PhD Preliminary Qualifying Examination: Applied Mathematics (6710/20) August, 2005

Let \(A\in \mathbb {R}^{n\times m}\), \(B\in \mathbb {R}^{r\times m}\), and \(b\in \mathbb {R}^n\) be given. Assume \(R(B^*)=N(A)\). For fixed \(\mu \ge 0\), consider the problem \[ \min _{x\in \mathbb {R}^m} \|Ax-b\|_2^2+\mu \|Bx\|_2^2. \]

1. Find the necessary conditions (analogous to the normal equations for the standard least-squares problem) and prove that the necessary conditions have a solution when \(\mu =0\).
2. Prove that the necessary conditions have a unique solution if \(\mu >0\).

Consider the distribution \(g\) defined by \(\langle g,\phi \rangle =\int _{-\infty }^{\infty } e^{-|x|}\phi (x)\,dx\), for all test functions \(\phi \).

1. Calculate the second derivative \(g''\) explicitly. Prove \(g''\) is a distribution of order zero.
2. Find a differential equation for which \(g\) is a Green’s function.

Let \(K:H\to H\) be a compact linear operator on the infinite-dimensional Hilbert space \(H\).

1. Prove that \(K^{-1}\), if it exists, is not bounded.
2. Given \(T>0\), let \(H=L^2(0,T)\). For \(u\in H\), define \[ (Ku)(t)=\int _0^T e^{-st}u(s)\,ds,\qquad 0\le t\le T. \] Prove that \(K:H\to H\) is a compact operator. What are the consequences of this fact for the utility of the Laplace transform as a numerical method?

Consider the fourth-order differential equation \[ \begin {aligned} u^{(4)} &= f, && \text {on } (0,1),\\ u(0) &= u(1), && u'(0)=u'(1),\\ u''(1) &= -2, && u^{(3)}(1)=1. \end {aligned} \]

1. Find the least restrictive constraint(s) on \(f\in L^2(0,1)\) under which a solution to the problem exists.
2. With fixed \(f\) chosen such that a solution exists, is the solution unique? If the solution is unique, prove it. Otherwise provide a counterexample.

Prove that the nonlinear integral equation \[ u(x)-\int _0^1 \frac {u^2(y)}{4+x^2-y}\,dy=\frac {x^3}{3},\qquad 0\le x\le 1, \] has exactly one continuous solution \(u\) satisfying \(\max _{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure for obtaining an approximation to this solution.

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(0<\alpha <1\).

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2}, \qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=f(t). \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\widehat {F}(s)\sinh sx}{\sinh sl} \] where \(\widehat {F}(s)\) is the Laplace transform of \(f(t)\).
2. Suppose that \(\phi _0(x,t)\) is the solution when \(f(t)=1\). Show that the solution for general \(f\) is given by \[ \phi (x,t)=\int _0^t \frac {\partial \phi _0}{\partial t'}(x,t')f(t-t')\,dt'. \]

Discuss the flow pattern associated with the complex potential \[ \Omega (z)=Q_0z+\frac {\overline {Q_0}a^2}{z}+\frac {i\gamma }{2\pi }\log z \] with \(a>0\), \(\gamma \) real and \(Q_0=U_0+iV_0\). Sketch the flow when \(\gamma =0\). Given that the complex force on a cylindrical obstacle with boundary \(C\) is \[ \overline {F}=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz, \] where \(\rho \) is the density of fluid, determine the lift when \(\gamma \ne 0\).

1. Introducing the discrete Fourier transform \(U(k,t)=\sum _{n=-\infty }^{\infty }e^{ikn}u_n(t)\), \(0\le k<2\pi \), show that the general solution to the discretized heat equation \[ \frac {du_n}{dt}=\frac {1}{h^2}\{u_{n+1}-2u_n+u_{n-1}\},\qquad -\infty <n<\infty , \] can be written in the form \[ u_n(t)=\int _0^{2\pi } U(k,0)e^{-\epsilon (k)t}e^{-ikn}\,\frac {dk}{2\pi } \] and calculate the function \(\epsilon (k)\).
2. Using the fact that \(U(k,t)\) and \(u_n(t)\) form a transform pair for fixed \(t\), prove the identity \[ \sum _{n=-\infty }^{\infty }e^{i(k-k')n}=2\pi \delta (k-k'). \] Hence, find \(u_n(t)\) given the initial data \(u_n(0)=\sin (2\pi n/p)\) for some fixed integer \(p\).

Suppose that \(\psi _+(z)\) is analytic on the upper half plane and \(\psi _-(z)\) is analytic on the lower half plane such that on the real axis \[ \psi _+(x)-\psi _-(x)=f(x) \] Assuming that the integrable function \(f(x)\) can be analytically extended in a neighborhood of the real axis, show that a solution to this problem is given by the Cauchy integral \[ \Psi (z)=\frac {1}{2\pi i}\int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -z}\,d\xi \] with \(\psi _\pm (x)=\lim _{\epsilon \to 0+}\Psi (x\pm i\epsilon )\). Hence, establish that \[ \psi _+(x)+\psi _-(x)=\frac {1}{\pi i}\operatorname {PV} \int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -x}\,d\xi \] where \(\operatorname {PV}\) denotes the principal value.