Partial Differential Equations

Partial differential equations are an extension of ODE where the evolution depends not only on time but takes into account the environment values in a neighborhood.

The most prominent example of which is the diffusion of temperature in space which in its simplest form (one dimensional) is given by

\[ \partial_t u = \frac{1}{2}\partial_{xx} u \]

This equation involves the second derivative for which we will consider the central difference:

\[ \begin{equation*} f^{\prime\prime}(x) \approx \frac{f(x+h) + f(x-h) - f(x)}{2 h^2} \end{equation*} \]