ABSTRACT
We will derive Green’s functions for Laplace’s and Helmholtz’s equations. In previous sections we have used a source (of strength +1) for parabolic and hyperbolic equations, but now we will use a sink (of strength −1) for elliptic equations. The method is to first develop Green’s function G (x) for the sink (source) at the origin, and then obtain the general Green’s function G (x− x′) ≡ G (x;x′) for the sink (source) at a point x′ = 0 by simply replacing x in G(x) by x − x′. Note that in R the Laplacian operator is an ordinary differential operator, which has already been studied in Chapter 3. We will, therefore, determine Green’s function only in R2 and R
3 by different methods.
