For the taut string with fixed ends

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

Conditions of Validity, Ch Green functions: inverting the operator

Problem

For the taut string with fixed ends, −u′′=δ(x−ξ)-u''=\delta(x-\xi) on [0,L][0,L] with u(0)=u(L)=0u(0)=u(L)=0, the closed-form Green function is G(x,ξ)=x<(L−x>)/LG(x,\xi) = x_<(L-x_>)/L, and its bilinear eigenfunction expansion is G(x,ξ)=∑n=1∞yn(x)yn(ξ)λn,yn(x)=2Lsin⁡nπxL,λn=(nπL)2.G(x,\xi) = \sum_{n=1}^{\infty}\frac{y_n(x)y_n(\xi)}{\lambda_n}, \qquad y_n(x) = \sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}, \qquad \lambda_n = \left(\frac{n\pi}{L}\right)^2. ((a)) Evaluate the closed form at x=ξ=L/4x=\xi=L/4, as a multiple of LL. ((b)) Evaluate the truncated bilinear sum at x=ξ=L/4x=\xi=L/4 using only the terms n=1,2,3,4n=1,2,3,4, as a multiple of LL, and state the percentage error against part (a). ((c)) Repeat with the terms n=1,…,8n=1,\ldots,8, and state the new percentage error.

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