  We present direct methods and symbolic software for the computation of
conservation laws of nonlinear partial differential equations (PDEs) and
differential-difference equations (DDEs).The methods are applied to nonlinear
PDEs in (1+1) dimensions with polynomial nonlinearities which include the
Korteweg-de Vries (KdV), Boussinesq, and Drinfel'd-Sokolov-Wilson equations. An
adaptation of the methods is applied to PDEs with transcendental
nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville
equations.
  With respect to nonlinear DDEs, our methods are applied to Kac-van Moerbeke,
Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the
undetermined coefficients method, we designed a new direct method which uses
leading order analysis. That method is applied to discretizations of the KdV
and modified KdV equations, and a combination thereof. Additional examples
include lattices due to Bogoyavlenskii, Belov-Chaltikian, and
Blaszak-Marciniak.
  The undetermined coefficient methods for PDEs and DDEs have been implemented
in Mathematica. The code "TransPDEDensityFlux.m" computes densities and fluxes
of systems of PDEs with or without transcendental nonlinearities. The code
"DDEDensityFlux.m" does the same for polynomial nonlinear DDEs. Starting from
the leading order terms, the new Maple library "discrete" computes densities
and fluxes of nonlinear DDEs.
