UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Applied Mathematics August 15, 2018

Consider the nonlinear integral equation \[ f(x)-\lambda \int _a^b k(x,y;f(y))\,dy=\phi (x) \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on \([a,b]\), \(k\) is continuous on \(C=[a,b]\times [a,b]\times \mathbb {R}\), and \(k\) satisfies a Lipschitz condition of the form \[ \lvert k(x,y;z_1)-k(x,y;z_2)\rvert \le L\lvert z_1-z_2\rvert \] in its ‘functional argument’. Show that the nonlinear integral equation has a unique solution, \(f\), for any \(\lambda \) such that \[ \lvert \lambda \rvert <\frac {1}{L(b-a)}. \]

Let \(T:X\to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \ne 0\), the null space of \(T_\lambda =T-\lambda I\) is finite dimensional.

Let \(a\in \ell ^2\setminus \{0\}\). Consider the bounded linear operator \(T:\ell ^2\to \ell ^2\) defined by \[ Tu=\langle u,a\rangle a. \]

1. Show that the operator \(T\) is compact and self-adjoint.
2. Use the Fredholm Alternative Theorem to find a simple condition on \(a\) guaranteeing that the equation \[ (T-I)u=v \] has a unique solution \(u\) for all \(v\in \ell ^2\). What is the solution?

Consider the operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \((Tx)(t)=tx(t)\).

1. Show that \(T\) is self-adjoint and satisfies \(\langle x,Tx\rangle \ge 0\) for every \(x\in L^2[0,1]\).
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Let \(\delta \in \mathcal {D}'(\mathbb {R})\) denote the ‘delta function’. Find \(u\in \mathcal {D}'(\mathbb {R})\) that solves the following linear partial differential equation: \[ \partial ^2 u=\delta . \] Is the solution unique?

Hint: You can solve this equation by guessing the distribution and then verifying that is solves the equation.

Function \(f(x)\) (see below) is expanded in powers of \(x-x_0\), i.e. in the Taylor or Laurent series with center \(x_0\) (\(x\) is a real variable, \(x_0\) is a real number). Find the radius of convergence of this series. \[ \begin {array}{rcll} \text {(a)}\quad f(x)&=&\dfrac {\sin x^2}{(x-3)^2(x^2+1)},& x_0=3,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x},& x_0=5,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x}{\sin x+3},& x_0=8. \end {array} \]

Solve Laplace’s equation \(\phi _{xx}+\phi _{yy}=0\) in the domain \[ D=\{(x,y)\in \mathbb {R}^2:x^2+y^2>1,\ (x+2)^2+y^2<9\} \] with boundary conditions: \(\phi (x,y)=a\) when \(x^2+y^2=1\) and \(\phi (x,y)=b\) when \((x+2)^2+y^2=9\).

Calculate the following integrals (\(a\) and \(b\) are positive parameters) \[ \begin {array}{rcl} \text {(a)}\quad I(a)&=&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx,\\[2.2ex] \text {(b)}\quad I(a,b)&=&\displaystyle \int _{-\infty }^{\infty } \frac {\cos ax-\cos bx}{x^2}\,dx. \end {array} \]

\(F(\omega )\) is the Fourier transform of \(f(x)\) (which is continuous and not identical zero), \(x\) and \(\omega \) are real variables. Explain why it is impossible that both functions \(f(x)\) and \(F(\omega )\) have finite support.

Consider integral \[ I(s)=\int _C \frac {e^{s z^2}}{z^2-1}\,dz, \] \(C\) is the vertical line \(\operatorname {Re} z=3\), from \(z=3-i\infty \) to \(z=3+i\infty \), with large (real) parameter \(s\). Find the three-term asymtotic expansion (approximation with three non-zero terms) of this integral as \(s\to +\infty \).