Spectral Numerical Methods

Consider a PDE of the form:

Consider the finite dimensional space of functions
spanned by .
We seek a solution of the form

In general, this does not exist,
because there is no reason for to be in this span.
So we only seek an approximation which satisfies:

Now we just need to pick our
and ideally an inner product where they are orthogonal.
We can use Fourier Series or Chebyshev Polynomials

For Chebyshev expansion, in general, boundary conditions may be a problem.
We might need to use if .
If there is time dependence on boundary conditions,
we may run into serious stability issues.