A rod of length is held at zero temperature at both ends
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Function spaces and operator theory
Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient
Problem
A rod of length L is held at zero temperature at both ends, so its temperature profiles are expanded in the Dirichlet eigenfunctions yn(x)=sin(nπx/L), weight w=1, λn=(nπ/L)2. Two candidate initial profiles both vanish at x=0 and x=L:
((a)) f(x)=x(L−x)2, a smooth (infinitely differentiable) cubic. Apply the boundary-matching chain: check whether f satisfies the Dirichlet conditions, then whether Lf=−f′′ also satisfies them, and hence predict the power-law decay of f's sine coefficients.
((b)) g(x)=2L−x−2L, a triangular "tent" profile that is continuous but has a discontinuous first derivative (a corner) at x=L/2. Explain why the boundary-matching chain of part (a) cannot be carried past its first step for g, and state, from the standard Fourier decay rate of a triangular wave, whether g's coefficients decay faster or slower than f's.
((c)) f is infinitely differentiable, yet its coefficients fall only as a fixed power of n. State briefly what this shows about the claim that the decay rate of an eigenfunction expansion is set by the smoothness of the expanded function alone.