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.

Abundant polynomial type conservation laws are constructed for a seventh order nonlinear evolution equation with six arbitrary parameters. From the parameters constraints that leading to the existence of conserved densities, a new unnamed seventh order KdV type equation that may be integrable is reported. By introducing nonlinear transformation, the new soliton solution as well as the solitary wave solution to the new unnamed seventh order KdV type equation are obtained.

By using the general solutions of a coupled Riccati equations and the relations between them a direct algebra method is described to construct several kinds of closed-form travelling wave solutions for some nonlinear differential equations. By this method three important nonlinear partial differential equations are studied and, in addition to rederiving all known solutions, several new solutions are explicitly obtained with the aid of symbolic computation.

A software package wkptest written in Maple is presented, which can carry out the traditional Painlev

Nature of physical problem: Our program provides exact traveling wave solutions, which describe various phenomena in nature, and thus can give more insight into the physical aspects of problems. These solutions may be easily used in further applications. Restriction on the complesity of the problem: The program can handle system of nonlinear evolution equations with any number of independent and dependent variables, in which each equation is a polynomial (or can be converted to a polynomial) in the dependent variables and their derivatives. Typical running time: It depends on the input equations as well as the degrees of the desired polynomial solutions. For most of the equations we have computed, the running time is no more than 100s. In Maple 8, by taking advantage of the package RIF contained in DEtools, we developed a package RAEEM which is a comprehensive and complete implementation of such methods as the tanh-method, the extended tanh-method, the Jacobi elliptic function method and the elliptic equation method. RAEEM can entirely automatically output a series of exact traveling wave solutions, including those of polynomial, exponential, triangular, hyperbolic, rational, Jacobi elliptic, Weierstrass elliptic type. The effectiveness of the package is illustrated by applying it to a large variety of equations. In addition to recovering previously known solutions, we also obtain more general forms of some solutions and new solutions.

A constructive algorithm for the completion of arbitrary algebraic partial differential equation systems to involutive ones are presented. Based on it, an implementation ICS for computing the involutive characteristic set of an algebraic partial differential equation systems is given. Some examples are tested with ICS and some experimental results are obtained which may show a hint for computing involutive characteristic set of arbitrary algebraic partial differential equation systems thereafter.