4.3 Boundary Value Problems

Examples:

- Poisson's and Laplace's equations, $d^{2} \phi /dx^{2}=-\rho(x)$, or

$$\frac{d \phi}{dx} = - e$$

$$\frac{de}{dx} = \rho(x)$$

where $\rho(x)$ is a charge density. (Laplace: $\rho(x)=0$). Another physical problem described by the same equation is the temperature distribution along a thin rod: $d^{2}T/dx^{2}=0$.

- Time independent Schroedinger equation for a particle of mass $m$ in a potential $U(x)$:

$$\frac{d^{2}\psi}{dx^{2}} = -g(x)\psi \,, \;\;\; {\rm with}\;\;g(x)=\frac{2m}{\hbar^{2}}[E-U(x)]$$

The general 1-dimensional BVP reads

$$\frac{dy_{i}}{dx} = f_{i}(x,y_{1},\dots y_{N})\,;\;\;\; i=1,\dots N$$

with $N$ boundary values required. Typically there are

$n_{1}$     boundary values $a_{j}\;(j=1,\dots n_{1})$ at $x=x_{1}$, and
$n_{2} \equiv N-n_{1}$ boundary values $b_{k}\;(k=1,\dots n_{2})$ at $x=x_{2}$.

The quantities $y_{i},a_{j}$ and $b_{k}$ may simply be higher derivatives of a single solution function $y(x)$. Two methods are available:

- Shooting method
- Relaxation technique