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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Partial Differential Equations III - The Cauchy Problem. Qualitative Theory of Partial Differential Equations (Paperback, Softcover reprint of the original 1st ed. 1991)
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Partial Differential Equations III - The Cauchy Problem. Qualitative Theory of Partial Differential Equations (Paperback, Softcover reprint of the original 1st ed. 1991)
Series: Encyclopaedia of Mathematical Sciences, 32
Expected to ship within 10 - 15 working days
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,h In the XIX century, mathematical physics continued to be the
main source of new partial differential equations and ofproblems
involving them. The study ofLaplace's equation and ofthe wave
equation had assumed a more systematic nature. In the beginning of
the century, Fourier added the heat equation to the aforementioned
two. Marvellous progress in obtaining precise solution repre-
sentation formulas is connected with Poisson, who obtained formulas
for the solution of the Dirichlet problem in a disc, for the
solution of the Cauchy problems for the heat equation, and for the
three-dimensional wave equation. The physical setting ofthe problem
led to the gradual replacement ofthe search for a general solution
by the study of boundary value problems, which arose naturallyfrom
the physics ofthe problem. Among these, theCauchy problem was of
utmost importance. Only in the context of first order equations,
the original quest for general integralsjustified itself. Here
again the first steps are connected with the names of D'Alembert
and Euler; the theory was being intensively 1h developed all
through the XIX century, and was brought to an astounding
completeness through the efforts ofHamilton, Jacobi, Frobenius, and
E. Cartan. In terms of concrete equations, the studies in general
rarely concerned equa- tions of higher than second order, and at
most in three variables. Classification 'h ofsecond orderequations
was undertaken in the second halfofthe XIX century (by Du
Bois-Raymond). An increase in the number of variables was not sanc-
tioned by applications, and led to the little understood
ultra-hyperbolic case.
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