ODE's & PDE's: Homogenous piecing vs Eigenexpansion vs Green functions I don't know if i'm within rules of the forum to ask this question. If i'm not please comment before downvoting. If you know of a source that answers these questions, please suggest. It would be benificial if the source is concise, may be upto 70 pages max.
I'm getting confused repeatedly regarding this
Consider a inhomogenous linear operator $Lu = f$ for a boundary value problem(BVP). Now there seem to be three ways of solving these equations.
a) One solves for inhomogenous in terms of a particular solution & a complementary solution(i.e. solving $Lu=0$). If the boundary conditions are linear one solves for two equations $Lu = f $ for homogenous boundary & $Lu =0$ for inhomogenous boundary.
b)One solves the eigenvalue problem $Lu =\lambda u$, expands $f$ in terms of the eigenfunctions $u_n$ say $f = \Sigma (c_nu_n)$ and $u = \Sigma (a_nu_n)$ where $a_n$ are undetermined coefficients determined by the boundary conditions.
c) One solves the fundamental equation $Lg = \delta$ & then $u = g * f$ where '$*$' means convolution here.
My questions are as as follows
1) What is the justification behind b). Is $f$ representable in the form $f = \Sigma c_nu_n$. What happens when $\lambda _n$'s are continous & not discrete.
2) Are approaches a) & b) equivalent. What are the situations where one preferable over the other.
3) Does the Green function approach give all solutions including the particular & complementary. Is it equivalent to b), if so how?
 A: The first idea always works; this is basically a linear algebraic fact, since the BVP is essentially the system of linear equations $Lu=f,Ru=g$ where $R$ is the restriction to the boundary (or similar in the non-Dirichlet case) and $g$ is the given boundary function.
Provided a Green's function exists, that approach will also always work. A Green's function may not always exist, although it has been shown that it always exists for operators with constant coefficients (at least on the full space). One nice thing about this approach is that it furnishes a general recipe for writing down the solution, even when that solution is not an elementary function.
By contrast, the eigenfunction expansion idea is rather special. It works provided the space with the boundary condition adjoined has a complete (Schauder) basis consisting of eigenfunctions of the differential operator. Whether this happens depends on both the operator and the boundary condition. The main property that is used is that the operator is self-adjoint on the space with the boundary conditions in question. For example, this does not happen for  all kinds of boundary conditions for the Laplacian, but it does in the special cases of homogeneous Dirichlet and homogeneous Neumann boundary conditions.
