PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Special functions and orthogonal polynomials
Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator
Problem
Heat conduction along a thin circular cooling fin, with heat lost sideways proportional to temperature, gives the dimensionless equation xy′′+y′−(x+xν2)y=0,0<x≤X, where x is a dimensionless radial coordinate and ν>0 is a fixed (dimensionless) constant. Its two independent solutions are Iν(x), which behaves like c1xν (finite, in fact tending to zero) as x→0, and Kν(x), which behaves like c2x−ν (diverging) as x→0, for constants c1,c2=0 and ν>0.
((a)) Write this equation in Sturm-Liouville form −(py′)′+qy=0 and identify p(x) and q(x).
((b)) Show that p(0)=0, so x=0 is a singular endpoint of exactly the same character as the ordinary Bessel equation's endpoint at the axis.
((c)) Using Abel's identity for the Wronskian of two solutions of a self-adjoint second-order equation, show that p(x)(fg′−f′g) is the same constant for every x>0 when f,g are any two independent solutions. Hence show that a self-adjoint domain cannot contain both Iν and Kν with independent coefficients, and state the condition, with its physical reason, that decides which of the two is kept.