Lowering of order and general solutions of some classes of partial differential equations
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.
scientific article; zbMATH DE number 1293891