PhD Preliminary Qualifying Examination: Applied Mathematics (6710/20) August, 2003

Suppose that \(A\) is an \(n\times n\) matrix with \(n\) distinct eigenvalues.

1. How are the eigenvalues of \(A\) and \(A^*\) related?
2. Show that the eigenvectors of \(A\) and \(A^*\) form a biorthogonal set, that is, if \(\{\phi _i\}_{i=1}^n\) and \(\{\psi _i\}_{i=1}^n\) are the eigenvectors of \(A\) and \(A^*\), appropriately ordered, then \(\langle \phi _i,\psi _j\rangle =0\) if \(i\ne j\).
3. Establish that \(\langle \phi _i,\psi _i\rangle \ne 0\).

1. Determine the conditions on \(f(x)\), \(\alpha \) and \(\beta \) for which there solutions of \[ \frac {d^2u}{dx^2}=f(x),\qquad u(0)=\alpha ,\qquad u'(1)-u(1)=\beta . \]
2. Find the “best” solution of \[ \frac {d^2u}{dx^2}=1,\qquad u(0)=1,\qquad u'(1)-u(1)=1. \]

Consider the non-local differential operator \[ Lu=Du''-u(g(x)) \] with boundary conditions \(u(0)=u(1)=0\) and where \(g(x)\) is a monotone increasing function with \(g(0)=0\), \(g(1)=1\), and \(g(x)<x\) for \(0<x<1\).

1. Find the adjoint of the operator \(L\).
2. Prove that the operator \(L\) is invertible.
3. Devise an iteration scheme that could be used to solve \(Lu=f(x)\). Under what conditions on \(D\) is this iteration scheme a contraction mapping?

Find the weak formulation for the boundary value problem \[ u''-\delta (x)u=1,\qquad u(-1)=u(1)=0, \tag {0.1} \] and then solve the problem. Verify that your solution is a weak solution. Why is it not a strong solution?

Use piecewise linear finite element basis functions with uniform grid size \(h\) and the Galerkin method to find an approximate representation for the differential equation \(a\frac {d^2u}{dx^2}+b\frac {du}{dx}+u(x)=f(x)\), with \(u(0)=0\), \(u(1)=0\), on the interval \([0,1]\). (It is only necessary to find the approximating equations, not to solve them.)

1. Evaluate the following real integrals by contour integration: \[ \int _0^{2\pi }\frac {d\theta }{(5-3\sin \theta )^2}. \]
2. Let \[ e^{t/2(z-1/z)}=\sum _{n=-\infty }^{\infty }J_n(t)z^n. \] Show from the definition of Laurent series and properties of complex integration that the Bessel function \[ J_n(t)=\frac {1}{\pi }\int _0^\pi \cos (n\theta -t\sin \theta )\,d\theta . \]

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 \(\gamma \ne 0\).

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2},\qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=1. \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\sinh sx}{s\sinh sl}. \]
2. Obtain the solution \(\phi (x,t)\) by inverting the Laplace transform using a Bromich integral and show that it can be expressed as the infinite series \[ \phi (x,t)=\frac {x}{l}+\sum _{n=1}^{\infty }\frac {2(-1)^n}{n\pi } \sin \left (\frac {n\pi x}{l}\right )\cos \left (\frac {n\pi t}{l}\right ). \]

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.