PhD Preliminary Qualifying Examination: Applied Mathematics (6710/20) January 2004

1. Prove that \(x\delta (x)=0\) in the sense of distribution.
2. Formulate the differential equation \(xu'-u=0\) as a problem in the sense of distribution and find its general solution, also in the sense of distribution.

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 \(\lambda \).

The Chebyshev functions are given by \(T_n(x)=\cos (n\cos ^{-1}x)\), and are eigenfunctions of the differential operator \[ Lu=-\sqrt {1-x^2}\left (\sqrt {1-x^2}u'\right )' \] on the interval \(-1<x<1\).

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

The “best” least squares solution of the boundary value problem \(u''+u=0\), \(u(0)=\alpha \), \(u(\pi )=0\) is the unique exact solution of what problem?

Prove that if \(L\) is a bounded linear operator in a Hilbert Space \(H\), then the adjoint operator \(L^*\) exists.

1. Using Jordan’s lemma show that \[ \int _0^\infty \frac {\cos ax}{1+x^2}\,dx=\frac {\pi }{2}e^{-a}. \]
2. Show that \[ \int _0^{2\pi }\frac {d\theta }{A+B\sin \theta }= \frac {2\pi }{\sqrt {A^2-B^2}} \] where \(A^2>B^2\), \(A>0\).

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

Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \] Show that \(r=a\) is a streamline. Determine the asymptotic velocity when \(z\to \infty \). Calculate the stagnation points. Sketch the flow when \(\gamma =0\). Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \ne 0\).

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Consider the bilinear transformation \[ \omega =\frac {az+b}{cz+d},\qquad ad-bc\ne 0. \]

1. Show that this transformation maps circles to circles.
2. Show that the bilinear transformations form a group.
3. In the case \((a,b,c,d)=(1,-i,1,i)\) show that the upper-half plane is mapped on to the unit disc.