The integration of systems of linear PDEs using conservation laws of syzygies

A new integration technique is presented for systems of linear partial differential equations (PDEs) for which syzygies can be formulated that obey conservation laws. These syzygies come for free as a by-product of the differential Gröbner Basis computation. Compared with the more obvious way of integrating a single equation and substituting the result in other equations the new technique integrates more than one equation at once and therefore introduces temporarily fewer new functions of integration that in addition depend on fewer variables. Especially for high order PDE systems in many variables the conventional integration technique may lead to an explosion of the number of functions of integration which is avoided with the new method. A further benefit is that redundant free functions in the solution are either prevented or that their number is at least reduced. 1 A critical look at conventional integration In this paper a new integration method is introduced that is suitable for the computerized solution of systems of linear PDEs that admit syzygies. In the text we will call the integration of single exact differential equations, i.e. equations which are total derivatives, the ‘conventional’ integration method (discussed, for example, in [11]). To highlight the difference with the new syzygy based integration method we have a closer look at the conventional method first. About notation: To distinguish symbolic subscripts from partial derivatives we indicate partial derivatives with a comma, for example, ∂xyei = ei,xy. To solve, for example, the system 0 = f,xx (1) 0 = xf,y +f,z (2) for f(x, y, z) one would, at first, integrate (1) with 2 new functions of integration g(y, z), h(y, z), then substitute f = xg + h (3)