Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 4: 4.6 only 4.6 Helmholtz and Poisson Equations on a Disk ==== HELMHOLTZ EQUATION DEF. u=u(r,w), r=radial var, w=theta var, 0 < r < a, 0 < w < 2Pi u_rr + (1/r) u_r + (1/r^2) u_ww = -k u [Helmholtz PDE] u(a,w)=0 for 0 < w < 2Pi [u=zero on the edge of the disk] PRODUCT SOLUTION u=R(r)W(w) THEOREM mu = separation constant W'' + mu W = 0, and mu=m^2 because W must be 2Pi-periodic r^2 R'' + r R' + (kr^2 - m^2) R = 0, R(a)=0 LEMMA. For k<0, the R-equation is the modified Bessel equation of order m. The only bounded solution in this case with R(a)=0 is R=0. LEMMA For k >= 0, the R-equation is the parametric form of Bessel's equation of order m. This has a solution for m = 0,1,2,3, ... which is bounded if and only if R is a multiple of J_m(alpha_mn r/a), corresponding to k=(alpha_mn/a)^2. DEF. Symbols alpha_m1, alpha_m2, alpha_m3, ... are the positive zeros of the equation J_m(x)=0, for m=0,1,2,... THEOREM The product solutions u=R(r)W(w) are cos(m w) J_m(alpha_mn r/a), m >= 0, sin(m w) J_m(alpha_mn r/a), m >= 1, indexed for n=1,2,3, ... They each satisfy the Helmholtz PDE u_rr + (1/r) u_r + (1/r^2) u_ww = -k u where k=(alpha_mn/a)^2 DEF. The inner product for the product solutions is defined by = double integral of f(r,w)g(r,w)r over the rectangle 0 = 0 for two different product solutions f,g > 0 for any product solution f In short, the system of product solutions is an orthogonal set relative to the inner product <*,*>. THEOREM 2 [section 4.6, Eigenfunction Expansion Theorem] Under smoothness assumptions for f(r,w), defined on rectangle 0