Lowering of order and general solutions of some classes of partial differential equations
type: Item
instance of: scholarly article
title: Lowering of order and general solutions of some classes of partial differential equations
zbMATH Open document ID: 0934.35009
DOI: 10.1016/S0034-4877(98)80018-9
published in: Reports on Mathematical Physics
publication date: 1999-08-16T00:00:00Z
review text: The authors consider the following partial differential equation \[ L(D[u])+F(D[u])=0, \] where \(u=u(x)\), \(x=(x_0,x_1,\dots,x_k)\), \(L\) is a first-order differential operator of the form \(L\equiv a^i(x,u)\partial_{x_i}\), \(i=0,1,\dots,k\), \(a^i(x,u)\) and \(F\) are arbitrary smooth functions, and \(D[u]\) is an \(n\)-order differential expression. Using a change of variables they transform the initial equation in the new form \[ \partial_\tau(\widetilde{D}[z])+F(\widetilde{D}[z])=0, \] where \(\widetilde{D}[z]\) is \(D[u]\) expressed in the new variables \(\tau=f^0(x,u)\) and \(z(\tau, \omega^\alpha)\), \(\omega^\alpha=f^\alpha(x,u)\). The change is determinated from the conditions \(L(f_0)=1\), \(L(f^\alpha)=0\). Simple examples are considered. Generalizations to higher order partial differential equations and systems are presented.
reviewed by: S. B. Medvedev
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 1293891
zbMATH Keywords: exact solution
zbMATH Keywords: implicit form of solution
zbMATH Keywords: change of variables
zbMATH Keywords: closed form of solution
author: V. M. Bojko
author: W. I. Fushchich
MaRDI profile type: Publication
cites work: General Solution of the Universal Equation in n-Dimensional Space
full work available at URL: s0034-4877(98)80018-9
OpenAlex ID: W2004472171
description: scientific article; zbMATH DE number 1293891