18.305 - Advanced Analytic Methods (Fall, 2019)

Problem Sets

Problem Set 1 (due Sep. 9, Mon)

  1. Chapter 1, Prob 1a, 1b;
  2. Chapter 1, Prob 2b, 2d, 2e.

Problem Set 2 ( due Sep 16, Mon)

  1. Consider the Laplace equation which holds in a two dimensional strip:

    $(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}})u(x,y)=0,$ $-\infty$ < $x$ < $\infty,$ $0$ < $y$ < $a,$

    The function $u(x,y)$ satisfies the boundary conditions

    $u(x,0)=f(x),$ $u(x,a)=0,$ $-\infty$ < $x$ < $\infty,$

    ($f(x)$ is a given function) as well as

    $u(\pm\infty,y)=0.$

    Find $u(x,y).$

  2. Consider the equation

    ($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $\Psi(x,t)=\rho(x,t),$

    where $\rho(x,t)$ is a given function and where $\Psi(x,t)$ vanishes at the infinity of space, i.e,

    $\Psi(\pm\infty,t)=0.$

    $\Psi(x,t)$ is required to satisfy the initial condition

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

    Find $\Psi(x,t).$

Problem Set 3 (due Sep 25, Wed)

  1. The wavefunction of the electron $\Psi(x,t)$ satisfies the Schrodinger equation

    ($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $\Psi(x,t)=U(x,t)\Psi(x,t),$

    where $U(x,t)$ is a given function (physically, it is the potential which acts on the electron). The wavefunction $\Psi(x,t)$ satisfies the same boundary conditions and initial condition as those given in prob 2.

    Use the result in prob 2 to convert the Schridinger equation into an integral equation.

    Find the perturbtion series for $\Psi(x,t).$

  2. Consider the heat equation

    $\dfrac{\partial T(x,t)}{\partial t}-\dfrac{\partial^{2}T(x,t)}{\partial x^{2}}=\rho(x,t),$ $-\infty$ < $x$ < $\infty,$ $t$ > $0,$

    where $T(x,t)$ is the temperature of a one-dimensional rod and $\rho(x,t)$ is a given source function. Let the initial temperture be

    $T(x,0)=f(x)$.

    Find the Green function $G(x-x^{\prime},t-t^{\prime})$ for the heat equation above in a closed form (not as an integral) and express $T(x,t)$ with it.

  3. The Green function for the wave equation is given by the integral

    $G(x,t)=\int_{-\infty}^{\infty}\dfrac{dk}{(2\pi)}e^{ikx}\cos kt.$

    Evaluate this integral. What does this result mean physically?

Problem Set 4 (due Sep 30, 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 16, Wed)

  1. Evaluate the integral

    $F_{\epsilon}(x)\equiv\int_{-\infty}^{\infty}e^{ikx}e^{-\epsilon k^{2}} \dfrac{dk}{2\pi}.$

    Show that in the limit $\epsilon\rightarrow0,$ $F_{\epsilon}(x)$ becomes the Dirac delta function.

  2. Consider the Schrodinger equation

    $[\dfrac{d^{2}}{dx^{2}}+\lambda^{2}(E-g\left\vert x\right\vert )]\Psi(x)=-0,$ $-\infty$ < $x$ < $\infty,$

    where

    $\lambda^{2}=m/(2\pi^{2}h^{2}),$

    $h$=$6.626x10^{-27}erg-\sec,$ and $g$ is a constant. Use the WKB method to determine, approximately, the energy eigenvalues$.$

  3. Find the approximate energy eigenvalues for the radial Schrodinger equation

    $[\dfrac{d^{2}}{dr^{2}}+\lambda^{2}(E+\dfrac{e^{2}}{r})]\Psi(r)=-0$, $\infty>r>0.$

    In the above, $e$ is the electric charge. Also, $-\dfrac{e^{2}}{r}$ is the attractive Coulomb potential the the positively charged hydrogen nucleus

    provides the negatively charged electron. Note that the energy $E$ is negative and that the boundary condition at $r=0$ is

    $\Psi(0)=0$. The exact result is $E=-\dfrac{e^{4}\lambda^{2}}{4(n+1)^{2}},n=0,1,2\cdot\cdot\cdot.$ Which of the eigenvalues you find are good approximations of the exact result?

Problem Set 6 (due Oct 21, Mon)

  1. Evaluate the following integrals:

    1. $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
    2. $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
  2. The Green function for the Schrodinger equation of one spatial dimension satisfies

    ($i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})$ $G(x-x^{\prime},t-t^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime}),$

    where $G(x-x^{\prime},t-t^{\prime})$ vanishes for $t$ < $t^{\prime}.$ Find $G(x-x^{\prime},t-t^{\prime})$ by expressing the Dirac delta functions with their Fourier integrals, i.e.

    $\delta(t-t^{\prime})=\int_{-\infty}^{\infty}e^{-ik_{0}(t-t^{\prime})} dk_{0}/(2\pi)$

    etc. and carrying out the integration over $k_{0}$.

Problem Set 7 (due Oct 28, Mon)

  1. Find the leading term for the following integrals for $\lambda>>1$:

    1. $\int_{0}^{\pi/2}e^{-\lambda\cos t}dt,$
    2. $\int_{-1}^{1}e^{\lambda t^{2}}dt,$
    3. $\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$

    Use the computer to evaluate the numerical values of these integrals and compare them with the numerical values of their leading terms as a function of $\lambda$.

Problem Set 8 (due Nov 4, Mon)

  1. As we know, the Gamma function is defined by

    $\Gamma(1+\lambda)=\int_{0}^{\infty}e^{-t}t^{\lambda}dt..$

    Find the asymptotic form of $\Gamma(1+\lambda)$ when $\lambda>>1.$

    Hint: you will need to change the variable of integration (why?)

  2. The Laplace transform of the function $f(x),$ $0$ < $x$ < $\infty$, is defined as

    $L(s)=\int_{0}^{\infty}dx$ $e^{-sx}f(x).$

    Find the asymptotic form of $L(s)$ when $s>>1.$ Verify your result with the examples of $f(x)=\sin x$ and $f(x)=\cos x.$

  3. Find the asymptotic form for

    $I(k)=\int_{-\infty}^{\infty}e^{-ikx}e^{-x^{4}},$ $k>>1,$

    (The integral above is the Fourier transform of $e^{-x^{4}}).$

Problem Set 9 (due Nov 13, Wed)

  1. Find the leading asymptotic form of the integral

    $I(\lambda)=\int_{-\infty}^{\infty}e^{i\lambda t}e^{-it^{3}/3}dt, \lambda>>1.$

  2. Determine the unshaded regions of $f(z)=e^{iz^{5}}$ in the infinity of which $f(z)$ vanishes.

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

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

    Where is the boundary layer and what is its width?

Problem Set 10 (due Nov 18, Mon)

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

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

    Where is the boundary layer and what is its width?

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

    $\epsilon y^{"}+2y^{\prime}+(1+x^{3})y=0,$ $0$ < $x$ < $1,$ $y(0)=1,$ $y(1)=0.$

    It is assumed that $\epsilon<<1.$ Where is the boundary layer?

Problem Set 11 (due Nov 25, Mon)

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

    $y(0)=1,y(1)=2.$

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

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

Problem Set 12 (due Dec 2, Mon)

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

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

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

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