Theory of two-dimensional differential equations and systems of second order (Q1080053)

We consider the construction of exact and approximate solutions of homogeneous and inhomogeneous, linear and nonlinear, partial differential equations of second order with two independent variables t (time) and z (depth), of the general type \[ \Phi \equiv U_{zz}+pU_{tz}+qU_{tt}+\alpha U_ t+\beta U_ z+eU-f=0, \] and also of nonlinear systems of equations of the type \({\vec \Phi}={\vec \Psi}_ z+P{\vec \Psi}_ t+Q{\vec \Psi}-f=0\), in which the square second-order matrices P and Q and the vector f are assumed to be given, while the vector \(\Psi\) has to be determined.