A bar of length is held at zero temperature at and thermally insulated at

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

A bar of length L=1.5 mL = 1.5\,\mathrm{m} is held at zero temperature at x=0x=0 and thermally insulated at x=Lx=L. Its steady temperature u(x)u(x), with a distributed heat source f(x)f(x) and a term μu\mu u proportional to the local temperature, obeys −u′′(x)=μ u(x)+f(x),u(0)=0, u′(L)=0.-u''(x) = \mu\,u(x) + f(x), \qquad u(0) = 0,\ u'(L) = 0. The boundary-value Green function for −d2/dx2-\mathrm{d}^2/\mathrm{d}x^2 with these conditions is the standard result K(x,ξ)=min⁡(x,ξ)K(x,\xi) = \min(x,\xi). It is given that for this Green function ∬0LK(x,ξ)2 dx dξ=L46.\iint_0^L K(x,\xi)^2\,\mathrm{d}x\,\mathrm{d}\xi = \frac{L^4}{6}. ((a)) Write the Fredholm equation of the second kind for uu. ((b)) Compute ∥K∥HS\|K\|_{\rm HS} and the radius within which the Neumann series for uu is guaranteed to converge. ((c)) State whether convergence is guaranteed for μ=0.8 m−2\mu = 0.8\,\mathrm{m^{-2}}, and what this says about whether μ=0.8 m−2\mu = 0.8\,\mathrm{m^{-2}} can be a characteristic value.

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