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

$$ \mathcal{S}_{\Delta}=\left\{\Delta=0 \right\} $$

for a smooth function

$$ \Delta: J^n(\mathbb{R}^p,\mathbb{R})\rightarrow \mathbb{R} $$

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

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.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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