Problem Sets

Problem Set 1 (due Sep. 11, Mon)

  1. Chapter 1, Prob 1a, 1b.
  2. Chapter 1, Prob 2b, 2d, 2e.
  3. Express the general solution of
    $(\dfrac{d^{3}}{dx^{3}}-1)y=e^{-e^{x}}$
    with incomplete Gamma fuctions.

Problem Set 2 ( due Sep 18, Mon)

  1. Express the general solution of
    $(\dfrac{d^{3}}{dx^{3}}-1)y=e^{-e^{x}}$
    with incomplete Gamma functions.
    1. Consider the (inhomogenious) Schrodinger equation with a source function $\rho:$
      $(i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})\Psi(x,t)=\rho(x,t),$ \ $-\infty < x < \infty,$ $t>0$
      where $\Psi$ vanishes at $x=\pm\infty$ and satisifes the initial condition
      $\Psi(x,0)=f(x).$
      Solve this problem with the use of the Green's function method.
    2. $\Psi(x,t)$, the wave function of an electron, satisfies
      $i\dfrac{\partial\Psi}{\partial t}-\dfrac{\partial^{2}\Psi}{\partial x^{2}}=U(x,t)\Psi,$
      where $U=U(x,t)$ is the potential. Convert this equation into an integral equation.

Problem Set 3 (due Sep 25, Mon)

  1. Problem 4a, 4c, and 4d in Chapter 2.
  2. Evaluate $I=\int_{0}^{\infty}\dfrac{\cos x}{1+x^{2}}dx.$

Problem Set 4 (due Oct 2, Mon)

  1. Let $y$ satisfy the equation

    $y"+x^{2}y=0$

    and the initial conditions $y(x_{0})=1,$ $y^{\prime}(x_{0})=0$ where $x_{0}>0.$

    1. Find the WKB approximation of $y(x)$ for $x>x_{0}$. For what values of $x$ do you expect it be a good approximation?
    2. Use the computor to obtain the numerical values of $y(x)$ as a function of $x$ for $x_{0}=1,5,10.$ Compute also the numerical values of the WKB approximation of $y(x)$. Compare the computor result with the approximate result.
  2. Problem 4, Chapter 7.

Problem Set 5 (due Oct 11, Wed)

  1. Problem 10, Chapter 7.

Problem Set 6 (due Oct 16, Mon)

  1. Find the leading term for the following integrals for $\lambda>>1$:
    1. $\int_{0}^{\pi/2}e^{-\lambda\cos t}dt,$
    2. $\int_{-1}^{1}e^{\lambda t^{5}}dt,$
    3. $\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$
    Use the computer to evalaute the numerical values of these integrals and compare them with the numerical values of their leading terms as a function of $\lambda$.

Problem Set 7 (due Oct 23, Mon)

  1. Compute the leading asymptotic term of the following integrals:
    1. $I(\lambda)=\int_{0}^{\infty}e^{-\lambda x}e^{-1/x^{3}}dx,$

    2. $I(\lambda)=\int_{-\infty}^{\infty}e^{-i\lambda x^{3}-x^{4}}dx,$

      where $\lambda$ is very large.

  2. The Bessel function of order $\nu,$ denoted by $J_{\nu}(z),$ has the integral representation

    $J_{\nu}(x)=\dfrac{(x/2)^{\nu}}{\sqrt{\pi}\Gamma(\nu+1/2)}\int_{-\pi/2}^{\pi/2}e^{ix\sin\phi}(\cos\phi)^{2\nu}d\phi.$

    If $J_{\nu}(x_{0})=0,$ we call $x_{0}$ a zero of the Bessel function $J_{\nu}(x).$ Determine these zeroes of very large magnitude.

  3. Find the leading asymptotic form of the integral

    $I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda x}e^{ix^{4}/4}dx,$ $\ \lambda>>1.$

    Compare this asymptotic form with the numerical value of the integral.

Problem Set 8 (due Oct 30th, Mon)

    1. Determine the unshaded regions of $f(z)=e^{iz^{5}}$ in the infinity of which $f(z)$ vanishes.
    2. Express $J\equiv\int_{-\infty}^{\infty}e^{ix^{5}}dx$ by Gamma functions.
    3. Find the values of $\int_{-\infty}^{\infty}\cos x^{5}$ $dx$ and $\int_{-\infty}^{\infty}\sin x^{5}$ $dx.$

  2. Find the leading asymptotic form of the integral

    $I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda(x-x^{5}/5)}dx,$ $\lambda >> 1.$

  3. Prob 1 of Chapter 8.

Problem Set 9 (due Nov 6th, Mon)

  1. Find both $y_{in}$ and $y_{out}$ for the equation

    $\epsilon y^{"}-y^{\prime}+(1+x^{3})y=0,$   $0$ < $x$ < $1,$   $y(0)=1,$   $y(1)=3.$

    It is assumed that $\epsilon << 1.$ Where is the boundary layer and what is its width?

Problem Set 10 (due Nov 13th, Mon)

  1. Solve approximately $\epsilon y^{\prime\prime}-(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ <$1$. The boundary conditions are

    $y(0)=1,y(1)=3.$

  2. Solve the differential equation in Problem 1 which holds for $-1$ < $x$ < $1$. The boundary conditions are

    $y(-1)=1,y(1)=3.$

Problem Set 11 (due Nov 20th, Mon)

  1. Solve the differential equation

    $\epsilon y^{\prime\prime}-(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ < $1,$ $-1$ < $x$ < $1$, with the boundary conditions

    $y(-1)=1,y(1)=3.$

  2. Solve the differential equation

    $\epsilon y^{\prime\prime}+(3\sin x)y^{\prime}+(\cos x)y=0,0$ < $x$ < $1,$ $-1$ < $x$ < $1$, with the boundary conditions

    $y(-1)=1,y(1)=3.$

Problem Set 12 (due December 4th, Mon)

  1. Consider the equations

    $\overset{\cdot\cdot}{x}+\omega_{0}^{2}x=-\dfrac{\epsilon}{3}(x+y)^{3},$

    $\overset{\cdot\cdot}{y}+W_{0}^{2}y=-\dfrac{\epsilon}{3}(x+y)^{3}.$

    The initial conditions are $x(0)=2a,$ $y(0)=2A,$ $\overset{\cdot}{x}(0)=\overset{\cdot}{y}(0)=0.$

    Find an approximate solution for this problem. For what values of $t$ is your solution a good approximation?

  2. Find with the two-scale method an approximate solution of

    $\overset{\cdot\cdot}{y}+(1+\epsilon\overset{\cdot}{y}^{2})y=0,$

    satisfying

    $y(0)=0,$ $\overset{\cdot}{y}(0)=1.$

    For what values of $t$ is this approximate solution good?