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D’Alembert and the notion of solution of differential equations with partial derivatives. (D’Alembert et la notion de solution des équations différentielles aux dérivées partielles.) (French. English summary) Zbl 1225.01026

Serge Demidov tells the story of a grand debate with characters as strong and powerful as d’Alembert and Euler as well as Lagrange and Laplace. It is the story of different notions of solutions for partial differential equations with initial and boundary conditions, namely classical (considered by d’Alembert ) and generalized or weak (entertained by Euler). The issue at stake being the requirement by d’Alembert that certain initial condition function must be twice differentiable and hence continuous. This severely limited the applicability of such differential equations to physical situations that did not satisfy this continuity condition and provoked Euler to counter d’Alembert’s solution with a solution of his own that worked for any such initial function, albeit with incorrect reasoning, and thus started the debate to which Lagrange and Laplace later contributed.
Demidov raises an interesting point in this article, namely he wonders how it is that neither Euler nor d’Alembert did understand that the subject matter of their debate was a chimera? How is it that they could not see that it was a matter of different notions of solution, namely classical for d’Alembert and weak for Euler which was at the heart of their debate? Demidov thinks that Euler understood d’Alembert’s objection to his views, however he thought that his intuitive understanding of physics and his common sense was nevertheless guiding him towards the correct solution.
It appears that later on, in the 1780’s, d’Alembert relaxed the continuity condition and rigorously described the solution where discontinuous functions were involved. He did this by eliminating all methods and notions that he thought were not sufficiently justified. This, Demidov concludes correctly, proves d’Alembert as the precursor of the rigorist mathematicians of the century to come: Bolzano, Cauchy, Abel and Weierstrass.

MSC:

01A50 History of mathematics in the 18th century
35-03 History of partial differential equations
35A09 Classical solutions to PDEs
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