Heat Equation

The heat equation (also called the diffusion equation) describes how a quantity $u$ (e.g. temperature) diffuses through space over time.

In 1D:

$$\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$$

where $\alpha >0$ is the diffusion coefficient.

Initial and Boundary Conditions

The heat equation requires:

• An initial condition: $u(0,x)=f_0(x)$ specifying the profile at $t=0$.
• Boundary conditions: fixing $u$ (Dirichlet) or $\partial u/\partial x$ (Neumann) at the spatial boundaries for all $t$.

Numerical Schemes

The heat equation is discretised in space using the centred second derivative and in time using either:

• FTCS scheme: forward difference in time (explicit, conditionally stable)
• BTCS scheme: backward difference in time (implicit, unconditionally stable)

The solution of the Wiener process (Brownian motion) is also a diffusion equation solution: the PDF of $W(t)$ is Gaussian with variance $t$, which satisfies the heat equation with $\alpha =1/2$ and a delta-function initial condition.

FTCS scheme | BTCS scheme | Boundary conditions | Finite differences | Laplace equation | Wiener process