Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSWe may treat the diffusion equationκ∇ 2 u = ∂u∂tin a similar way. Separating the spatial and time dependences by assuming asolution of the form u = F(r)T (t), and taking the separation constant as k 2 ,wefind∇ 2 F + k 2 dTF =0,dt + k2 κT =0.Just as in the case of the wave equation, the spatial part of the solution satisfiesHelmholtz’s equation. It only remains to consider the time dependence, whichhas the simple solutionT (t) =A exp(−k 2 κt).Helmholtz’s equation is clearly of central importance in the solutions of thewave and diffusion equations. It can be solved in polar coordinates in much thesame way as Laplace’s equation, and indeed reduces to Laplace’s equation whenk = 0. Therefore, we will merely sketch the method of its solution in each of thethree polar coordinate systems.Helmholtz’s equation in plane polarsIn two-dimensional plane polar coordinates, Helmholtz’s equation takes the form(1 ∂ρ ∂F )+ 1 ∂ 2 Fρ ∂ρ ∂ρ ρ 2 ∂φ 2 + k2 F =0.If we try a separated solution of the form F(r) = P (ρ)Φ(φ), and take theseparation constant as m 2 , we findd 2 Φdφ 2 + m2 φ =0,d 2 Pdρ 2 + 1 ( )dPρ dρ + k 2 − m2ρ 2 P =0.As for Laplace’s equation, the angular part has the familiar solution (if m ≠0)Φ(φ) =A cos mφ + B sin mφ,or an equivalent form in terms of complex exponentials. The radial equationdiffers from that found in the solution of Laplace’s equation, but by making thesubstitution µ = kρ it is easily transformed into Bessel’s equation of order m(discussed in chapter 16), and has the solutionP (ρ) =CJ m (kρ)+DY m (kρ),where Y m is a Bessel function of the second kind, which is infinite at the origin738