(a) A rod of length with fixed ends satisfies on , , with boundary-value Green…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Function spaces and operator theory

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

Problem

(a) A rod of length

L=2\,\mathrm{m}

with fixed ends satisfies

-u''(x)=\mu u(x)+f(x)

[0,2]

u(0)=u(2)=0

, with boundary-value Green function

G(x,\xi)=x_<(2-x_>)/2

. Write the equivalent Fredholm equation of the second kind, and give the first two characteristic values

\mu_n=(n\pi/2)^2

and kernel eigenvalues

\lambda_n=(2/n\pi)^2

for

n=1,2

, to three significant figures. (b) Separately, on

[0,1]

(a dimensionless interval), consider the symmetric kernel

K(x,\xi)=1+x^2\xi^2

. Expanding

\phi(x)=\alpha+\beta x^2

, find the

2\times2

symmetric matrix representing

K

on this basis, and its two eigenvalues and (unnormalised) eigenfunctions. (c) Confirm

\lambda_1^2+\lambda_2^2=\|K\|_{\rm HS}^2

for the kernel of part (b), and write the general Hilbert-Schmidt expansion formula for the solution of

\phi=f+\mu K\phi

in terms of the normalised eigenfunctions found.

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