PhD Preliminary Qualifying Examination: Applied Mathematics August, 2010

1. State and prove the Fredholm Alternative Theorem for a linear transformation \(A:\mathbb {R}^n\to \mathbb {R}^m\).
2. Find and give a graphical interpretation of the pseudo-inverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2 \end {pmatrix}. \]

This problem examines features of the differential operator \[ Lu=u'' \] subject to boundary conditions \(u(-1)+u(1)=0\) and \(u'(-1)=u'(1)\).

1. What is the adjoint of this operator? Is it self-adjoint?
2. What is the nullspace of the operator and what is the nullspace of the adjoint operator?
3. Under what conditions on \(f\), \(a\), and \(b\) does the boundary value problem \[ u''=f,\qquad u(-1)+u(1)=a,\qquad u'(-1)-u'(1)=b \] have a solution?
4. Find the smallest least squares solution for the boundary value problem \[ u''=1,\qquad u(-1)+u(1)=1,\qquad u'(-1)-u'(1)=1. \] What equation does this least squares solution satisfy?

1. Find all the eigenvalues and eigenfunctions for the operator \[ (Lu)(x)=\int _0^1 xyu(y)\,dy. \]
2. Do these eigenfunctions form a complete set? If so, how do you know (state a relevant theorem) and on what Hilbert space, and if not, why not?

1. Find the weak formulation for the boundary value problem \[ u''-\delta (x)u=1 \] subject to boundary conditions \(u(-1)=u(1)=0\).
2. Find the solution to this boundary value problem.

Let \(P_n(x)\), \(n=0,1,\cdots \), be the Legendre polynomials, \[ P_n(x)=\frac {1}{2^n n!}\frac {d^n}{dx^n}(x^2-1)^n, \] which are also eigenfunctions for the linear differential operator \[ Lu=-\frac {d}{dx}\left ((1-x^2)\frac {du}{dx}\right ) \] on the interval \(-1\le x\le 1\).

1. Determine the eigenvalue associated with \(P_n\)
2. Show that the Legendre polynomials are orthogonal.
3. Suppose \(f(x)\) is square integrable on the interval \(-1<x<1\). Use that the Legendre polynomials are orthogonal to prove that \[ g(x)=\sum _{n=0}^{\infty }\alpha _nP_n(x) \] where \[ \alpha _n= \frac {\int _{-1}^{1}f(x)P_n(x)\,dx}{\int _{-1}^{1}P_n^2(x)\,dx}, \] is also square integrable.

For a function \(f(z)\) which is analytic within a simple, nonintersecting, closed curve \(C\) a corollary of the Cauchy integral formula implies that for any point \(z\) inside \(C\), \[ f'(z)=\frac {1}{2\pi i}\int _C\frac {f(\xi )}{(\xi -z)^2}\,d\xi . \] Use this to prove Liouville’s theorem that any function which is bounded and analytic in the entire complex plane must be a constant, and hence prove the fundamental theorem of algebra that an \(n\)th order polynomial \(P_n(z)\) has \(n\) roots (counting multiplicity) in the complex plane.

The fourth quadrant \(x\ge 0\), \(y\le 0\) is occupied by an impermeable solid, around which air flows. Treating the flow as incompressible and irrotational use conformal mapping to map to a simpler problem and give one non-trivial solution for the complex potential \(f(z)\) in the original coordinates. Find the equation in polar coordinates for the streamline passing through \(z=-1\). What happens to the fluid velocity as one approaches the corner?

Use antiderivatives to evaluate the contour integral \[ I=\int _C\frac {1}{z+i} \] where \(C\) is a contour (possibly with loops) starting at \(z=0\) and ending at \(z=1\). Express your answer in the form \(I=a+ib\) where \(a\) and \(b\) are real. The answer is multivalued. Explain the interpretation of the different values, and account for the difference between successive values using the residue theorem.

The operator \(Lu=-u''\) on \(L^2(-\infty ,\infty )\) has a Green’s function defined by \(-G''-\lambda G=\delta (x-\xi )\) given by \[ G(x,\xi ;\lambda )=-\frac {e^{i\sqrt {\lambda }|x-\xi |}}{2i\sqrt {\lambda }}. \] Assume the representation \[ -\frac {1}{2\pi i}\int _{C_\infty }G(x,\xi ;\lambda )\,d\lambda =\delta (x-\xi ) \] and show how it leads to the Fourier Transform Theorem that \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-i\mu x}F(\mu )\,d\mu , \qquad F(\mu )=\int _{-\infty }^{\infty }e^{i\mu x}f(x)\,dx. \]

Use the method of steepest descent to estimate the integral \[ I(s)=\int _0^1 e^{s(-z^3+iz^3)}\,dz \] for large positive real values of \(s\).