Department of Mathematics University of Utah\ 1.5 ] Written Qualifying Examination in Applied Mathematics

Prove the following statement. In an arbitrary real Hilbert space \(H\) an orthonormal system \((\phi _1,\phi _2,\ldots ,\phi _n,\ldots )\) is complete if and only if the equality \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }\langle f,\phi _n\rangle ^2 \] holds for any \(f\in H\).

Consider the integral equation \[ u(x)=f(x)+\frac {1}{2}\int _0^1 xtu(t)\,dt,\qquad \text {for }0\le x\le 1 \] where \(f(x)\) is a prescribed function.

1. Show that the equation has at most one solution.
2. Derive a formula for the solution.
3. Calculate the solution for the special case \(f(x)=5x/6\).

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

Using complex integration, evaluate the integral \[ I=\int _0^{2\pi }\frac {1}{1+\frac {1}{3}\cos \phi }\,d\phi . \]

Evaluate by contour integration \[ I=\int _0^\infty \frac {x\sin 3x}{4+x^2}\,dx. \]

Using conformal mapping find the electrostatic potential \(\phi (x,y)\) in the region between the circumferences \[ \gamma =\{z:\ |z|=1\}\qquad \text {and}\qquad \Gamma =\{z:\ |z-1|=2.5\} \] satisfying the boundary conditions \[ \phi \equiv 0\text { on }\gamma ,\qquad \phi \equiv 1\text { on }\Gamma . \]

The so called “Brachistochrone problem” admits such formulation. “A particle is sliding along a trough (in a vertical plane from the point \((0,0)\) to some given point \((x_1,y_1)\) \((x_1>0,\ y_1<0)\) under the effect of gravity without friction (having zero initial velocity). Find the form \(y=Y(x)\) of the trough which minimizes the time of descent of this particle.”

1. Find the functional that ought to be minimized.
2. Using this functional, write the boundary value problem that ought to be solved for finding the solution of the “Brachistochrone problem”. You do not need to solve this boundary value problem.