Category:Parabolic: Difference between revisions

This category contains all control problems which are governed by a parabolic partial differential equation.

A second order linear partial differential equation can be written as ${\displaystyle {\displaystyle \sum _{i,j=1}^n}{a}_{ij}\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+\text{lower-order terms}=0}$.

If the matrix ${\displaystyle A=(a_{ij}{)}_{ij}}$ is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic.

An example is the heat equation: ${\displaystyle \frac{\partial u}{\partial t}-\mathrm{\Delta }u=f}$, where ${\displaystyle \mathrm{\Delta }}$ denotes the Laplace operator, ${\displaystyle u}$ is the unknown, and the function ${\displaystyle f}$ is given.