Basis functions for Galerkin approximation of BVP

Question

Given the following boundary-value problem (BVP) for $\lambda$: $$y^{(4)}+ay''+\lambda b y'+\lambda^2 y=0,\quad y=y(x),\quad 0\le x\le 1,\\ y(0)=y''(0)=y''(1)=y^{(3)}+ay'(1)=0$$ how can we systematically determine a suitable orthonormal basis for approximating the solutions $y$ of the BVP using Galerkin's scheme? Simple $\sin()$-functions won't work here. Any help is appreciated.

Answer 1

Since your domain is bounded, it makes sense to use an orthonormal basis with a bounded domain. Examples include Legendre and Chebyshev polynomials. While both have a domain $[-1,1]$ you can map it to $[0,1]$.

However, this won't guarantee a very good convergence. Usually, good basis sets for a certain problem are those that solve similar problems. In such a case, you can consider the new problem as a perturbation to the old problem. You can look, for example, at the equations that standard basis sets solve and see whether any looks similar to your problem. See also discussion in spectral method for an eigenvalue problem, what basis functions to choose?.