Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

An algorithm is presented to linearize nonlinear partial differential equations by non-invertible mappings. The algorithm depends on finding nonlocal symmetries of the given equations which are realized as appropriate local symmetries of a related auxiliary system. Examples include the Hopf-Cole transformation and the linearizations of a nonlinear heat conduction equation, a nonlinear telegraph equation, and the Thomas equations.

This note deals with a new model of elastic–perfectly plastic materials in which the yield stress is regarded as a threshold above which plastic flow occurs rather than a constraint which cannot be violated. This modelling change allows us to treat a number of signalling and impact problems not solvable within the classic framework of elastic–perfectly plastic materials.

The coupled nonlinear partial differential equations obeyed by the electrical potential and temperature distribution for a medium undergoing steady state electrical heating are applied to a one-dimensional rod having its surface temperature held constant as current is conducted along its length due to a potential difference maintained between its ends. Extension is given to the previously discussed class of solutions by the inclusion of a thermal conductivity which varies linearly with temperature. The resulting electrical current and resistance are found to be significantly influenced by the thermal conductivity of the medium. Molybdenum is identified as a material exemplifying such a thermal conductivity and the general effects are then numerically illustrated.

We prove the existence of a weak solution for a two-phase continuous casting Stefan problem with a general monotone nonlinear cooling condition. We establish a sufficient condition for stability, which yields uniqueness and comparison results for the evolutionary and the steady- state solutions. We also discuss the asymptotic behaviour as t←∞ of the corresponding temperatures and enthalpies.

In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.