Riley hobson bence · Problem 21.28

Partial differential equations: separation of variables

Mathematics — Methods · greens-function-general-pde-operator

Problem

Consider the PDE Lu(r)=ρ(r)\mathcal{L}u(\mathbf{r}) = \rho(\mathbf{r}), for which the differential operator L\mathcal{L} is given by L=∇⋅[p(r)∇]+q(r),\mathcal{L} = \nabla \cdot [p(\mathbf{r}) \nabla] + q(\mathbf{r}), where p(r)p(\mathbf{r}) and q(r)q(\mathbf{r}) are functions of position. By proving the generalised form of Green's theorem, ∫V(ϕLψ−ψLϕ) dV=∮Sp(ϕ∇ψ−ψ∇ϕ)⋅n^ dS,\int_V (\phi \mathcal{L} \psi - \psi \mathcal{L} \phi) \, dV = \oint_S p(\phi \nabla \psi - \psi \nabla \phi) \cdot \hat{\mathbf{n}} \, dS, show that the solution of the PDE is given by u(r0)=∫VG(r,r0)ρ(r) dV(r)+∮Sp(r)[u(r)∂G(r,r0)∂n−G(r,r0)∂u(r)∂n]dS(r),u(\mathbf{r}_0) = \int_V G(\mathbf{r}, \mathbf{r}_0) \rho(\mathbf{r}) \, dV(\mathbf{r}) + \oint_S p(\mathbf{r}) \left[ u(\mathbf{r}) \frac{\partial G(\mathbf{r}, \mathbf{r}_0)}{\partial n} - G(\mathbf{r}, \mathbf{r}_0) \frac{\partial u(\mathbf{r})}{\partial n} \right] dS(\mathbf{r}), where G(r,r0)G(\mathbf{r}, \mathbf{r}_0) is the Green's function satisfying LG(r,r0)=δ(r−r0)\mathcal{L} G(\mathbf{r}, \mathbf{r}_0) = \delta(\mathbf{r} - \mathbf{r}_0).

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