PhD Preliminary Qualifying Examination Applied Mathematics August 13, 2001

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 \(Ux=b\).

The dilation equation is \[ \phi (x)=\sum _k c_k\phi (2x-k). \] Show that the function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere} \end {cases} \] satisfies the dilation equation. What are the values of \(c_k\)? Are the wavelets formed from \(N_2(x)\) an orthogonal set? Why or why not?

1. State the Riesz Representation Theorem.
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

1. What is the weak formulation for the boundary value problem \(u''=f(x)\) subject to boundary conditions \(u(0)=u(1)=0\)?
2. Let \(\phi (x)=N_2(x+1)\) where \(N_2(x)\) is the piecewise linear function given in Problem A.2. Divide the unit interval \([0,1]\) interval \(N\) uniform subintervals with endpoints at \(x_k=\frac {k}{N}\), \(k=0,\ldots ,N\). Use \(\phi (x)\) to define a finite element basis for continuous piecewise linear functions on the unit interval which interpolates function values at \(x_k\). Suppose \(u_k=u(x_k)\) is known. Express \(u(x)\) in terms of \(\phi (x)\).
3. Use the Galerkin method to find equations for \(u_k=u(x_k)\) that approximately solve the boundary value problem in part (a), using the piecewise linear finite element basis.

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

Evaluate \[ \int _0^\infty \frac {\sqrt {x}}{x^2+1}\,dx. \]

Remark: For this problem you may use without proof that \[ \frac {1}{2\pi }\int _{-\infty }^{\infty }e^{i\mu x}\,d\mu =\delta (x). \]

1. Suppose \(f(x)\) has the Fourier transform \(F(\mu )\). Evaluate \[ \int _{-\infty }^{\infty } f(x)\overline {f(x-k)}\,dx \] in terms of \(F(\mu )\).
2. Use this identity to prove that the set \(\{\operatorname {sinc}(t-k)\}_{k=-\infty }^{\infty }\) forms an orthonormal set.

1. Derive the “diffusion” or “heat” equation in several spatial dimensions.
2. In the derivation of this equation, it is routinely assumed that \[ \frac {d}{dt}\int _\Omega u(x,t)\,dx=\int _\Omega \frac {\partial u}{\partial t}\,dx. \] Under what conditions is this interchange of order valid? Prove your result. Hint: Examine the function \[ w(t)=\int _0^t\left (\int _\Omega \frac {\partial u(x,t)}{\partial t}\,dx\right )\,dt. \]

Find an integral representation for the solution of the telegrapher’s equation \[ u_{tt}+au_t+bu=u_{xx} \] with \(u(0,t)=f(t)\) and \(u_x(0,t)=g(t)\) specified for all time.

1. Find the leading order behavior of \[ G(x)=\int _0^\infty \cos \left (x\left (\frac {t^3}{3}-t\right )\right )\,dt \] for large \(x\).
2. Outline the method to find more terms in the asymptotic expansion of \(G(x)\).

Useful information: \[ \int _{-\infty }^{\infty }\cos (ax^2)\,dx =\int _{-\infty }^{\infty }\sin (ax^2)\,dx =\sqrt {\frac {\pi }{2a}}. \]