PDE and BC problems often require that the boundary and initial conditions be given at certain evaluation points (usually in which one of the variables is equal to zero). Using linear changes of variables, however, it is possible to change the evaluation points of BC, obtaining the solution for the new variables, and then changing back to the original variables. This is now automatically done by the pdsolve command.
Example 1: A heat PDE & BC problem in a semi-infinite domain:
Note the evaluation points A and B. The method typically described in textbooks requires the evaluation points to be . The change of variables automatically used in this case is:
so that pdsolve's task becomes solving this other problem, now with the appropriate evaluation points
and then changing the variables back to the original {x, t, u} and giving the solution. The process all in one go:
Example 2: A heat PDE with a source and a piecewise initial condition
Example 3: A wave PDE & BC problem in a semi-infinite domain:
Example 4: A wave PDE & BC problem in a semi-infinite domain:
Example 5: A wave PDE with a source: