We present several algorithms, executable in a finite number of steps, which have been implemented in the symbolic language maple. The standard form algorithm reduces a system of PDEs to a simplified standard form which has all of its integrability conditions satisfied (i.e. is involutive). The initial data algorithm uses a system's standard form to calculate a set of initial data that uniquely determines a local solution to the system without needing to solve the system. The number of arbitrary constants and arbitrary functions in the general solution to the system is directly calculable from this set. The taylor algorithm uses a system's standard form and initial data set to determine the Taylor series expansion of its solution about any point to any given finite degree. All systems of linear PDEs and many systems of nonlinear PDEs can be reduced to standard form in a finite number of steps. Our algorithms have simple geometric interpretations which are illustrated through the use of diagrams. The standard form algorithm is generally more efficient than the classical methods due to Janet and Cartan for reducing systems of PDEs to involutive form.

There are symbolic programs based on heuristics that sometimes, but not always, explicitly integrate the determining equations for the infinitesimal Lie symmetries admitted by systems of differential equations. We present a heuristic-free algorithm ‘Structure constant’, which can always determine whether the Lie symmetry group of a given system of PDEs is finite- or infinite-dimensional. If the group is finite-dimensional then ‘Structure constant’ can determine the dimension and structure constants of its associated Lie algebra without the heuristics of integration involved in other methods. If the group is infinite-dimensional, then ‘Structure constant’ computes the number of arbitrary functions which determine the infinite-dimensional component of its Lie symmetry algebra and also calculates the dimension and structure of its associated finite-dimensional subalgebra. ‘Structure constant’ employs the algorithms ‘Standard form’ and ‘Taylor’, described elsewhere. ‘Standard form’ is a heuristic-free algorithm which brings any system of determining equations to a standard form by including all integrability conditions in the system. ‘Taylor’ uses the standard form of a system of differential equations to calculate its Taylor series solution. These algorithms have been implemented in the symbolic language MAPLE. ‘Structure constant’ can also automatically determine the dimension and structure constants of the Lie symmetry algebras of entire classes of differential equations dependent on variable coefficients. In particular, we obtain new group classification results for some physically interesting classes of nonlinear telegraph equations depending on two variable coefficients, one representing a nonlinear wave speed and the other representing a nonlinear dispersion.

We investigate the Brusselator reaction–diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method. For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.

The two-dimensional flow of a viscous incompressible fluid in a channel with accelerating walls is analysed by use of Hiemenz's similarity solution. Steady flows and their instabilities are calculated, and unsteady flows are computed by solving the initial-value problem for the governing partial-differential system. Thereby, these exact solutions of the Navier–Stokes equations are found to exhibit turning points, pitchfork bifurcations, Hopf bifurcations and Takens–Bogdanov bifurcations along the route to chaos. The substantial physical result is that the chaos previously found for flows with symmetrically accelerating walls is easily destroyed by a little asymmetry.