A thin bar of length has its left end held at zero temperature and its right end…

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 Green functions: inverting the operator

Problem

A thin bar of length L=4.0 mL = 4.0\,\text{m} has its left end held at zero temperature and its right end perfectly insulated. A steady point source of unit strength at x=ξx=\xi gives the temperature u(x)u(x) satisfying −u′′(x)=δ(x−ξ)-u''(x) = \delta(x-\xi) on [0,L][0,L] with u(0)=0u(0) = 0 and u′(L)=0u'(L) = 0. ((a)) Construct G(x,ξ)G(x,\xi) by matching a homogeneous solution satisfying y(0)=0y(0)=0 to one satisfying y′(L)=0y'(L)=0. ((b)) Evaluate G(1.0 m,2.5 m)G(1.0\,\text{m}, 2.5\,\text{m}).

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