PROBLEM SETS

Problem Set 1 (due 02/12/2012, Monday)

  1. Chapter 1 (Problems 1a, 1b, 1c, 2a, 2c, 2f).
  2. Solve y'''' - y = x + 2cosx + sinx .
  3. Solve y''' + y'' - y' - y = xex.

Problem Set 2 (due 02/14/2011, Monday)

  1. Find the eigenvalues as well as the corresponding orthonormal eigenfunctions of the Sturm-Liouville problem:
    d2u⁄dx2 = -λu, 0 < x < L, with the boundary conditions u' (0) = 0 and u(L) = 0.
  2. Find the values of (1 + √3 i ) 1/2 and i i?
  3. Chapter 2 (Problem 2).

Problem Set 3 (due 02/27/2012, Monday)

  1. Let $f(z)=z^{2}.$
    1. Find $u(x,y)$ and $v(x,y),$ the real part and the imaginary part of $f(z).$
    2. The curve given by $g(x,y)=c,$ with $c$ a constant and $g$ a real-value function, is called a level cuve of $g(x,y).$ Plot the level curves of $u(x,y)$ with $c=0,1,2,-1,-2.$ Do the same with $v.$ Can you see from your graph how a level curve of $u$ intersects a level curve of $v?$
    3. Prove that $\overset{\rightarrow}{\nabla}g(x,y)$ is a vector parallel to the normal of the level curve of $g(x,y)$ passing through $(x,y).$
    4. Prove that the way the level curves of $u$ and $v$ intersect is a consequence of the Cauchy-Riemann equations.
  2. Let $u$ satisfies the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$.
    1. Let $R$ be the disk $x^{2}+y^{2}<4$ and let $u$ satisfies the boundary condition $u=1+2\cos4\theta$ at $r=2$. Find $u$.
    2. Let $R$ be the annulus between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1.$
      Let $u$ satisfies the boundary condition $u=1+2\cos4\theta$ at $r=2$ and $u=0$ at $r=1$. Find $u$.
  3. Problem 3 in chapter 2.

Problem Set 6 ( due 03/19/2012, Monday)

  1. Problem 11a in Chapter 2.
  2. Problem 11b in Chapter 2.
  3. Find the solution of $u_{x}+(e^{2x}+y)u_{y}=0$, $u(0,y)=\dfrac{1}{1+y^{2}%};$
  4. Find the solution of $u_{x}+(2x+y)u_{y}=0,$ $u(0,y)=\dfrac{1}{4+y^{2}}.$

Problem Set 7 (due 04/02, 2012, Monday)

  1. Problem 1 in Chapter 3.
  2. Problem 16 in Chapter 2.
  3. Find the Fourier transform of 1/$\cosh x.$ Show explicitly that Fourier's inversion formula is valid for this example.
  4. Problem 17 in Chapter 2.

Problem Set 8 (due 04/09/2012, Monday)

  1. As we have learned the week before the spring vacation,
    the convolution of $f(t)$ and $g(t)$, -${\infty}$ < t < $\infty$, is defined as
    2. $C(t)\equiv\int_{-\infty}^{\infty}f(t-t^{\prime})g(t^{\prime})dt^{\prime},$
    Let $f(t)=g(t)=0$ for $t$ negative.
    (a) Prove that
    $C(t)=$ $\int_{0}^{t}f(t-t^{\prime})g(t^{\prime})dt^{\prime}$ for $t>0,$
    and that
    $C(t)=0$ for $t<0.$
    (b) Prove that
    $L_{C}(s)=L_{f}(s)L_{g}(s)$,
    where
    $L_{C}(s)$, $L_{f}(s)$ and $L_{g}(s)$ are, respectively, the Laplace
    transforms of $C(t),f(t)$ and $g(t)$.
  2. Apply the result in problem 1 to find $y(t)$ which satisfies the integral equation
    3. $1=\int_{0}^{x}\dfrac{1}{\sqrt{x-x^{\prime}}}y(x^{\prime})dx^{\prime},$ $x>0.$
  3. Find the Fourier coefficient $a_{n}$ for the function $1/(a+b\cos\theta)$,
    4. $-\pi<\theta<\pi$, where $a>b.$
  4. Let $f(\theta)=e^{\theta},$ $0<\theta<\pi$. Find the coefficients of its Fourier sine series.

    5. Problem Set 9 (due 04/18/2012)

    1. Let $f(x)=\theta,$ $0<\theta<\pi.$
      2. (a) Find the coefficients of $f(\theta)$ in the Fourier cosine series $f(\theta)=a_{0}+\underset{n=1}{\overset{\infty}{% %TCIMACRO{\dsum }% %BeginExpansion {\displaystyle\sum} %EndExpansion }}A_{n}\cos (n\theta).$
      (b) By setting $\theta$ to zero, prove from the series in (a) that $\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\cdot\cdot\cdot.$
      From the formula $\dfrac{1}{\pi}\int_{0}^{\pi}f^{2}(\theta)d\theta=a_{0}^{2}$ $+\dfrac{1}{2}\underset{n=1}{\overset{\infty}{% %TCIMACRO{\dsum }% %BeginExpansion {\displaystyle\sum} %EndExpansion }}A_{n}^{2}$ $,$ prove that $\dfrac{\pi^{4}}{96}=1+\dfrac{1}{3^{4}}+\dfrac{1}{5^{4}}+\cdot\cdot\cdot.$
      Compare how rapidly the two series in (b) and (c) converge.
    2. With the use of Laplace transform, find the solution of the differential equation $\dfrac{d^{2}x}{dt^{2}}-x=e^{-t},$ $t>0,$ which satisfies the initial conditions $x(0)=A,$ $\overset{\cdot}{x}(0)=B.$ Show explicit that your solution satisfies the initial conditions as $t\rightarrow0^{+}$. What is the value of the inverse Laplace integral for $x(t)$ at $t=0$?

    Problem Set 10 (due 04/25/2012)

    1. Let $u$ satisfy $u_{xx}+u_{yy}=0$ inside the rectangle with vertices $(0,0)$, $(a,0)$, $(a,b)$, $(0,b).$ The boundary conditions are: $u_{x}=0$ at the vertical sides of the rectangle and $u(x,0)=\cos(\pi x/a)$ and $u(x,b)=\cos(2\pi x/a).$ Find $u(x,y)$.
    2. The equation $u_{xx}+u_{yy}=u$ holds in the rectangle with vertices $(0,0)$, $(a,0)$, $(a,b)$, $(0,b).$ At the right vertical side of the rectangle, $u(a,y)=\sin(\frac{\pi y}{b})$, $0 < y < b$. The function $u$ vanishes at the other three sides of the rectangle. Find $u.$
    3. Solve for $u(x,t)$ which satisfies the heat equation $u_{t}=u_{xx}$, $t>0$, $-\infty < x < \infty$, and the initial condition $u(x,0)=f(x)$ by performing the Laplace transform with respect to $t$ first before making the Fourier transform with respect to $x$.


    Problem Set 11 (due 05/02/2012)

    1. Let $u(r,\theta)$ satisfy the Laplace equation $\nabla^{2}u=0$ inside the circle $x^{2}+y^{2}=1$, and let the boundary condition be $u(1,\theta )=f(\theta)$, where $f(\theta)$ has the Fourier series expansion:
      $f(\theta)=a_{0}+\sum_{n=1}^{\infty}(A_{n}\cos( n\theta)+B_{n}\sin( n\theta)).$
      We have shown in one of the lectures that $u(r,\theta )=a_{0}+\sum_{n=1}^{\infty}(A_{n}\cos( n\theta)+B_{n}\sin (n\theta))r^{n}.$
      Express the Fourier coefficients $a_{0},$ $A_{n},$ and $B_{n}$ above by integrals over $f(\theta)$ as explicitly given by (2.72) and (2.91) in the textbook and show that
      $u(r,\theta)=\int_{-\pi}^{\pi}d\theta^{\prime}$ $G(\theta-\theta^{\prime },r)f(\theta^{\prime})d\theta^{\prime},$ $r<1,$
      where $G(\theta,r)$ is the Green function:
      $G(\theta,r)=\dfrac{1}{2\pi}[-1+\dfrac{1}{1-e^{i\theta}r}+\dfrac {1}{1-e^{-i\theta}r}]=\dfrac{1}{2\pi}\dfrac{1-r^{2}}{1-2r\cos\theta+r^{2}}$.
    2. Let $f(\theta)=\dfrac{1}{2+\cos\theta}.$ Find $u(r,\theta)$ of problem 1 in a closed form.
    3. Find $u(x,y,t)$ which satisfies
      $u_{xx}+u_{yy}=u_{t}$, $-\infty < x,y < \infty,$
      and the boundary condition
      $u(x,y)\rightarrow0$ as $x$ or $y$ go to infinity,
      as well as the initial condition
      $u(x,y,0)=f(x,y).$
      Find the Green function for this problem.

    Problem Set 12 (due 05/07/2012)

    1. Consider the partial differential equation
      $[\dfrac{1}{r^{2}}\dfrac{\partial}{\partial r}r^{2}\dfrac{\partial}{\partial r}+\dfrac{1}{r^{2}\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\phi^{2}}% +\dfrac{1}{r^{2}\sin\theta}\dfrac{\partial}{\partial\theta}\sin\theta \dfrac{\partial}{\partial\theta}]u=V(r,\theta,\phi)u,$
      where
      $V(r,\theta,\phi)=V_{1}(r)+\dfrac{V_{2}(\phi)}{r^{2}\sin^{2}\theta}% +\dfrac{V_{3}(\theta)}{r^{2}}.$
      a. Perform separation of variables and reduce the equation into three ordinary differential equations.
      b. Prove that the form of $V(r,\theta,\phi)$ given above is the only one which allows the application of the separation of variables to the PDE above.
    2. The function $y$ is the solution of the Bessel equation
      $y"+\dfrac{1}{x}y^{\prime}+(1-\dfrac{n^{2}}{x^{2}})y=0,$
      where $n$ is a constant. Let this equation hold for $0 < y < L.$ Cast this differential equation into that of the Sturm-Liouville problem:
      $\dfrac{d}{dx}[p(x)\dfrac{dy}{dx}]+[q(x)+\lambda\omega(x)]y=0.$
      Identify $p(x),$ $q(x),\omega(x)$ and $\lambda.$ Give a set of boundary conditions which will make the eigenfunctions of this Sturm-Liouville problem orthogonal to one another. Spell out the condition of orthogonality, particularly the weighting function in this condition.
    3. Find the Green function which satisfies the partial differential equation
      $(\nabla^{2}-m^{2})G(\overrightarrow{x})=\delta^{(3)}(\overrightarrow{x}),$ all $\overrightarrow{x}.$
      The boundary condition is $G(\overrightarrow{x})$ vanishes at infinity. The $\overrightarrow{x}$ space is of three dimensions.