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The solution of Cauchy's Problem for linear partial differential equations, with constant coefficients, by means of integrals involving complex variables

Published online by Cambridge University Press:  20 January 2009

C. A. Stewart
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The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 2 , March 1928 , pp. 94 - 103
Copyright
Copyright © Edinburgh Mathematical Society 1928

References

page 94 note 1 Darboux, : Théorie générate des surfaces, II, pp. 75et seq.Google Scholar

page 94 note 2 The Riemannian method of integration has been extended by the writer to equations of higher order: Proc. Land. Math. Soc., 26 (1927). pp. 81–94.Google Scholar

page 95 note 1 If there is no term in Vxx in the original equation, take the second order terras as

page 97 note 1 Any linear boundary can be changed to x = h, by a suitable linear transformation of the independent variables.Google Scholar

page 102 note 1 CfZeilon, : Arkev für Matematik, Astronomi och Fysik, 6 (1910).Google Scholar