Computer Algebra Recipes

Richard H Enns, George C McGuire

Chapter 5 Linear PDE Models. Part 1 - all with Video Answers

Educators

Chapter Questions

Problem 1

In the text recipe, make the following modifications:
(a) The region $x>0$ is composed of solid wood for which $K=0.15 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K})$, $\rho=700 \mathrm{~kg} / \mathrm{m}^3$, and $C=1800 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})$.
(b) The mean daily temperature is $30^{\circ} \mathrm{C}$ and the amplitude of the temperature variation is $20^{\circ}$.
Run the animation and determine the approximate depth in centimeters at which the temperature variation is essentially zero.

Ajay Singhal

Numerade Educator

01:19

Problem 2

The surface of a sphere of radius $r=a$, surrounded by an ideal compressible fluid, pulsates radially with frequency $\omega$. The radial velocity of the surface is given by $V=U \cos (\omega t)$. It is stated in an advanced calculus text that the steady-state fluid velocity at an arbitrary point $r>a$ is of the form
$$
V=\frac{U a^2}{\left(c^2+a^2 \omega^2\right) r^2}\left[\left(c^2+a r \omega^2\right) \cos (\theta)+c \omega(r-a) \sin (\theta)\right],
$$
where $\theta \equiv \omega(r-a) / c-\omega t$ and $c$ is the speed of sound in the fluid.
(a) Check that the solution satisfies the boundary condition.
(b) The velocity $V$ is related to the velocity potential $\Phi$ by $V=-\partial \Phi / \partial r$. Determine the radial dependence of the velocity potential.
(c) Since $\Phi$ depends only on the distance $r$ from the center of the sphere, it satisfies the wave equation in the form
$$
\frac{\partial^2(r \Phi)}{\partial r^2}=\frac{1}{c^2} \frac{\partial^2(r \Phi)}{\partial t^2} .
$$
Verify that $\Phi$ satisfies the wave equation, thus ensuring that it is the correct solution to the pulsating sphere problem.
(d) Taking the nominal values $U=1, a=1, c=1$, and $\omega=1$, animate the analytic formula for $V$ in the region outside the spherical surface.
(e) How far from the surface does the velocity oscillation amplitude drop to $5 \%$ of the value at the surface?

Dominador Tan

Numerade Educator

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Problem 3

An elastic string fixed between $x=0$ and $L$, and initially at rest, is "plucked," its initial shape being given by the following symmetric triangular profile,
$$
\psi(x, 0)= \begin{cases}2 h x / L, & 0 \leq x \leq L / 2, \\ 2 h(L-x) / L, & L / 2 \leq x \leq L\end{cases}
$$
Verify that the motion for $t>0$ may be described by the Fourier series solution
$$
\psi(x, t)=\frac{8 h}{\pi^2} \sum_{n=1}^{\infty} \frac{\sin (n \pi / 2)}{n^2} \sin (n \pi x / L) \cos (n \pi c t / L)
$$
and animate the solution for parameter values of your own choosing.

Lainey Roebuck

Numerade Educator

05:53

Problem 4

A piano string fixed between $x=0$ and $a$ is struck by a piano hammer in a region of width $d$ centered at $x=x_0$. Its initial velocity distribution is
$$
\dot{\psi}(x, 0)= \begin{cases}v \cos \left(\pi\left(x-x_0\right) / d\right), & \left|x-x_0\right|<d / 2, \\ 0, & \left|x-x_0\right|>d / 2 .\end{cases}
$$
Neglecting stiffness and assuming $\psi(x, 0)=0$,
$$
\psi(x, t)=\frac{4 v d}{\pi^2 c} \sum_{n=1}^{\infty} \frac{1}{n} \frac{\sin \left(n \pi x_0 / a\right) \cos (n \pi d /(2 a))}{\left(1-(n d / a)^2\right)} \sin (n \pi x / a) \sin (n \pi c t / a)
$$
is the shape of the string at time $t$. Verify that this series solution is correct and animate it for parameter values of your own choosing.

Khoobchandra Agrawal

Numerade Educator

02:44

Problem 5

If $\epsilon_1=1, \epsilon_2=16, k_1=1$, and $L=2$, what fraction of the incident plane wave energy is transmitted through the middle segment into the third region?

Ajay Singhal

Numerade Educator

06:40

Problem 6

Consider a plane wave of frequency $\omega$ traveling from $x=-\infty$ along an infinitely long string with three regions of different density, the middle one being located between $x=0$ and $L$. Suppose that the wave number in region $1(x<0)$ is $k_1=1$, in region $2(0<x<L)$ is $k_2=2$, and in region $3(x>L)$ is $k_3=3$.
(a) Confirm that the reflection coefficient is
$$
R=\frac{17+15 \cos (4 L)}{113+15 \cos (4 L)} .
$$
(b) Because $k_3 \neq k_1$, the transmission coefficient $T$ is equal to $k_3|d|^2 /\left(k_1|a|^2\right)$, where $a$ and $d$ are the incident and transmitted amplitudes. Determine the analytic form of $T$ and confirm that $R+T=1$.
(c) Plot $R$ and $T$ on the same graph.
(d) Is there an $L$ value for which $100 \%$ transmission is possible?
(e) What is the maximum value of $T$, and for what $L$ value does this transmission occur?

Nicole Neumann

Numerade Educator

06:40

Problem 7

Consider an infinitely long string that has a linear density $\epsilon_1=1$ in region 1 $(x<0)$, density $\epsilon_2=4$ in region $2(0<x<L)$, density $\epsilon_3=1$ in region 3 $(L<x<2 L)$, density $\epsilon_4=4$ in region $4(2 L<x<3 L)$, and density $\epsilon_5=1$ in region $5(x>3 L)$. For a plane wave of frequency $\omega$ coming from $x=-\infty$ :
(a) Show that the transmission coefficient $T$ into region 5 is
$$
T=2048 /\left(\sum_{n=0}^5 b_n \cos (2 n L)\right)
$$
with $b_0=3686, b_1=-882, b_2=-1800, b_3=1611, b_4=162, b_5=-729$.
(b) Show that the sum $T+R$ equals 1 , where $R$ is the reflection coefficient.
(c) Plot the transmission and reflection coefficients in the same graph.
(d) At what values of $L$ is there $100 \%$ transmission? Discuss your answer.

Nicole Neumann

Numerade Educator

04:47

Problem 8

Schrödinger's equation describing one-dimensional motion of a particle of mass $m$ and energy $E$ in a potential $V(x)$ is
$$
-\frac{\hbar^2}{2 m} \frac{d^2 \psi(x)}{d x^2}+V(x) \psi(x)=E \psi(x) .
$$
Here $\psi(x)$ is the probability amplitude for finding the particle at $x$ and $\hbar=$ $h /(2 \pi)$, where $h$ is Planck's constant. Consider a particle with energy $E<V_0$ incident on a rectangular barrier $V(x)=V_0>0$ located in the region $x=0$ to $L$. The potential $V(x)$ equals 0 outside the barrier.
(a) Show that the transmission coefficient $T$, which may be calculated in a similar manner to that for the 3-piece string, is
$$
T=\left(1+\frac{\sinh ^2(K L)}{\left(4 E / V_0\right)\left(1-E / V_0\right)}\right)^{-1},
$$
with $K=\sqrt{2 m\left(V_0-E\right) / \hbar^2}$.
(b) Plot $T$ as a function of $K L$ for some representative values of $E / V_0<1$.
(c) Discuss the behavior of the transmission coefficient and contrast it with what would be expected classically.

João Gabriel Alencar Caribé

Numerade Educator

02:20

Problem 9

In the text recipe replace the complex function with $w=u+I v=z^2$, with $z=x+I y$. Confirm that $u(x, y)$ and $v(x, y)$ satisfy the Cauchy-Riemann conditions and Laplace's equations. Confirm by making a suitable plot that the equipotentials and electric field lines are appropriate to the quarter-space bounded by two semi-infinite conducting plates intersecting at right angles. Make another plot that shows the electric field vectors and equipotentials.

Nick Johnson

Numerade Educator

05:46

Problem 10

Consider the function $w=u+i v=\sqrt{z^2-1}$, with $z=x+i y$. By creating a suitable figure in the $x-y$ plane, show that the constant- $u$ curves can represent the streamlines (tracks of the fluid particles) for fluid flow around an infinitely long plate of finite width, lying between $x=-1$ and +1 , inserted perpendicular to a previously uniform fluid flow in the $y$ direction. Include the plate and the equipotentials (constant $v$ curves) in your figure.

Aman Gupta

Numerade Educator

01:32

Problem 11

For the heated rod discussed in the text, at what time is the temperature at the center of the rod equal to one-third of its initial value?

Kian Manafi

Numerade Educator

01:00

Problem 12

The temperatures at the ends $x=0$ and $x=100$ of a rod (insulated on its sides) $100 \mathrm{~cm}$ long are held at $0^{\circ}$ and $100^{\circ}$, respectively, until steady state is achieved. Then at the instant $t=0$, the temperatures of the two ends are interchanged. Determine the resultant temperature distribution $T(x, t)$. Animate your solution and discuss its behavior.

Hast Aggarwal

Numerade Educator

05:27

Problem 13

An infinitely long hollow conductor has a rectangular cross section with sides $L$ and $2 L$. One of the longer sides is charged to a potential $V=V_0$ and the other three sides are held at zero potential. By solving Laplace's equation, determine the potential distribution in the interior region. Taking $V_0=1$ and $L=1$, plot the equipotential lines. Calculate the electric field $\vec{E}=-\nabla V$ in the interior region. Plot $\vec{E}$ on the same graph as the equipotential lines.

Vishal Gupta

Numerade Educator

01:06

Problem 14

Suppose that a rectangular barbecue plate has a finite thickness $c=0.01 \mathrm{~m}$ in the $z$-direction and dimensions $a=0.5 \mathrm{~m}$ and $b=0.5 \mathrm{~m}$ in the $x$ - and $y$-directions, respectively. Assuming that the bottom of the plate is uniformly heated and is $100^{\circ}$ hotter than the other five sides, determine the steady-state temperature distribution $T(x, y, z)$ inside the plate and make a suitable plot.

Raj Bala

Numerade Educator

03:55

Problem 15

Using the textplot command, add appropriate potential values to Figure 5.8 so that the equipotential lines are clearly identified.

Charles Magnusen

Numerade Educator

02:07

Problem 16

Along the circumference of a circle of radius $r=1$, the potential $\phi(r, \theta)$ has the value $\phi=1$ for the angular range $0<\theta<\pi$ and $\phi=0$ when $\pi<\theta<2 \pi$. Determine $\phi(r, \theta)$ for $r<1$ and for $r>1$. Plot the equipotentials inside and outside the circle.

Christopher Stanley

Numerade Educator

01:57

Problem 17

A plate has the form of an annular region bounded by two concentric circles $r_1=1 \mathrm{~m}$ and $r_2=2 \mathrm{~m}$. The temperature in degrees Celsius along $r_1$ is $T_1=$ $75 \sin \theta$ and along $r_2$ is $T_2=60 \cos \theta$. Determine the steady-state temperature at each point of the annular region and plot the isotherms.

Anand Jangid

Numerade Educator

02:39

Problem 18

An infinitely long hollow conducting cylinder of radius $r=a$ is divided into equal quarters, alternate segments being held at the potentials $+V$ and $-V$. Using the two-dimensional form of Laplace's equation, determine the potential inside the cylinder. Taking $a=1$ and $V=1$, plot the equipotential surfaces.

Dading Chen

Numerade Educator

02:10

Problem 19

For the two-dimensional bipolar coordinate system, use ?coords to find out how bipolar and Cartesian coordinates are related. Plot the bipolar coordinate system, using options of your own choice. Suggest an electrostatic boundary value problem in which this coordinate system would be useful.

Linda Hand

Numerade Educator

01:07

Problem 20

Use coordplot3d to plot the surfaces corresponding to holding each coordinate equal to a constant value for the following three-dimensional systems:
(a) cylindrical;
(b) spherical;
(c) paraboloidal;
(d) sixsphere.

Carson Merrill

Numerade Educator

03:21

Problem 21

Making use of the VectorCalculus package, determine the form of Laplace's equation for the scalar field $\psi$ in the following coordinate systems:
(a) cylindrical, with $\psi=\psi(r, \theta, z)$;
(b) spherical, with $\psi=\psi(r, \theta, \phi)$;
(c) bispherical, with $\psi=\psi(\xi, \eta, \phi)$;
(d) paraboidal, with $\psi=\psi(\eta, \xi, \phi)$;
(e) prolate spheroidal, with $\psi=\psi(u, v, \phi)$;
(f) elliptic, with $\psi=\psi(u, v)$.

James Kiss

Numerade Educator

04:06

Problem 22

Solve Laplace's equation in the following coordinate systems. In each case, look at what Maple does as each additional option is included, ending up with the complete product solution:
(a) cylindrical, with $\psi=\psi(r, \theta, z)$;
(b) spherical, with $\psi=\psi(r, \theta, \phi)$;
(c) paraboloidal, with $\psi=\psi(\eta, \xi, \phi)$.

Eduard Sanchez

Numerade Educator

02:28

Problem 23

Carry out a calculation similar to that in the text recipe for a fissionable spherical mass of radius $r=a$ and an initial neutron density $f(r)=1-r^2 / a^2$. Make an animated plot of the neutron density inside the sphere when the mass slightly exceeds the critical mass.

Jennifer Hudspeth

Numerade Educator

01:14

Problem 24

A thin semicircular plate of radius $r=1$ has its edges held at zero temperature and its flat faces insulated. If the initial temperature distribution inside the plate is $T(r, \theta, 0)=100 r^2 \cos (2 \theta)$, determine the temperature $T(r, \theta, t)$ for $t>0$. Taking the diffusion constant $d=1$, animate the temperature profile.

Raj Bala

Numerade Educator

Problem 25

A solid has the shape of an infinitely long quarter-cylinder of radius $r=1$ and diffusion constant $d$. The flat sides are insulated, so no heat flows through them, while the curved surface is kept at $100^{\circ} \mathrm{C}$. Assuming that the temperature initially varies as the fourth power of the distance from the axis, find the temperature distribution at any point inside the solid for $t>0$. Choosing nominal parameter values, create an animated plot of the temperature profile.

Check back soon!

01:46

Problem 26

A cylinder of unit radius and unit height has its circular ends $z=0$ and $z=1$ kept at $T=0$ and $T=1$, respectively, while the circular surface is kept at $T=1$. Determine the steady-state temperature profile inside the cylinder and plot the contours of equal temperature.

Hast Aggarwal

Numerade Educator

02:11

Problem 27

An infinitely long bar of elliptic cross section has its curved surface, $x^2+4 y^2=1$, kept at $T=0^{\circ}$. Determine the temporal evolution of $T$ in the cylinder if initially $T=100^{\circ}$ everywhere inside and the diffusion constant $d=1$. Animate the result.

Lucas Finney

Numerade Educator

05:36

Problem 28

On the surface of a hollow sphere of radius $a$, the electric potential is
$$
\Phi(a, \theta)= \begin{cases}+V, & 0 \leq \theta<\pi / 2, \\ -V, & \pi / 2<\theta \leq \pi .\end{cases}
$$
By solving Laplace's equation in spherical coordinates, determine $\Phi(r, \theta)$ for both the regions $r<a$ and $r>a$. Plot the equipotential lines and electric field vectors for both regions, taking $a=1$ and $V=1$.

Suzanne W.

Numerade Educator

01:57

Problem 29

Two thin concentric spherical shells of radius $a$ and $b>a$ are each divided into two hemispheres by the same horizontal plane. The potential on the top hemisphere of the inner shell is $V$ and the potential on the bottom hemisphere is zero, whereas the potential on the top hemisphere of the outer shell is zero and the potential on the lower hemisphere is $V$. Using Laplace's equation in spherical polar coordinates, determine the potential in the region between the spheres. Choosing your own parameter values, plot the equipotential lines in this region.

Manik Pulyani

Numerade Educator

04:04

Problem 30

A uniform solid iron sphere of radius $20 \mathrm{~cm}$ is heated to a temperature of $100^{\circ} \mathrm{C}$ throughout. Its surface is to be kept at the constant temperature $0^{\circ} \mathrm{C}$.
(a) Explicitly determine $T(\vec{r}, t)$.
(b) If the heat diffusion coefficient $d=0.185 \mathrm{cgs}$ units, find the temperature of the center of the sphere 15 minutes after the cooling has begun.
(c) Plot the constant-temperature profiles at this time.
(d) Animate the temperature profile inside the sphere.

Surendra Kumar

Numerade Educator

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Problem 31

The temperature of the surface of a solid sphere of radius $a$ is prescribed to be $T=T_0(1-\cos \theta)$. Find the steady-state temperature distribution at any point inside the sphere. Plot the constant-temperature profiles inside the sphere, taking $a=1$ and $T_0=100$.

Victor Salazar

Numerade Educator