Background for Linear Systems

The Gauss Algorithm reduces systems of linear algebraic equations to a triangular or standard form. A first guess for an analogous method for linear systems of differential equations is to apply the Gauss algorithm to such systems with their derivatives and dependent variables regarded as unknowns. An example of such a Gauss-reduced form is the system $u_x = y u$, $u_y = u$, for $u(x,y)$ where $u_x$ denotes $\partial u / \partial x$ and $u_y$ denotes $\partial u / \partial y$. Differentiation of the first equation with respect to $y$ and the second with respect to $x$ yields the integrability condition $u_{xy}-u_{yx} = y u_y + u - u_x$ $=0$. When reduced subject to the system this condition becomes $u=0$. Thus generalizing the Gauss Algorithm to systems of linear PDEs, requires the consideration of their differential consequences. Such a generalization was given by Riquier [69] early this century. In rough terms, for linear systems his algorithm repeats the steps of Gauss reducing the system, calculating and reducing its integrability conditions, adjoining nontrivial integrability conditions to the system, until there are no nontrivial integrability conditions. A number of such algorithms have been implemented (see [77,89,9] and the extensive bibliography of our paper [65]). Our standard form algorithm [57] uses similar principles to produce a coordinate dependent standard form of linear system of differential equations. The reduced form produced by ours and Riquier's algorithms does not solve such systems. However it does provide a count of the number of parameters in their local analytic solutions [78]. It can substantially simplify them, and can be used to test whether a differential expression is a consequence of the system. It also yields a form of the system to which local existence-uniqueness theorems can be applied. These theorems generalize the Cauchy-Kovaleskaya Theorem [87,88,41].

Such algorithms have proved their worth in simplifying complicated overdetermined systems of linear PDEs; especially those arising in the Lie symmetry analysis of physical PDEs. They are becoming a standard part of computer algebra packages for simplifying and solving linear PDEs (see the references in [65] and especially those in the review article [34]).