Editing Dirichlet problem (section)

==Example: equation of a finite string attached to one moving wall==

Consider the Dirichlet problem for the [[wave equation]] describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. the [[d'Alembert equation]] on the triangular region of the [[Cartesian product]] of the space and the time:
: <math>\frac{\partial^2}{\partial t^2} u(x, t) - \frac{\partial^2}{\partial x^2} u(x, t) = 0,</math>
: <math>u(0, t) = 0,</math>
: <math>u(\lambda t, t) = 0.</math>

As one can easily check by substitution, the solution fulfilling the first condition is
: <math>u(x, t) = f(t - x) - f(x + t).</math>

Additionally we want
: <math>f(t - \lambda t) - f(\lambda t + t) = 0.</math>

Substituting

: <math>\tau = (\lambda + 1) t,</math>

we get the condition of [[self-similarity]]

: <math>f(\gamma \tau) = f(\tau),</math>

where

: <math>\gamma = \frac{1 - \lambda}{\lambda + 1}.</math>

It is fulfilled, for example, by the [[composite function]]
: <math>\sin[\log(e^{2 \pi} x)] = \sin[\log(x)]</math>

with

: <math>\lambda = e^{2\pi} = 1^{-i},</math>

thus in general

: <math>f(\tau) = g[\log(\gamma \tau)],</math>

where <math>g</math> is a [[periodic function]] with a period <math>\log(\gamma)</math>:

: <math>g[\tau + \log(\gamma)] = g(\tau),</math>

and we get the general solution

: <math>u(x, t) = g[\log(t - x)] - g[\log(x + t)].</math>