Mathematical Methods in the Physical Sciences · Problem 8.22

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · green-functions-nonzero-boundary

Problem

Show that our results can be extended to find the following solution of (8.22) which satisfies given nonzero boundary conditions: u(r)=∭G(r,r′)f(r′) dτ′+∬u(r′)∂G(r,r′)∂n′ dσ′u(\mathbf{r}) = \iiint G(\mathbf{r},\mathbf{r}')f(\mathbf{r}')\,d\tau' + \iint u(\mathbf{r}')\frac{\partial G(\mathbf{r},\mathbf{r}')}{\partial n'}\,d\sigma' where G(r,r′)G(r,r') is the Green function (8.28) which is zero on the surface σ\sigma, and ∂G/∂n′=∇G⋅n′\partial G/\partial n' = \nabla G \cdot \mathbf{n}' is the normal derivative of GG (see Chapter 6, Section 6). Hints: In Green's second identity (Chapter 6, Problem 10.16) let ϕ=u(r)\phi = u(\mathbf{r}) and ψ=G(r,r′)\psi = G(\mathbf{r},\mathbf{r}'), and use (8.22) and (8.23) to find ∇2ϕ\nabla^2\phi and ∇2ψ\nabla^2\psi. Comment: Although we derived the divergence theorem and so Green's identities only for bounded regions in Chapter 6, they are valid for unbounded regions if the functions involved tend to zero sufficiently rapidly at infinity.
Intuitive explanation

Green's second identity relates two functions and their Laplacians through a surface integral; the key is choosing ϕ=u\phi = u and ψ=G\psi = G so that their Laplacians encode the source term ff and the delta function respectively. Since GG vanishes on the boundary by design, the surface integral simplifies to depend only on uu and ∂G/∂n′\partial G/\partial n'—exactly the boundary data you're given. This transforms a difficult PDE with nonzero boundary conditions into an explicit integral formula where the Green function does all the heavy lifting.

Intuitive explanation

Green's second identity relates two functions and their Laplacians through a surface integral; the key is choosing ϕ=u\phi = u and ψ=G\psi = G so that their Laplacians encode the source term ff and the delta function respectively. Since GG vanishes on the boundary by design, the surface integral simplifies to depend only on uu and ∂G/∂n′\partial G/\partial n'—exactly the boundary data you're given. This transforms a difficult PDE with nonzero boundary conditions into an explicit integral formula where the Green function does all the heavy lifting.

Related theory

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