Wolfram Mathematica: an effective tool for solutions to partial differential
equations and for visualisation of the solutions.

Abstract:

The latest versions of the Wolfram Mathematica give us a possibility to perform
both analytical solutions and numerical simulations using own commands of the
program. Also the codes may be developed to realize both finite difference and
spectral numerical algorithms. The command NDSolve provides numerical
solutions of partial differential equations and the method of lines is employed
for that. According to this method, only a spatial discretization is used while
the built-in ODE solver is employed for temporal variations choosing the most
suitable numerical method for the solution.

One can note an important educational application of the program based on the
ability of the creation of the demonstration projects. These projects
dynamically describe various physical phenomena or mathematical solutions.

Of special interest will be an analysis of the Navier-Stokes equations
including their analytical and numerical solutions.