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An expansion method for parabolic partial differential equations

1953; US government; Volume: 51; Issue: 3 Linguagem: Inglês

10.6028/jres.051.016

2376-5305

J.W. Green,

Differential Equations and Boundary Problems

The aim of t hif! paper is t o adapt to certain parabolic partial differen tial equations an expanRion m ethod of solution d eveloped by S. Faedo 3 for hyperbolic equat ions.In order to make possible a moderately co mpac t presentation , the equat ion s t rea ted are no t t he most general to which t he m ethod is applicable, but are t he simplest nontl'ivial relatives of t he heat equation .Similarly, t he boundary values and initial conditions are not the most general, but are assu med to be in a canonical form to which others, if su fficiently smooth, can be red uced .The method of solution not only shows the existence of a solution , but d escribes a d efinite procedure for appro ximatin g it.Some remarks are made on t he possibility of estimatin g t he error.1 Tonelli, L'estremo assoluto degli in tegrali doppi, Annali della Scuola Normale di Pisa (1933).Suppose that for some positive Xo, ~K(xo) > ~K(O ) + (a/2)xo.Then there exists Xl such that ~K(XI) =~K(O) + (a/2)xl, but ~K(x) > ~K (O ) + (a/2)x