<html> <head> <title>Math on the Web Example</title> </head> <body bgcolor=#ffffff> Use the method of separation of variables to show that the 1-D heat conduction problem: find $u = u(x,t)$ so that \[ \frac{\partial u}{\partial t}(x,t) - \alpha \frac{{\partial}^2 u}{\partial x^2}(x,t) = 0,\thicksp 0 < x < l,\thicksp t \geq 0 \] \[ u(0,t) = 0,\thicksp t \geq 0 \] \[ u(l,t) = 0,\thicksp t \geq 0 \] \[ u(x,0) = f(x),\thicksp 0 \leq x \leq l \] has the solution \[ u(x,t) = \sum_{n=1}^{\infty} c_n \sin \frac{n \pi x}{l} \exp \left[ -\alpha \left( \frac{n \pi}{l} \right)^2 t \right] \] where \[ c_n = \frac{2}{l} \int_0^l f(x) \sin \frac{n \pi x}{l} dx. \] </body> </html>