Problem Sets
Problem Set 1 ( due Sep 13, Mon)
- Chapter 1, Prob 2a,2e;
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Chapter 1, Find the general solution of
(D-1)(D-2)(D-3)y=2e$^{\cos x}.$
Problem Set 2 ( due Sep 20, Mon)
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Find the solution of the partial different equation
$(\dfrac{\partial^{2}}{\partial t^{2}}-\dfrac{\partial^{2}}{\partial x^{2} }+m^{2})\phi(x,t)=0,$ $-\infty$ < $x$ < $\infty,$ $t$ > $0,$ $m$ a constant,
The solution satisfies the initial conditions
$\phi(x,0)=f(x),$ $\phi_{t}(x,0)=g(x),$
and vanishes at the spatial infinities. Identify from this solution the Green functions for this equation.
Problem Set 3 ( due Sep 27, Mon)
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Consider the heat equation
$\dfrac{\partial}{\partial t}T(x,t)=\dfrac{\partial^{2}}{\partial x^{2}}T(x,t)+\rho(x,t),$
where $-\infty$ < $x$ < $\infty$, $t$ > $0$, with $\rho(x,t)$ a given source function.
The temporature $T(x,t)$ satisfies the boundary condition
$T(\pm\infty,t)=0$
and the initial condition
$T(x,0)=f(x),$
with $f(x)$ a given function. Find $T(x,t)$ as well as the Green function of the partial differential equation.
Problem Set 4 ( due Oct 4, Mon)
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One of the Green functions for the wave equation in one spatial dimension is given by
$G(x,t)=\int_{-\infty}^{\infty}\dfrac{dk}{(2\pi)}e^{ikx}\cos kt.$
Evaluate this integral. What does this result mean physically?
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Let $u$ satisfy the Laplace equation in the upper-half plane of the two dimensional space, i.e.,
$\dfrac{\partial^{2}u(x,y)}{\partial x^{2}}+\dfrac{\partial^{2}u(x,y)} {\partial y^{2}}=0,$ $ y$ > $0,$ $-\infty$ < $x$ < $\infty.$
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Find $u$ if it satisfies the following boundary conditions:
$u(\pm\infty,y)=u(x,\infty)=0,$ $u(x,0)=\delta(x-1).$
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Show that no solution exists if the conditions above are changed to $u(\pm\infty,y)=0,$ $u_{y}(x,0)=\delta(x-1),$ and $u(x,0)=\delta(x-1).$
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Problem Set 5 ( due Oct 13, Wed)
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Find the eigenfunctions and the corresponding eigenvalues of the Sturm-Liouville problem
$\dfrac{d^{2}\phi(\theta)}{d\theta^{2}}=-\lambda\phi(\theta),$ $0$ < $\theta $ < $2\pi,$
with the homogenious (periodic) boundary conditions
$\phi(0)=\phi(2\pi),$ $\phi^{\prime}(0)=\phi^{\prime}(2\pi).$
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Problem 4 of the Homework Problems in Chapter 5 of the textbook (p.199).
Problem Set 6 ( due Oct 18 Mon)
Prob 6 in Chapter 5 (p.199).
Problem Set 7 ( due Oct 25 Mon)
Prob 1 in Chapter 2 (p.111).
Prob 2 in Chapter 2 (p.111).
Problem Set 8 (due Nov 1, Mon)
Prob 1. Evaluate the following integrals:
- $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
- $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
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Prob 2. Evaluate the integrals in Problem 4c and 4d on p. 112.
Problem Set 9 (due Nov 8, Mon)
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Let $y$ satisfy the equation
$y"+x^{2}y=0$
and the initial conditions $y(x_{0})=1,$ $y^{\prime}(x_{0})=0$ where $x_{0}$ > $0.$
- Find the WKB approximation of $y(x)$ for $x$ > $x_{0}$. For what values of $x$ do you expect it be a good approximation?
- 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.
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Problem 4, Chapter 7.
Problem Set 10 (due Nov 22, Mon)
- Problem 6b, Chapter 7.
- Problem 10, Chapter 7.
