Partial Differential Equation

Particular case of a system of DEs.

Following Lychagin_2021 page 11, we consider a PDE a submanifold of the jet bundle $J^n({\mathbb{R}}^p,\mathbb{R})$ (where $n$ is the order of the PDE and $p$ the number of independent variables) given by

for a smooth function

A solution is any integral manifold of the Cartan distribution of this jet bundle such that is contained in ${\mathcal{S}}_{\mathrm{\Delta }}$.

Symmetries

When the PDE possesses a symmetry can be reduced, sometimes even to an ODE. The solutions of the original PDE obtained from the ODE are called similarity solutions

Other facts

• In some cases they can be seen like an ODE, see characteristics of first-order partial differential equation.

• Very important cases are the homogeneous linear PDEs and theinhomogeneous linear PDEs.

• A kind of particular solutions are travelling wave solutions. When they exist, the PDE can also be reduced to an ODE.

• An important particular case can be seen as evolution equations.