Boundary Conditions

Boundary conditions (BCs) specify the behaviour of a solution at the boundary of the domain. For PDEs, they are required to obtain a unique solution.

Dirichlet Boundary Conditions

The value of $u$ is fixed at the boundary:

$$u{|}_{\text{boundary}}=f$$

Neumann Boundary Conditions

The normal derivative of $u$ is fixed at the boundary:

$$\frac{\partial u}{\partial n}=d$$

In 1D, this means $u^{\prime }$ is fixed at the endpoints. For a left boundary with derivative $d$, an imaginary point technique is used: introduce a fictitious point $u_{-1}$ and use the centred difference

$$\frac{u_1-u_{-1}}{2dx}=d⟹u_{-1}=u_1-2dx\cdot d$$

This allows the interior equations to be applied at the boundary node too. The special case $d=0$ (insulation) gives $u_{-1}=u_1$, i.e. $u_0=u_1$.

Mixed Boundary Conditions

A combination of Dirichlet and Neumann conditions applied at different parts of the boundary.

Boundary conditions appear in both the Heat equation (combined with initial conditions) and the Laplace equation (BCs alone determine the solution, with no initial condition).

Heat equation | Laplace equation | FTCS scheme | BTCS scheme