Chapter 17 - Applications of mechanical methods to partial differential equations

Abstract

This chapter reports the research work on unifying and extending various special methods for solving partial differential equations (PDEs). It includes a mechanical approach for constructing the analytical solutions of PDEs and Wu-Ritt's characteristic set method and other computational tools for constructing analytical solutions for PDEs. The applications of the differential characteristic set method to compute the symmetry groups for PDEs and automatic reasoning by the differential characteristic set method are elaborated. The constructing analytical solutions of nonlinear PDEs are also explained in this chapter. The Clarkson–Kruskai method, which involves no group-theoretic techniques, is extended to investigate similar reductions of the variable coefficient KdV (VCKdV) equation. It is shown that its reduction equations are Painleve I and II, respectively, which are the same as those of the KdV equations. By this property, the transformation from the VCKdV equation to the KdV equation is established. The symmetries and Hamiltonian structure of nonlinear PDEs are also elaborated.