A string of length

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 string of length L=2 mL = 2\,\mathrm{m}, fixed at both ends and driven by a distributed transverse load f(x)f(x), has small transverse displacement satisfying −u′′(x)=μ u(x)+f(x),u(0)=u(2)=0.-u''(x) = \mu\, u(x) + f(x), \qquad u(0) = u(2) = 0. The boundary-value Green function for −d2/dx2-\mathrm{d}^2/\mathrm{d}x^2 with these homogeneous conditions is K(x,ξ)=x<(2−x>)/2K(x,\xi) = x_<(2 - x_>)/2, where x<=min⁡(x,ξ)x_< = \min(x,\xi) and x>=max⁡(x,ξ)x_> = \max(x,\xi). It is given that ∬02K(x,ξ)2 dx dξ=2490=845.\iint_0^2 K(x,\xi)^2\,\mathrm{d}x\,\mathrm{d}\xi = \frac{2^4}{90} = \frac{8}{45}. ((a)) Write the Fredholm equation of the second kind that uu satisfies after applying KK to the equation. ((b)) Compute the Hilbert-Schmidt norm ∥K∥HS\|K\|_{\rm HS} and the radius within which the Neumann series for uu is guaranteed to converge. ((c)) State whether the Neumann series is guaranteed to converge for μ=2 m−2\mu = 2\,\mathrm{m^{-2}}, and what this says about whether μ=2 m−2\mu = 2\,\mathrm{m^{-2}} can be a characteristic value of the string.

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