Written Qualifying Examination in APPLIED MATHEMATICS September 12, 1994

Prove that an orthonormal sequence \(\{\phi _1,\phi _2,\ldots ,\phi _n\}\) in a Hilbert space \(\mathcal {H}\) is complete if and only if Parseval’s relation \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }|\langle f,\phi _n\rangle |^2 \] holds for all vectors \(f\in \mathcal {H}\).

Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\) matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).

1. State the conditions that define \(A'\).
2. Show that \(A'\) always exists.
3. Construct \(A'\) explicitly for the \(2\times 2\) matrix \[ A=\begin {pmatrix} 1 & 1\\ 0 & 0 \end {pmatrix}. \]

Consider the boundary value problem \(u''=f(x)\), \(u'(0)=0\), \(u(1)=0\) where \(f(x)\) is a prescribed function.

1. Show that the problem has at most one solution.
2. Show that the solution can be written as \[ u(x)=\int _0^1 k(x,y)f(y)\,dy \] for all continuous functions \(f(x)\) where \[ k(x,y)= \begin {cases} x-1, & \text {for }0\le y\le x\le 1,\\ y-1, & \text {for }0\le x\le y\le 1. \end {cases} \]

Find the general solution, in the sense of distribution theory, of the differential equation \[ x^2u'(x)=\delta (x^2-1). \]

Construct the Green’s function for the equation \(u''(x)=f(x)\) on the interval \(0\le x\le 1\) if \(u\) satisfies the auxiliary conditions \[ \int _0^1 xu(x)\,dx=0\qquad \text {and}\qquad u(0)+u'(1)=0. \]

Find the adjoint operator, its domain and the least squares solution of the problem \(Lu\equiv 4u''+u=e^x\) for \(0\le x\le \pi /2\) if \(u(0)=\alpha \) and \(u(\pi /2)=\beta \).

Find the spectral representation of the delta function for the operator \(L\) defined by \(Lu=-u''\) on \(x\in (-\infty ,0]\) with boundary condition \(u(0)=0\). Then solve the problem \(Lu=f\) if \(f(x)=1\) for \(-2\le x\le -1\) and \(f(x)=0\) elsewhere.

Find the Green’s function for the Poisson equation \(\Delta u=f\) in the infinite strip \(0\le y\le 1\), \(-\infty <x<\infty \) with boundary data \(u(x,0)=0\) and \(u(x,1)=1\). Then solve the problem (in quadrature) if \(f(x,y)=y(1-y)e^{-x^2}\).

Describe quantitatively the diffusion of 3 units of a diffusing material which is placed at the initial moment \(t=0\) at the origin of an infinite two-dimensional plane. Assume that the plane has unit conductivity.