PhD Preliminary Qualifying Examination: Applied Mathematics Jan. 3, 2013

The \(n\) data points \((x_i,y_i)\), \(i=1,2,\ldots ,n\) are believed to lie on an exponential curve \[ y_i=A\exp (\lambda x_i). \] Estimate \(\lambda \).

1. Specify the weak formulation for the differential equation \[ u''+\lambda \delta (x-1/2)u=f(x) \] subject to boundary conditions \(u(0)=u(1)=0\).
2. Find all values of \(\lambda \) for which this differential equation with \(f(x)=0\) has a nontrivial solution and verify that the corresponding solution is a weak solution.
3. Suppose \(\lambda \) is such that this differential equation with \(f(x)=0\) has only the trivial solution. Find the solution with \(f(x)=1\).

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

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt=\cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \). Why is a unique solution guaranteed to exist?
2. Show that there is no solution when \(\lambda =2/\pi \).
3. Show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=f_p(x)+\alpha f_h(x). \] Determine \(f_h(x)\).

1. Describe the Gram-Schmidt orthogonalization procedure for a set of \(n\)-vectors \(\{u_1,u_2,\ldots ,u_k\}\).
2. Suppose \(k<n\) and that the vectors \(\{u_1,u_2,\ldots ,u_k\}\) are linearly independent. Show that the Gram-Schmidt procedure is equivalent to factoring the matrix \(U\) with column vectors \(\{u_1,u_2,\ldots ,u_k\}\) as \(U=QR\) where \(R\) is triangular. What is the structure of \(Q\)?
3. Use this factorization of \(U\) to find the least-squares solution of \(Ux=b\).

1. Formulate and derive the argument principle [which determines the difference between the number of zeros \((N_0)\) and poles \((N_\infty )\) of an analytic function \(f(z)\)].
2. Applying this principle, determine the number of zeros located inside the first quadrant \(\{z=x+iy:\ x>0,\ y>0\}\) of the function \(f(z)=z^5+1\).

Find the image of the half-strip \[ \{z=x+iy:\ x>0,\ 0<y<1\} \] under the mapping \(w=1/z\).

Calculate the integral \[ I=\int _0^\infty \frac {x^\alpha }{1+x}\,dx, \] where \(\alpha \) is a real number for which the integral converges.

Formulate and derive the uncertainty principle. Is its inequality optimal?

Find at least three terms of the asymptotic expansion of the integral \[ I(s)=\int _0^1 \ln t\, e^{ist}\,dt, \qquad s\text { is real and }s\to +\infty . \]