In this paper, an algorithm for computing the Janet bases of linear differential equations is described, which is the differential analogue of the algorithm JanetBasis improved by Gerdt. An implementation of the algorithm in Maple is given. The implemented algorithm includes some subalgorithms: Janet division, Pommaret division, the judgement of involutive divisor and reducible, the judgement of conventional divisor and reducible, involutive normal form and conventional normal form, involutive autoreduction and conventional autoreduction, PJ-autoreduction and so on. As an application, the Janet Bases of the determining system of classical Lie symmetries of some partial differential equations are obtained using our package.

An algorithm of constructing the polynomial type conservation laws of nonlinear evo-lution equations (either parameterized or non-parameterized) is described and a Maple package CONSLAW to automate the computation is presented. For parameterized equa-tion(s), CONSLAW can determine the parameters constraints automatically such that a sequence of conservation laws exist.

The master equation of a one-dimensional lattice-gas model with order preservation where the occupation probabilities of sites corresponding to Bose statistcs as a consequence of the prescribed dynamics is studied with the potential symmetry method. The infinite-parameter potential symmetry and a new exact solution are obtainde. The result illustrates that there remains the possibility of the above nonlinear equation to a linear partial differential equation by a non-invertible mapping.

The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.