Reduction of systems of nonlinear partial differential equations to simplified involutive forms

Abstract

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

Computing representations for radicals of finitely generated differential ideals

“…They describe a method for computing characteristic sets of prime differential ideals different from our methods given in [Ollivier, 1990], [Boulier, 1994] and [Boulier et al, 1995, section 5, page 164]. Reid et al [1996] and Reid et al [2001] developed algorithms for studying systems of PDE and computing Taylor expansions of their solutions. These methods are based more on differential geometry than on algebra.…”

Similarity: generalizations, applications and open problems

“…[10,42,[48][49][50][51][52][53][54][55]); -efficiently solve the (overdetermined) linear systems of determining equations for symmetries or conservation law multipliers and solve the nonlinear systems of determining equations for the nonclassical and related methods through the development of symbolic manipulation software (cf. [42,[56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71]); -develop numerical schemes that effectively use symmetries and/or conservation laws for ODE's (cf. [72]), for difference equations (cf.…”

“…[63]) to the linear infinitesimal determining equations of the system. It is shown that the invariance of the classification under the action of the equivalence group can be tested algorithmically knowing only the determining equations of the equivalence group.…”

Self-Invariant Contact Symmetries

“…Reliable packages are available that will reduce overdetermined systems of PDEs to a differential Gröbner basis that is far easier to solve than the original system [15,19]. However, our example is easily solved by hand, because (4.3) can be factorized into…”