PhD Preliminary Qualifying Examination: Applied Mathematics Jan. 7, 2010

Let \[ A= \begin {pmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end {pmatrix}. \]

1. Find the eigenvalues and eigenvectors of \(A\), and the range of the function \[ \phi (x)=\frac {\langle Ax,x\rangle }{\langle x,x\rangle } \] where \(x=(x_1,x_2,x_3)\) is a real-valued vector.
2. Compute \(\exp (A)\).
3. Consider the equation \[ Ax=\mu x \] Find all solutions for \(\mu =1\) and for \(\mu =3\).

Consider linear operator \[ H=-\frac {d^2}{dx^2}, \] acting on \(\psi (x)\in L^2[0,\pi ]\) with periodic boundary conditions \(\psi (0)=\psi (\pi )=0\). Find the eigenvalues and eigenfunctions of \(H\). Show that the eigenvectors of \(H\) corresponding to distinct eigenvalues are orthogonal. Are the eigenvectors of \(H\) complete, and if so, in what sense?

(b) Solve the equation \[ H\psi (x)=x^2, \] representing the solution as an expansion in eigenfunctions of \(H\).

Let \[ f(x)= \begin {cases} \frac {100}{n}, & x\ \text {rational},\ x=\frac {m}{n},\\ -2, & x\ \text {irrational}. \end {cases} \] Does the Riemann integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Does the Lebesgue integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Why is the Lebesgue integral used in defining the Hilbert space \(L^2[0,1]\), and not the Riemann integral?

Let \[ f(x)=\operatorname {signum}(x)= \begin {cases} 1, & x>0,\\ 0, & x=0,\\ -1, & x<0. \end {cases} \]

1. Find its first and second derivatives using the theory of distributions.
2. Compute \[ \begin {aligned} I_0&=\int _{-1}^{\infty }\log (x+10)f(x)\,dx,\\ I_1&=\int _{-1}^{\infty }\log (x+10)f'(x)\,dx,\\ I_2&=\int _{-1}^{\infty }\log (x+10)f''(x)\,dx. \end {aligned} \]

Using Green’s functions, solve the problem \[ \frac {d^2u}{dx^2}=\frac {1}{1+x^2},\qquad u(-1)=u(1)=0 \] for \(u(x)\), \(x\in [-1,1]\) (obtain an integral representation for the solution, but do not evaluate the integral).

Find a solution \(u(x,y)\) of Laplace’s equation on the domain \(-\infty <x<\infty \), \(0<y<\infty \) for which \(u(x,0)=x^{1/2}\) for \(0<x<\infty \). What is \(u(x,0)\) for \(-\infty <x<0\)?

Use Jordan’s Lemma (and describe how Jordan’s Lemma is used) to evaluate the integral \[ I=\int _0^\infty \frac {\cos ax}{x^2+1}\,dx. \]

Use Fourier transforms to 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.

Use the \(z\)-transform to 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 order 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.