Math 401-501/502 Assignments — Spring 2013
Assignment 1 — Due Tuesday, 1/29/13.
- Problems
- For $\varepsilon$ small, find the first order approximations in
$\varepsilon$ for roots of $x^2-(2+\varepsilon)x+\varepsilon
=0$.
- Use the quadratic formula to solve the equation in the previous
problem exactly. For the approximation to the root that is 2 at
$\varepsilon=0$, plot $\Delta x = x_{exact}-x_{approx}$ versus
$\varepsilon$, for $0\le \varepsilon\le 1$. In addition, plot the
relative percent error, $(\Delta x/x_{exact})*100\%$. For what value
of $\varepsilon$ is the relative error less than 0.5%, 1%, 3%?
- For $\varepsilon$ small, find the approximation to
the large root of $\varepsilon x^2+x-2=0$ that is valid to
order $\varepsilon^1$.
- Throughout our calculations we have assumed that if
$a_0+a_1\varepsilon=\mathcal O(\varepsilon^2)$ then $a_0=0$ and
$a_1=0$. Show that this is in fact true. (Recall that
$f(\varepsilon)= \mathcal O(\varepsilon^n)$ means that there is a
constant $C$ that is independent of $\varepsilon$ such that
$|f(\varepsilon)| \le C|\varepsilon|^n$, for $\varepsilon\to 0$.)
- Consider the equation $\varepsilon^2x^4-x^2+2x - 1=0$. Find the
approximation to the roots that is valid to order $\varepsilon^1$.
Assignment 2 — Due Thursday, 2/7/13.
- Problems
- In class, we discussed Duffing's equation: $m\!\frac{d^2
y}{dt^2}+ky+ ay^3=0$. Assume that $y(0)=0$, that $y'(0) = V$. The
units of the variables $m,t,y,k,a$ are, respectively mass, time,
length, mass/time2 and mass/(length2 ×
time2).
- Scale $t$ and $y$ to dimensionless variables $\tau$ and $u$, and
put Duffing's equation in the dimensionless form $\frac{d^2
u}{d\tau^2}+u+\varepsilon u^3 =0$, $u(0) = 0$, $\frac{d
u}{d\tau}\!\!(0)=1$. Find $\varepsilon$ in terms of the quantities
$m$, $k$, $V$, etc.
- For $\varepsilon$ small and $t$ fixed, but not large, find the
first order perturbation expansion $u(t) = u_0(t)+\varepsilon
u_1(t)+\mathcal O(\varepsilon^2)$. (In class, we did the problem for
$u(0)=1$ and $\frac{d u}{d\tau}\!\!(0)=0$.)
- The gamma function is $ \Gamma(x) := \int_0^\infty
t^{x-1}e^{-t}dt $ and the factorial function is then defined as $x! =
\Gamma(x+1)$, for $0\le x$. Note: $x$ does not have to be an
integer.
- Use integration by parts to that $x! =x(x-1)!$, where $x\ge
1$.
- Find the values of $R(x) := \sqrt{2\pi}x^{x+1/2}e^{-x}/x!$ for
$x=5,10, 20, 30$ and $50$.
- Let $E_{\alpha}(x) = \int_{x}^\infty \frac{e^{-t}}{t^\alpha}$,
for all $\alpha\ge 1$. Show that for $x\gg 1$, $E_{3/2}(x) =
\frac{e^{-x}}{x^{3/2}}\big(1+\mathcal O(1/x)\big)$.
- In class we showed that for $x\gg 1$ the asymptotic error for
$xe^x E_1(x)$ satisfies $\big|xe^x E_1(x) - \sum_{k=0}^n
\frac{k!}{x^k}\big|\le \frac{(n+1)!}{x^{k+1}}$. Use this for $n=7$
to determine smallest value of $x$ for which this error is less than
4×10−4.
- Let $\varepsilon \ll 1$. Show that $\int_0^\infty
\frac{e^{-t}}{\varepsilon^2\,+\,t^2}dt < \varepsilon^{-2}$. Hint:
scale $t$ and, in a way similar to what we did in class, estimate the
resulting integral.
Assignment 3 — Due Thursday, 2/14/13.
- Problems
- Scale both $t$ and $x$ in the IVP $ m\ddot x + c\dot x + kx=0,\
x(0)=0,\ \dot x(0)=V, $ to obtain the desired the scaled form of the
equation given below. In each case express $\varepsilon$ in terms of
$m,c,k,V$. (The parameter $\varepsilon$ will be different in each
case.)
- $u''+u'+\varepsilon u=0, \ u'(0)=1$.
- $u''+\varepsilon u'+u=0, \ u'(0)=1$.
- $\varepsilon u''+u'+u=0, \ u'(0)=1$.
- Consider the two perturbation series $x(\varepsilon) \sim a_0+
a_1\varepsilon +a_2\varepsilon^2 +\cdots+a_n\varepsilon^n + \mathcal
O\big(\varepsilon^{n+1}\big)$ and $y(\varepsilon) \sim b_0+
b_1\varepsilon +b_2\varepsilon^2 +\cdots+b_n\varepsilon^n + \mathcal
O\big(\varepsilon^{n+1}\big)$. Show that $x(\varepsilon)
y(\varepsilon) \sim a_0b_0+(a_0b_1+a_1b_0)\varepsilon
+\cdots+(a_0b_n+a_1b_{n-1}+a_2b_{n-2}+\cdots a_n b_0)\varepsilon^n
+\mathcal O\big(\varepsilon^{n+1}\big)$.
- In class we used the identity $\sin^3(x) =
(3\sin(x)-\sin(3x))/4$. In the next problem you will need $\sin^5(x)
= (\sin(5x) - 5\sin(3x)+10\sin(x))/16$. Derive both of these
identities using $\sin(x)= (e^{ix}-e^{-ix})/(2i)$ and the binomial
formula for $(u+v)^n$, with $n=3$ and $n=5$.
- Use the Poincaré Linstedt method to find a perturbation
solution to $y''+y+\varepsilon y^5=0$, $y(0)=0$, $y'(0)=1$ that is
valid through $\varepsilon^1$ and has no secular terms.
- Consider the pendulum shown below. (See
Wikipedia for the image copyright.) Suppose there is no
friction and that the length, mass, and $g$ have been scaled out of
the problem. The equation of motion is then $\ddot \theta +
\sin(\theta)=0$, plus initial conditions, which we take to be
$\theta(0)=\varepsilon$, where $0<\varepsilon \ll 1$ and $\dot
\theta(0)=0$. Our aim is to derive the small-angle (simple pendulum)
approximation.
- Do a scaling: $\theta=\varepsilon \varphi$. Show that
$\varphi(t)$ satisfies the equation $\ddot \varphi +
\sin(\varepsilon \varphi)/\varepsilon$, $\varphi(0)=1$, $\dot
\varphi=0$,
- Assume that $\varphi(t) = \varphi_0(t) + \mathcal
O \big(\varepsilon\big)$. Show that the $\varepsilon^0$ approximation,
$\varphi_0(t)$, satisfies $ \ddot \varphi_0+\varphi_0=0$, $\varphi(0)
= 1$, $\dot \varphi(0)=0$. (You will need to use $\sin(x) = x -
x^3/6 + \cdots$.)
- Finally, just multiply by $\varepsilon$ to show that $\theta_0(t)
= \varepsilon\varphi_0(t)$ satisfies $\ddot \theta_0+\theta_0=0$,
$\theta_0(0) = \varepsilon$, $\dot \theta_0(0)=0$, which is the initial
value problem for the simple pendulum.
Assignment 4 — Due Thursday, 2/21/13.
- Problems
- Consider the nonlinear pendulum $\ddot \varphi + \sin(\varepsilon
\varphi)/\varepsilon$, subject to the initial conditions
$\varphi(0)=1$, $\dot \varphi(0)=0$.
- Use the differential equation and the uniqueness of the solution
$\varphi(t,\varepsilon)$ to show that it is an even
function of $\varepsilon$. That is, $\varphi(t,\varepsilon) =
\varphi(t,-\varepsilon)$. (This is easy!)
- Use the evenness of the solution to show that if the perturbation
series $\varphi(t,\varepsilon) = \varphi_0(t)+ \varphi_1(t)
\varepsilon +\varphi_2(t) \varepsilon^2 + \cdots +\varphi_n(t)
\varepsilon^n + \mathcal O(\varepsilon^{n+1})$ holds, then terms
from the odd powers of $\varepsilon$ are all $0$. That is,
$\varphi_1(t)=0$, $\varphi_3(t)=0$, and so on.
- Use the Poincaré-Linstedt method with strained time $\tau=
\omega(\varepsilon)t$ to find a perturbation expansion of the form
$\varphi(\tau,\varepsilon) = \varphi_0(t) +
\varphi_2(t)\varepsilon^2 + \mathcal O(\varepsilon^4)$, with no
secular terms.
- Scale both $t$ and $x$ in the equation $m\ddot x +kx + c \dot
x^2=0$, $x(0)=A,\dot x(0)=0$ to obtain the equation $u''+u +
\varepsilon u'^2=0$, $u(0)=1,u'(0)=0$. Express $\varepsilon$ in terms
of $m,c,k,A$.
- For the initial value problem $u''+u + \varepsilon u'^2=0$,
$u(0)=1,u'(0)=0$, use a two-time method to obtain an order $0$
approximation. You may use the formulas for the first and second
derivatives obtained in class (Thursday, 2/14/13).
- Consider the equation $\varepsilon^2 y''+V(x)y=0$, where $
-\infty < x < \infty $ and, for all $x$, $1\le V(x)\le 2$. (This
condition is meant to keep $V$ strictcly greater than $0$ and
bounded.) Make a change of variable from $x$ to
$\tau=\varepsilon^{-1}\int_0^x \sqrt{V(s)} ds$. Find the equation
with the new time variable.
- Use the Poincaré-Linstedt method to find the terms up to
$\varepsilon^1$ in the perturbation expansion for the van der Pol
equation, $\ddot y +\varepsilon (1-y^2)\dot y +
y=0$, where $y(0)=1$ and $\dot y(0)=0$.
Assignment 5 — Due Thursday, 2/28/13.
- Problems (Below, 2 through 5 are review problems for perturbation
theory.)
- Consider the differential equation $(xy')'+\omega^2
xe^{-2x}y=0$ on an interval $1\le x \le 2$. Use $y(x,\omega) =
e^{i\omega g(x,\omega)}$ and repeat the derivation of the WKBJ
approximation to show that for $\omega \gg 1$ the function
$y(x,\omega) = x^{-1/2}\exp\big(\pm i\omega e^{-x}+x/2+ \mathcal
O(\omega^{-1})\big)$. Hint: use $h=xg'$ at an appropriate point in
the calculation. Calculate $g$ via $g=\int x^{-1}h(x) dx$.
- Find perturbation approximations to order $\varepsilon^1$ for the
roots of $\varepsilon x^3 + 4x -12 =0$. Hint: there are two large
roots. You will need complex coefficients. The expansions for these two
roots will also require positive and negative powers of
$\varepsilon^{1/2}$.
- Find a perturbation solution to order $\varepsilon^1$ for the
boundary value problem $x^2y"+\varepsilon xy' +\frac{4}{25}y=0$,
$y(1)=0, y(32) = 7$. (To do this problem, you will need to be able to
solve Cauchy-Euler equations. These are discussed in notes on
Cauchy-Euler equations that were written by
Professor Lynne Yengulap.)
- Let $\omega(\varepsilon) = 1+ \omega_1 \varepsilon +\mathcal
O(\varepsilon^2)$ and consider the initial value problem $\ddot y +y
+\varepsilon (y^2+y^3)=0$, $y(0)=0, \dot y(0)=1$, where $0<\varepsilon
\ll 1$. Using the Poincaré-Linsted method, find the
coefficient $\omega_1$ that eliminates secular terms in the order
$\varepsilon^1$ perturbation expansion for $y$. You do not
need to find the actual solution, just $\omega_1$.
- Let $0<\varepsilon\ll 1$ and consider the initial value problem
$\ddot y + y + \varepsilon \dot y^2=0$, $y(0)=0,\dot y(0) = 1$. Use the
two-time method, with $\tau=\varepsilon t$ and $\sigma=(1+\omega_1
\varepsilon+\mathcal O(\varepsilon^2))t$, to determine the value of
$\omega_1$ such there are no secular terms in the perturbation expansion
through terms of order $\varepsilon^1$ or to show that no such choice of
$\omega_1$ exists.
Assignment 6 — Due Thursday, 3/28/13.
- Problems
- Constanda, p. 25: 2, 7, 10, 11, 14, 15.
- Suppose that $f$ and $g$ have the Fourier series
\[
\begin{aligned}
f(x) &=a_0/2+\sum_{n=1}^\infty a_n\cos(n\pi x/L) + b_n\sin(n\pi x/L), \\
g(x) &=A_0/2+\sum_{n=1}^\infty A_n\cos(n\pi x/L) + B_n\sin(n\pi x/L).
\end{aligned}
\]
- Consider the integral $\frac{1}{L}\int_{-L}^L f(x)g(x)dx$. Replace $g$ by its series to get
\[
\begin{aligned}
\frac{1}{L}\int_{-L}^L f(x)g(x)dx &= \frac{1}{L}\int_{-L}^L f(x)\big\{ A_0/2+\sum_{n=1}^\infty A_n\cos(n\pi x/L) + B_n\sin(n\pi x/L)\big\}dx.
\end{aligned}
\]
In this formula, interchange $\sum$ and $\int$ and use the formulas for the Fourier coefficients (the $a$'s and $b$'s) of $f$ to show that
\[
\frac{1}{L}\int_{-L}^L f(x)g(x)dx = \frac{1}{2}a_0A_0 + \sum_{n=1}^\infty a_n A_n + b_nB_n.
\]
Note that if we set $f=g$ then we have Parseval's identity,
\[
\frac{1}{L}\int_{-L}^L f(x)^2dx = \frac{1}{2}a_0^2 + \sum_{n=1}^\infty a_n^2 + b_n^2.
\]
- In class we showed that the Fourier series for $f(x) =
\left\{\begin{array}{cl} 0 & -\pi \le x <0 \\ 1 & 0\le x \le \pi
\end{array}\right.$ is $ f(x) = \frac{1}{2} +
\frac{2}{\pi}\sum_{k=1}^\infty \frac{\sin((2k-1)x))}{2k-1}$. Use this
and Parseval's identity to show that $\sum_{k=1}^\infty
\frac{1}{(2k-1)^2} = \frac{\pi^2}{8}$.
Assignment 7 — Due Thursday, 4/4/13.
- Problems
- Constanda, p. 25-26: 17, 18, 21, 22.
- Recall that we solved the heat equation
$\frac{1}{k}\!\frac{\partial u}{\partial t} = \frac{\partial^2
u}{\partial x^2}$, subject to the initial conditions $u(x,0)=f(x)$,
$0\le x \le L$, and the boundary conditions $u(0,t)=u(L,t)=0$. The
result was
\[ u(x,t) = \sum_{n=1}^{\infty} e^{-\frac{n^2\pi^2 k
t}{L^2} kt} b_n\sin\big(\frac{n\pi x}{L}\big),
\]
where the $b_n$'s are the coefficients in the sine series for
$f(x)$. Find $u(x,t)$ for the problems below.
- $f(x)= 4 - x$, $0\le x \le 4$. ($L=4$).
- $f(x) = \cos(x)$, $0\le x \le \pi$. ($L=\pi$).
- Find both the Fourier sine series and the Fourier cosine series for
$f(x) = x$, $0\le x \le \pi$. Sketch the functions to which each of
these series converges for $-3\pi \le x \le 3\pi$.
- In class we defined uniform converge for a Fourier
series this way: We say that $S_N(x)$ converges uniformly to
$f(x)$ on $-L\le x \le L$ if and only if $\lim_{N\to
\infty}\max_{-L\le x \le L}|f(x) - S_N(x)| =0$. In addition, we
stated this condition, which is sufficient for $S_N$ to converge
uniformly to $f$: If the $2L$-periodic extension of $f$ is
piecewise smooth and continuous (i.e., the extension has only
corners, but no jumps), then $S_N(x)$ converges uniformly to
$f$. Moreover, we remark that the series is not
uniformly convergent if there are jump discontinuities in the
$2L$-periodic extension of $f$. For the functions in problems 2 and
3, use this test to determine whether the series that you found is
uniformly convergent. (You only need to look at sketches of the
extensions involved.)
- In Assignment 6, you showed that if $f(x) =a_0/2 +
\sum_{n=1}^\infty a_n\cos(n\pi x/L) + b_n\sin(n\pi x/L)$, then
Parseval's identity holds:
\[
\frac{1}{L}\int_{-L}^L f(x)^2dx = \frac{1}{2}a_0^2 +
\sum_{n=1}^\infty a_n^2 + b_n^2.
\]
Since $S_N(x) = a_0/2 + \sum_{n=1}^N a_n\cos(n\pi x/L) + b_n\sin(n\pi
x/L)$, we have that $f(x)=S_N(x)+\sum_{n=N+1}^\infty a_n\cos(n\pi x/L)
+ b_n\sin(n\pi x/L)$, or, equivalently, $f(x)-S_N(x)=
\sum_{n=N+1}^\infty a_n\cos(n\pi x/L) + b_n\sin(n\pi x/L)$. (The
function $f(x) - S_N(x)$ is the "residual signal.") Use this
and Perseval's identity to show that
\[
\frac{1}{L}\int_{-L}^L (f(x)-S_N(x))^2dx = \sum_{n=N+1}^\infty
a_n^2 + b_n^2.
\]
We remark that the physical significance of this equation is that the
energy in the residual signal is the sum of the energies in the modes
that are left over.
Assignment 8 — Due Thursday, 4/11/13.
- Problems
- Constanda, p. 125-128: 18, 21, 35, 55. Do intergals by
hand. Problems 55 refers to $\nabla^2 u= 0$, where
$u(\alpha,\theta)=f(\theta)$, $u(r,\theta) = u(r, \theta+2\pi)$, and
$u(r,\theta)$ is "nice" as $r\downarrow 0$. In solving the
eigenvalue problems, find the Rayleigh quotient as we did in class.
Assignment 9 — Due Thursday, 4/18/13.
- Problems
- Find the Fourier transform of the funtion $f(x)$. You may use
table A.2 (p. 314) in the text.
- $f(x) = \left\{\begin{array}{cc} e^{-x}& x \ge 0\\ 0 & x<0
\end{array} \right.$
- $f(x) = \left\{\begin{array}{cc} xe^{-x}& x \ge 0\\ 0 & x<0
\end{array} \right.$
- Show that if $F(\omega)$ is the Fourier transform of $f(x)$, then
the Fourier transform of $F(x)$ is $f(-\omega)$. (Hint: this follows
straight from the definition of the Fourier transform and inverse
Fourier transform.) Use this and table A.2 to find the Fouier
transform of $f(x) = \frac{\sin{2x}}{x}$.
- Use the convolution theorem and A.2, #7, to find $f*g$, if $f(x)
= e^{-4x^2}$ and $g(x) =e^{-12x^2}$.
- Find the inverse Fourier transform of
\[
\hat f(\omega) =
\sin(\omega) e^{-2| \omega|}
\]
- Use the Fourier transform method for the infinite bar to solve
this convective heat flow problem:
\[
\frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2
u}{\partial x^2}+ \alpha \frac{\partial u}{\partial x}, \
u(x,0)=f(x),\ -\infty < x < \infty, \ \alpha>0.
\]
- For $f(x) = \left\{\begin{array}{cc} 1 &0 \le x \le 1\\ 0 &
x<0 \end{array}\right.$ and $\, k=\alpha =1$, find a solution to the
previous problem in terms of the error function, erf(x).
- Solve this wave equation for an infinite string:
\[
\frac{\partial^2 u}{\partial t^2} = 9\frac{\partial^2
u}{\partial x^2}, \ u(x,0) =0,\ \frac{\partial u}{\partial t}\!(x,0)
= xe^{-x^2},\ - \infty < x < \infty, \ t\ge 0.
\]
Updated 4/11/2013 (fjn)
