Riley hobson bence · Problem 21.27

Partial differential equations: separation of variables

Mathematics — Methods · greens-function-yukawa-equation

Problem

Consider the equation ∇2ϕ−m2ϕ=ρ(r)\nabla^2 \phi - m^2 \phi = \rho(\mathbf{r}), which arises in the theory of the screening of charges in a plasma. (a) Show that the solution ϕ(r)\phi(\mathbf{r}) in a volume VV bounded by a surface SS can be written as ϕ(r)=∫VG(r,r′)ρ(r′) dV′+∮S[G(r,r′)∂ϕ∂n′−ϕ(r′)∂G(r,r′)∂n′]dS′,\phi(\mathbf{r}) = \int_V G(\mathbf{r}, \mathbf{r}') \rho(\mathbf{r}') \, dV' + \oint_S \left[ G(\mathbf{r}, \mathbf{r}') \frac{\partial \phi}{\partial n'} - \phi(\mathbf{r}') \frac{\partial G(\mathbf{r}, \mathbf{r}')}{\partial n'} \right] dS', where the solution is sought in VV and takes the specified form ϕ=f\phi = f on SS, the boundary of VV. The Green's function, G(r,r′)G(\mathbf{r}, \mathbf{r}'), to be used satisfies ∇2G−m2G=δ(r−r′)\nabla^2 G - m^2 G = \delta(\mathbf{r} - \mathbf{r}') and vanishes when r\mathbf{r} is on SS. (b) When VV is all space, G(r,r′)G(\mathbf{r}, \mathbf{r}') can be written as G(t)=g(t)/tG(t) = g(t)/t, where t=∣r−r′∣t = |\mathbf{r} - \mathbf{r}'| and g(t)g(t) is bounded as t→∞t \to \infty. Find the form of G(t)G(t). (c) Find ϕ(r)\phi(\mathbf{r}) in the half-space x>0x > 0 if ρ(r)=δ(r−r1)\rho(\mathbf{r}) = \delta(\mathbf{r} - \mathbf{r}_1) and ϕ=0\phi = 0 both on x=0x = 0 and as r→∞r \to \infty.

Browse the problem library