PhD Preliminary Qualifying Examination: Applied Mathematics August 19, 2008

Find a singular value decomposition and pseudoinverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2\\ 0 & 0 \end {pmatrix}. \]

1. Explain what is meant by a Cauchy sequence, Completeness of a space, and a Hilbert space (you can assume the definitions of norm, inner product, and linear space are known).
2. Prove that the set of continuous functions on the interval \([a,b]\) is complete with respect to the uniform norm \[ \lVert f\rVert =\max _{x\in [a,b]} |f(x)|. \]

Prove that if a linear operator \(K\) can be approximated (in the operator norm) by a sequence of compact operators \(K_n\), with, \[ \lim _{n\to \infty }\lVert K_n-K\rVert =0, \] then \(K\) is a compact operator.

On the interval \([0,1]\) use eigenfunctions to find the resolvent of the integral equation \[ u(x)-\lambda \int _0^1 x^2y^2u(y)\,dy=f(x). \] When does a solution not exist? Write down the pseudoresolvent in this case.

1. Using Green’s functions, find the solution to the equation \[ u''=f(x),\qquad u(0)=0,\qquad u(1)=1. \]
2. Find the necessary and sufficient condition on \(f(x)\) for a solution to \[ u''+m^2\pi ^2u=f(x),\qquad u(0)=u(1)=0 \] to exist.

Find a solution \(u(x,y)\) of Laplace’s equation in 2 dimensions for which \(u(x,0)=|x|^{1/2}\).

Evaluate the integral \[ I=\int _0^\infty \frac {\cos ax}{(x+1)^2}\,dx. \]

Solve the integral equation \[ \int _{-\infty }^{\infty } k(x-y)u(y)\,dy-u(x)=f(x) \] with \(k(x)=H(x)\), the Heaviside function.

Solve the system of equations \[ \frac {du_n}{dt}=\frac {1}{h^2}\left (u_{n+1}-2u_n+u_{n-1}\right ), \qquad -\infty <n<\infty \] with \(u_n(t=0)=\sin \frac {2\pi n}{k}\), with \(k\) an integer.

Find the leading term of the asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty } e^{is\left (t+\frac {t^3}{3}\right )}\,dt, \qquad s\text { is real and }s\to +\infty , \] as well as a rigorous estimate of the error.