as what it means to define something and what sorts of purported definitions actually do define.
Article Media. Info Print Print. Indeed, the project has been thought as a transversal bridge across the mathematical communities, the common factor being the problems to study, that are the equations governing multiphase flows and their wide range of applications e.
In the recent years, new and promising results have been obtained separately in the field of optimal transportation, in the design and the study of advanced numerical methods, and in the development of adaptive strategies for solving numerically some PDEs based on a posteriori error analysis.
We aim to take advantage of all these new developments, and to apply this new material to the study, from a theoretical and numerical point of view, of nonlinear degenerated parabolic systems and equations. We claim that the gradient flow approach is relevant for studying multiphase flows in porous media. Indeed, the natural and physically consistent way to write the equations governing multiphase flows in porous media let some degenerate mobilities appear leading to the study of degenerate parabolic equations and making the gradient flow writing completely natural.
As a major advantage of this writing, only physically meaningful quantities appear, in contrary to the classical mathematical approaches. Keeping the physical relevance as a constant matter, we propose to design what we call nonlinear methods for approximating the solutions of such possibly degenerate parabolic equations and systems. The methods we have in mind are based on schemes that are widely used in oil-engineering despite very partially studied: they are robust and allow enrichments of the physics.
Taking advantage of the numerous new tools developed in the recent years we will design robust advanced numerical methods whose convergence will be proven.