Problem Sets

Problem Set 1 (due Sep. 12)

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

Problem Set 2 (due Sep. 19)

  1. Consider the wave equation for an electron

    $i\dfrac{\partial\Psi}{\partial t}-\dfrac{\partial^{2}\Psi}{\partial x^{2}}=U\Psi,$

    where $\Psi=\Psi(x,t)$ is the wavefunction of the electron and $U=U(x,t)$ is the potential.

    Let the initial condition be

    $\Psi(x,0)=f(x).$

    Use the Green function method to convert the wave equation into an integral equation.

  2. 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.

Problem Set 3 (due Sep. 26)

  1. Problem 4 in Chapter 7.

Problem Set 4 (due Oct. 3)

  1. Problem 10 in Chapter 7.

  2. 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 5 (due Oct. 12)

  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. $I(\lambda)=\int_{0}^{\infty}e^{-\lambda\sqrt{1+x}}xdx,$
      where $\lambda$ is very large.

  2. Find the leading term for each of the following integrals when $\lambda>>1:$

    1. $\int_{0}^{\pi}e^{i\lambda\sin x}dx,$
    2. $\int_{-2}^{2}e^{i\lambda/(1+x^{2})}dx,$

Problem Set 6 (due Oct. 19)

    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. Problem 4a in Chapter 2.

Problem Set 7 (due Oct. 24)

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

  3. Find the leading asymptotic form of the integral

    $I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda x}e^{ix^{3}/3}dx,$ $\ \lambda>>1.$

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

Problem Set 8 (due Oct. 31)

  1. Find the leading asymptotic form of the integral

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

    $\lambda>>1.$

  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. 7)

  1. Let $I(\lambda)=\int_{0}^{\infty}\dfrac{e^{i\lambda x-1/x}}{1+x}dx,$ $ \lambda>>1.$

    1. Calculate the leading term of this integral by evaluating the contribution from the relevant saddle point.
    2. Justify your result by deforming the contour of integration into one on which the integrand is the largest at the saddle point.

  2. 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?

  3. Find the next-order term for the problem above.

Problem Set 10 (due Nov. 21)

  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

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

  3. Solve approximately $\epsilon y^{\prime\prime}-2xy^{\prime}+(1+x^{4})y=0,0$ < $x$ < $1$. The boundary conditions are

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

  4. Solve approximately the differential equation in Problem 3 which holds for $-2$ < $x$ < $1.$ The boundary conditions are

    $y(-2)=1,y(1)=3.$

Problem Set 11 (due Dec. 5 - turn into grader's office 2-333C by the end of the day)

  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 of this problem valid for $t$ much less than $O(\epsilon^{-2})$.