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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 %{Class Time~}\underline{~~~~~~~~}
 \newline


\begin{center}
\bf Math 3150 Problems \\
{\small\bf Chapter 7}
\end{center}

{\small\sf
 {\bf Due date}: See the internet due date. Problems are
collected once a week. Records are locked when the stack is
returned. Records are only corrected, never appended.

{\bf Submitted work}. Please submit one package per problem.
 Label each problem with its corresponding problem number,
e.g., \fbox{Prob3.1-4} or \fbox{Xc1.2-4}. Kindly label extra credit
problems with label \fbox{\bf Extra Credit}. You may attach this
printed sheet to simplify your work.

{\bf Labeling}. The label \fbox{Probx.y-z} means the problem is for
chapter \fbox{x}, section \fbox{y}, problem \fbox{z}. When $y=0$,
then the problem does not have a textbook analog, it is a
\BF{background problem}. Otherwise, the problem number should match
a corresponding problem in the textbook. The same labeling applies
to extra credit problems, e.g., \fbox{Xc1.0-4}, \fbox{Xc1.1-2}. }

\section*{Chapter 7: 7.1 --     Fourier Integral Representation}

\Prob{Prob7.1-8a}{(Fourier Integral Formulas)} Find functions
$A(\omega)$, $B(\omega)$ for $f(x)=\left\{
\begin{array}{ll} 1, & 1<|x|<2, \\ 0 & \mbox{otherwise},
\end{array}\right.$ in the Fourier integral
representation
 $$f(x)=\int_0^\infty [A(\omega)\cos(\omega x)+B(\omega)\sin(
\omega x)]d\omega .$$
 Display all details, including computer algebra steps, in evaluating
 the integrals
$$
 A(\omega)=\frac{1}{\pi}\int_{-\infty}^\infty f(t)\cos(\omega t)dt,\quad
 B(\omega)=\frac{1}{\pi}\int_{-\infty}^\infty f(t)\sin(\omega t)dt.
$$

\Prob{Prob7.1-8b}{(Fourier Integral Convergence)} Given
$f(x)=\left\{
\begin{array}{ll} 1, & 1<|x|<2, \\ 0 & \mbox{otherwise},
\end{array}\right.$, report the values of $x$ for which $f(x)$
equals its Fourier integral.

\Prob{Prob7.1-19}{(Fourier Integral and Integration Formulas)}
Invent a function $f(x)$ such that the Fourier Integral
Representation implies the formula
$$ e^{-x} = \frac{2}{\pi} \int_0^\infty \frac{\cos(\omega
x)}{1+\omega^2}\,d\omega.$$


\section*{Chapter 7: 7.2-7.3 -- Fourier Transform}

\Prob{Prob7.2-20}{(Fourier Transform)}
 Let $f(x)=x$ for $|x|<1$ and $f(x)=0$ elsewhere.  Find the fourier
 Transform
 $$ \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty
 f(x)e^{-i\omega x}\, dx.$$

{\small\sf \BF{Reference}: See Asmar 1E problem 7.2-20 or Asmar 2E
problem 7.2-39.}

\clearpage
 \Prob{Prob7.2-18a}{(Fourier Transform Rules)}
  State in detail the following operational rules for the Fourier Transform.

\begin{tabular}{llll}
  (a) Linearity. 
  &(b) $x$-differentiation rule [parts].  
  &(c) $\omega$-differentiation rule.
  &(d) Convolution rule.
  \\
  (e) $x$-axis shifting rule. 
  &(f) $\omega$-axis shifting rule. 
  &(g) $t$-partial derivative rule.
\end{tabular}

{\small\sf \BF{Reference}: Asmar's text, {\em PDE and BVP}, section
7.2, Theorems 1,2,3,4,5 and problems 7.2-19, 7.2-20. Section 7.3 for
item (g). See also appendix B in Asmar. 

DETAILS. State a minimal rule, one which can easily generate all
variants of the rule. For example, all $\omega$-shifting rules arise
from the single identity
$\Ft[e^{iax}f(x)](\omega)=\Ft[f(x)](\omega-a)$.}

\Prob{Prob7.2-18b}{(Fourier Transform Table)}
 Derive the table below, considered to be a suitable minimal table
 for using the Fourier transform on applied problems. In the table,
 we use the Fourier Transform notation
 $$\Ft(f(x))= \hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty
 f(x)e^{-i\omega x}\, dx.$$
 Define the \BF{pulse function} by
 $$ \pulse(x,a,b) = \left\{ \begin{array}{ll} 1 & a < x < b, \\ 0 &
 \mbox{otherwise}. \end{array}\right.$$

$$
\begin{array}{|lcl|}
\hline
  \Ft(\pulse(x,a,b))(\omega) &=& \dd \frac{e^{-i\omega a} -
  e^{-i\omega b}}{i\omega\sqrt{2\pi}}
  \\[2ex] \hline
\Ft\left(\frac{1}{a^2+x^2}\right)(\omega) &=& \dd \frac{\pi
e^{-a|\omega|}}{a\sqrt{2\pi}} \quad (a>0)
  \\[2ex] \hline
\Ft\left(e^{-a|x|}\right)(\omega) &=& \dd
\frac{1}{\sqrt{2\pi}}\frac{2a}{a^2+\omega^2}\quad (a>0)
  \\[2ex] \hline
\Ft\left(e^{-ax^2}\right)(\omega) &=& \dd
\frac{1}{\sqrt{2a}}e^{-\frac{\omega^2}{4a}}\quad (a>0)
  \\[2ex] \hline
\end{array}
$$

{\small\sf \BF{Reference}: Asmar's text, {\em PDE and BVP}, 
section 7.2 and appendix B.}

 \Prob{Prob7.2-31a}{(Fourier Transform Calculus)} Use the Fourier
 transform rules and Fourier transform table to derive a formula for
 $\Ft(f(x))$.

\begin{tabular}{|l|l|l|}
\hline
  (a) $f(x)=\cos(x)\, e^{-x^2}$  & (b) $f(x)=\sin(x) \, e^{-|x|}$ &
  (c) $f(x) = \dd\cos(x)\,\frac{1}{1+x^2}$
  \\[2ex]
  \hline
\end{tabular}

DETAILS. The table entries are found from the $\omega$-shifting rule,
$\Ft[e^{iax}f(x)](\omega)=\Ft[f(x)](\omega-a)$, using identities that
write $\cos(bx)$ and $\sin(bx)$ in terms of complex exponentials 
$e^{ibx}$ and $e^{-ibx}$.

\Prob{Xc7.2-31a}{(Fourier Transform Calculus)} Use the Fourier
 transform rules and Fourier transform table to derive a formula for
 $\Ft(f(x))$.

\begin{tabular}{|l|l|l|}
\hline
  (a) $f(x)=\dd  \cos(2x)\frac{1}{4+x^2}$ & (b) $f(x)=\cos(x)\,\pulse(x,-1,1)$ & (c) $f(x)=\dd\sin(x)\pulse(x,-1,1)$
  \\[2ex]
  \hline
\end{tabular}

DETAILS. The same advice applies as in Problem 7.2-31a. 

\Prob{Prob7.2-45}{(Fourier Transform Convolution)}

 (a) Find the Fourier transform of the convolution of $xe^{-x^2/2}$
 and $e^{-x^2}$.

 (b) Solve for $h(x)$ in the equation $\dd \Ft(h(x))=e^{-\omega^2}\frac{\sin
 \omega}{\omega}$, using the convolution theorem.

\Prob{Xc7.2-47}{(Fourier Transform Convolution)} Write a proof for
the convolution identities
\begin{center}\begin{tabular}{|l|l|}
\hline
 (a) $f * g = g * f$ \hspace*{2pc}
\\
 (b) $f * (g * h) = (f * g) * h$ \hspace*{2pc}
 \\ \hline
\end{tabular}\end{center}
DETAILS. The convolution of $f$ and $g$
is defined by the equation
$\dd f*g(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x-t)g(t)dt$.

{\small\sf \BF{Reference}: See Asmar 1E problem 7.2-47, identical to
Asmar 2E problem 7.2-55.}

 \Prob{Prob7.3-1}{(Fourier Transform Method Wave Equation)} Solve the
 boundary value problem on $-\infty < x < \infty$, $t>0$.

$$
 \begin{array}{rcl}
 u_{tt}(x,t) &=& u_{xx}(x,t),\\[2ex]
 u(x,0) &=& \dd\frac{1}{1+x^2},\\[2ex]
 u_t(x,0)&=&0.
 \end{array}
$$

\Prob{Prob7.3-4}{(Fourier Transform Method Heat Equation)} Solve the
boundary value problem on $-\infty < x < \infty$, $t>0$.

$$
 \begin{array}{rcl}
 u_{t}(x,t) &=& \dd\frac{1}{100}u_{xx}(x,t),\\[2ex]
 u(x,0) &=& 100\pulse(x,-1,1).
 \end{array}
$$

\Prob{Xc7.3-17}{(Fourier Transform Method Infinite Beam)} Solve the
boundary value problem on $-\infty < x < \infty$, $t>0$.
$$
 \begin{array}{rcl}
 u_{tt}(x,t) &=& \dd\frac{1}{100}u_{xxxx}(x,t),\\[2ex]
 u(x,0) &=& 100\pulse(x,-2,2).
 \end{array}
$$

\section*{Chapter 7: 7.4-7.5 -- Heat Kernel and Poisson Integral Formula}

\Prob{Xc7.4-2}{(Heat Kernel)} Solve the boundary value problem on
$-\infty < x < \infty$, $t>0$.
$$
 \begin{array}{rcl}
 u_{t}(x,t) &=& \dd\frac{1}{100}u_{xx}(x,t),\\[2ex]
 u(x,0) &=& 100\pulse(x,-2,0)+50\pulse(x,0,1).
 \end{array}
$$

\Prob{Xc7.4-6}{(Heat Kernel)} Solve the boundary value problem on
$-\infty < x < \infty$, $t>0$.
$$
 \begin{array}{rcl}
 u_{t}(x,t) &=& \dd\frac{1}{100}u_{xx}(x,t),\\[2ex]
 u(x,0) &=& e^{-|x|}.
 \end{array}
$$

\end{document}
\endinput
 Zhu's problems
 • Section 2.1: 20;
 • Section 2.2: 8, 15;
 • Section 2.3: 5;
 • Section 2.4: 7(a), 15;
 • Section 3.3: 9, 12(a,b);
 • Section 3.4: 15;
 • Section 3.5: 13;
 • Section 3.6: 3;
 • Section 3.7: 5, 13;
 • Section 3.9: 3;
 • Section 4.4: 5, 15, 17;
 • Section 7.1: 8;
 • Section 7.2: 20, 31;
 • Section 7.3: 19.
