5.2 Comparison with D'Alembert

The example problem of the previous section was also solved in chapter 4 using D'Alembert. It is interesting to compare the two solutions.

The separation of variables solution took the form:

u =
f_n \cos\frac{(2n-1)\pi...
...2n-1)\pi at}{2\ell}
\cos\frac{(2n-1)\pi x}{2\ell}

Some of its nice features are:

The D’Alembert solution took the form

u(x,t) = \frac{\bar f(x-at) + \bar f(x+at)}{2}
+ \frac{1}{2a} \int_{x-at}^{x+at} \bar g(\xi) { \rm d}\xi

Some of its nice features are

In short, each method has its advantages and disadvantages.