PhD Preliminary Qualifying Examination Applied Mathematics August 21, 2006

1. State the Riesz Representation Theorem (a proof is not required).
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

1. What is the weak formulation for the boundary value problem \(u''=f(x)\) subject to boundary conditions \(u(0)=u(1)=0\)?
2. Let \(\phi (x)=N_2(x+1)\) where \(N_2(x)\) is the piecewise linear function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere}. \end {cases} \] Divide the unit interval \([0,1]\) into \(N\) uniform subintervals with endpoints at \(x_k=\frac {k}{N}\), \(k=0,\ldots ,N\). Use \(\phi (x)\) to define a finite element basis for continuous piecewise linear functions on the unit interval which interpolates function values at \(x_k\). Suppose \(u_k=u(x_k)\) is known. Express \(u(x)\) in terms of \(\phi (x)\).
3. Use the Galerkin method to find equations for \(u_k=u(x_k)\) that approximately solve the boundary value problem in part (a), using the piecewise linear finite element basis.

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

The \(n\) data points \((x_i,y_i)\), \(i=1,2,\ldots ,n\), are believed to lie on an exponential curve \(y_i=A\exp (\lambda x_i)\). Formulate a linear regression problem to estimate \(\lambda \).

The Legendre polynomials are given by \[ P_n(x)=\frac {1}{2^n n!}\frac {d^n}{dx^n}(x^2-1)^n, \] and are eigenfunctions of the differential operator \(Lu=-((1-x^2)u')'\) on the interval \(-1<x<1\).

1. Find the eigenvalue \(\lambda _n\) corresponding to the eigenfunction \(P_n(x)\).
2. In what sense, if any, are the functions \(\{P_n(x)\}\) complete?
3. Prove or disprove that \(L\) is a positive definite operator.

Evaluate the integral \[ \int _0^\infty \frac {\sin x}{x}\,dx. \] Explain your reasoning.

Solve the Laplace equation \(\Delta \phi =0\) subject to the boundary conditions: \[ \begin {aligned} \phi& =a &&\text {on } \{z: |z|=2,\ z\ne 2\},\\ \phi& =b &&\text {on } \{z: |z-1|=1,\ z\ne 2\}. \end {aligned} \] (\(a\) and \(b\) are real constants; \(z=x+iy\).)

Formulate and prove the Fundamental theorem of Algebra.

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha \,dx}{(x-1)^2+1}. \] (where \(\alpha \) is a constant, such that the integral converges: \(|\alpha |<1\)).

Find the two-term asymptotic expansion of the integral \[ I(s)=\int _0^3 \frac {e^{s(-t^2+4t-3)}}{5+t}\,dt, \] \(s\) is real, and \(s\to +\infty \).