Atendendo a algumas respostas apreciativas e inúmeras ignorativas, aqui está......

02 novembro 2012

Darboux's method

Darboux's method, as explained by Cartan, is a systematic way to find whether a partial differential equation can be solved using ordinary differential equations techniques. If so we may be able to solve a given equation point-by-point, or curve-by-curve, without the need to find the complete solution. Ordinary equations are easier to solve and, most important, the answers are a lot easier to understand, so Darboux's method should be widely used. That does not seem to be the case. Why? I have a number of explanations, possibly all equally wrong.

1 - The number of situations for which Darboux's method gives useful answers is too limited. As I understand, that happens when the exterior differential system admits so-called Monge characteristics, which, for many equations of practical importance, may not be the case.

2 - Darboux's method is used more often than my growing ignorance has been able to discern.

3 - Physicists, the people who most often solve partial equations, need complete solutions. In this case a complete numerical solution is less complicated and more useful than a piecewise solution along given curves.

4 - Numerical methods using computers are convenient for finding complete solutions. However they are not likely ever to be useful for finding real-time answers in engineering applications such as feedback control. This is of course the reason I am interested in Darboux's method.

5 - Cartan is difficult to read. There exist more recent texts, including ones by Burke; Arnold; Ivey & Landsberg; Stormark; Olver; and Bryant, Chern, Gardner, Goldschmidt & Griffiths. Except for Burke and Arnold, who wrote for physicists and are not complete explanations of Cartan, they are not that much easier going than the originals.

6 - Cartan wrote in French, as did Darboux. That should not be a serious problem, because everybody should be able to understand mathematics written in French. Or so thought our teachers when they began to translate all of science into English after Europe devolved into barbarity in 1933. They translated German and Russian, but of course translating French was not urgent - the literature in Italian and the other European languages was smaller. By now the subset of researchers who would consult a French original is a proper subset of the French-speaking scientists, possibly not even a very large one. So Cartan is still somewhat shrouded in mystery.

7 - Darboux's method requires working with complex characteristics, even for real differential equations. Of course imaginary and complex numbers are the delight of every self-respecting electrical engineer, may you have the same joy. The less fortunate might find them challenging.

I would not be very keen on alternative 1 being the main explanation. The others appear in the order of how amused I am by them.
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