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Computer Algebra Recipes

Richard H Enns, George C McGuire

Chapter 6

Linear PDE Models. Part 2 - all with Video Answers

Educators

Chapter Questions

03:12

Problem 1

A light homogeneous horizontal string of length $L$ fixed at its ends initially has a parabolic shape with height $h$ in the middle. If it is released from rest, solve the wave equation to determine its subsequent transverse displacement at arbitrary time $t$. Animate the vibrations for parameters of your own choice and choose enough frames to produce a smooth animation.

Mahendra K
Mahendra K
Numerade Educator
04:13

Problem 2

If a horizontal piano string fixed at $x=0$ and $x=L$ is struck in such a way that its initial displacement $\psi(x, 0)$ is zero and its initial transverse velocity is
$$
\dot{\psi}(x, 0)= \begin{cases}4 v x / L, & 0<x<L / 4, \\ (4 v / L)(L / 2-x), & L / 4<x<L / 2, \\ 0, & L / 2<x<L,\end{cases}
$$
solve the wave equation to determine the transverse displacement of the string for all times $t$. Taking $L=20 \mathrm{~cm}$ and $v=5 \mathrm{~cm} / \mathrm{s}$, animate the solution. Choose enough frames to produce a smooth animation. Discuss the results.

Keshav Singh
Keshav Singh
Numerade Educator
04:37

Problem 3

A square membrane whose sides are of unit length is given an initial transverse displacement $\psi(x, y, 0)=x y(1-x)(1-y)$ and then released. Determine the displacement $\psi(x, y, t)$ of the membrane for $t>0$ and animate the solution for nominal values of the parameters.

James Kiss
James Kiss
Numerade Educator
04:37

Problem 4

Consider a rectangular membrane having sides of length $a$ between $x=0$ and $x=a$ and sides of length $2 a$ between $y=0$ and $y=2 a$. The edges at $x=0$ and $x=a$ are fixed, while those at $y=0$ and $y=2 a$ are "free." At a free edge, the slope is zero. Explicitly determine $\psi(x, y, t)$ if the membrane is initially at rest and has the initial shape
$$
\psi(x, y, 0)= \begin{cases}2 x h / a, & 0 \leq x \leq a / 2, \\ (2 h / a)(a-x), & a / 2 \leq x \leq a .\end{cases}
$$
Animate the solution for $a=1, h=1$, and $c=1$.

James Kiss
James Kiss
Numerade Educator
02:50

Problem 5

A circular drumhead of radius $r=a$ and fixed on its perimeter is displaced a distance $h$ at its center at time $t=0$ so that it has a conical shape. Determine the displacement of the membrane for $t>0$. Taking $a=h=1$ and $c=1$, animate the motion of the membrane.

Anand Jangid
Anand Jangid
Numerade Educator
01:28

Problem 6

A circular drumhead of radius $a$ fixed on its outer edge has an initial displacement $\psi(r, \theta, 0)=\left(1-r^2 / a^2\right) \sin (4 \theta)$ and its initial velocity is zero. Determine the subsequent displacement $\psi(r, \theta, t)$, and animate the solution for parameters of your own choosing.

Arpit Gupta
Arpit Gupta
Numerade Educator
03:21

Problem 7

A sound wave of frequency $\omega$ is generated at one end $(z=0)$ of a very long straight cylindrical pipe of radius $a$ having rigid walls.
(a) Determine and discuss in detail the allowed modes of propagation and the cutoff frequency for wave propagation. The cutoff frequency is the minimum frequency for propagation of a specific mode.
(b) If the speed of sound in air is 1100 feet per second and if the frequency of the sound wave is $500 \mathrm{~Hz}$, show that only a plane wave will be propagated if $a<7.73$ inches.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:46

Problem 8

An infinitely long circular cylinder of radius $r=a$ is surrounded by an ideal compressible fluid. The cylinder's surface is vibrating with a radial velocity $V_0 \cos (\omega t)$. The fluid velocity is given by $\vec{v}=-\nabla \phi(r, t)$, where $\phi$ satisfies the wave equation in cylindrical coordinates and $r$ is measured from the cylinder axis. Assuming that the cylinder's surface is rigid, the fluid velocity must equal the velocity of the vibrating surface. Noting that far from the surface the waves in the fluid must be outgoing from the cylinder, analytically determine $\vec{v}(r, t)$ in the fluid and animate the solution for nominal values of the parameters. Note that the Bessel function of the second kind must be kept, since the origin of the cylindrical coordinates lies inside the cylinder and outside the fluid.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 9

Calculate the Fourier sine transform of each $f(x)$ below and plot the answers:
(a) $f(x)=e^{-3|x|}$;
(b) $f(x)=\cos (2 x)$;
(c) $f(x)=\sin (x)^2$;
(d) $f(x)=x /\left(x^2+1\right)$.

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01:12

Problem 10

Use the Fourier sine transform approach to solve the heat conduction problem for $d=10 \mathrm{~cm}^2 / \mathrm{s}$ in a semi-infinite rod $(0 \leq x \leq \infty)$ that has the boundary condition $T(0, t)=0$ and initial interior temperature distribution $T(x, 0)=$ $10 x /\left(x^2+1\right)$. Animate the plot.

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 11

Calculate the Fourier cosine transform of each $f(x)$ below and plot the answers where possible:
(a) $f(x)=e^{-3|x|}$;
(b) $f(x)=\cos (2 x)$;
(c) $f(x)=\sin (x)^2$;
(d) $f(x)=x /\left(x^2+1\right)$.

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Problem 12

Modify the text recipe to find the temperature distribution inside a semi-infinite $\operatorname{rod}(0 \leq x \leq \infty)$ that is insulated at $x=0$ and has the initial temperature distribution $\bar{T}(x>0,0)=25 x^2 /\left(x^2+25\right)$ and $d=1$. Animate $T(x, t)$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 13

Calculate the Laplace transforms of the following functions, simplifying the answer where necessary, and identifying any special functions that occur:
(a) $f(t)=e^{-a \sqrt{t}}$ with $a>0$;
(b) $f(t)=t \cos (a t)$;
(c) $f(t)=\arctan (t)$;
(d) $f(t)=t^n \ln (t)$ with $n>0$;
(e) $f(t)=\tanh (t)$;
(f) $f(t)=\tanh ^{-1}(t)$;
(g) $f(t)=\frac{\sin (3 \sqrt{t})}{t^{1 / 4}}$;
(h) $f(t)=J_0(t) J_1(t)$.

Victor Salazar
Victor Salazar
Numerade Educator
04:29

Problem 14

Calculate the inverse Laplace transforms of the following functions, identifying any special functions that occur in the answer:
(a) $F(s)=\frac{1}{s^2}$;
(b) $F(s)=\frac{a}{s^2+a^2}$;
(c) $F(s)=\frac{s^2}{\left(s^2+a^2\right)^{3 / 2}}$;
(d) $F(s)=\frac{1}{\sqrt{s^2+a^2}}$;
(e) $F(s)=e^{-a s}$;
(f) $F(s)=\frac{\sin (a s)}{s}$

Arpit Gupta
Arpit Gupta
Numerade Educator
01:17

Problem 15

Consider a semi-infinite rod spanning the range $x=0$ to $x=\infty$. The initial temperature of the rod is zero. For $t>0$, the temperature at $x=0$ is $T(x, 0)=T_0$. By Laplace transforming the temporal part of the diffusion equation, determine the temperature distribution inside the rod for $t>0$. Animate the temperature profile for nominal values of the parameters.

Aman Gupta
Aman Gupta
Numerade Educator
01:05

Problem 16

The heat flow along an insulated semi-infinite bar whose cross section varies exponentially is described by
$$
\frac{\partial}{\partial x}\left(e^{\alpha x} \frac{\partial T}{\partial x}\right)=e^{\alpha x} \frac{\partial T}{\partial t} .
$$
If $T(x, 0)=0$ for $x>0, T(0, t)=1$, and $T(\infty, t)=0$, use the Laplace transform approach to show that for $t>0$, the temperature distribution in the bar is
$$
T(x, t)=\frac{e^{-\alpha x}}{2}\left(\operatorname{erfc}\left(\frac{1}{2}\left(x t^{-1 / 2}-\alpha t^{1 / 2}\right)\right)+e^{\alpha x} \operatorname{erfc}\left(\frac{1}{2}\left(x t^{-1 / 2}+\alpha t^{1 / 2}\right)\right)\right) .
$$
Taking $\alpha=1$, animate $T(x, t)$ over the range $x=0$ to 5 for $t=0$ to 10 .

Manik Pulyani
Manik Pulyani
Numerade Educator
04:23

Problem 17

An important property of the Laplace transform is the convolution theorem. If $f_1(t)$ and $f_2(t)$ are two functions, their convolution is defined to be
$$
C(T)=\int_0^T f_1(T-t) f_2(t) d t
$$
If $F(s), F_1(s)$, and $F_2(s)$ are the Laplace transforms of $C(T), f_1(t)$, and $f_2(t)$, respectively, the convolution theorem states that
$$
F(s)=F_1(s) F_2(s) .
$$
Using the integral transform package, take the Laplace transform of $C(t)$ and confirm the convolution theorem.

Amany Waheeb
Amany Waheeb
Numerade Educator

Problem 18

Calculate the Fourier transforms of the following functions and plot the results:
(a) $f(x)=e^{-3|x|}$;
(b) $f(x)=\cos (2 x)$;
(c) $f(x)=x /\left(x^2+1\right)$.

Check back soon!
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Problem 19

Calculate the inverse Fourier transforms of the following functions, simplifying where necessary, and plot the results:
(a) $f(k)=\cos (\pi k / 2) /\left(1-k^2\right)$;
(b) $f(k)=-2 I k /\left(\pi\left(k^2+2\right)\right)$;
(c) $f(k)=(1 / \sqrt{2 \pi})(\sin k / k)^2$.

Victor Salazar
Victor Salazar
Numerade Educator
02:46

Problem 20

An approximately monochromatic plane wave packet in one dimension has the instantaneous form $u(x, 0)=f(x) e^{I k_0 x}$, with $f(x)$ the envelope function and $k_0$ the central wave number. Consider the following functions:
(a) $f(x)=2 e^{-3|x| / 2}$;
(b) $f(x)=4 e^{-x^2 / 4}$;
(c) $f(x)=5$ for $|x|<1$ and 0 otherwise.
For each function, perform the following:
$\bullet$ Calculate the wave number spectrum $|A(k)|^2$ where $A(k)$ is the Fourier transform of $f(x)$.
$\bullet$ Plot the intensities $|u(x, 0)|^2$ and $|A(k)|^2$.
$\bullet$ Explicitly evaluate the root mean square deviations from the means, $\Delta x$ and $\Delta k$, defined with respect to the above intensities.
$\bullet$ Show that in each case the bandwidth theorem (the optical analogue of the uncertainty principle) $\Delta x \Delta k \geq \frac{1}{2}$ is satisfied.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
01:02

Problem 21

By Fourier transforming the spatial part of the diffusion equation, determine the temperature distribution $T(x, t)$ in an infinite rod for $t>0$ when $T(x, 0)=T_0$ for $|x|<x_0$, and zero otherwise. Animate $T(x, t)$ for nominal values of the parameters.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:02

Problem 22

By Fourier transforming the spatial part of the diffusion equation, determine the temperature distribution $T(x, t)$ in an infinite rod for $t>0$ when $T(x, 0)=$ $T_0 e^{-\alpha^2 x^2}$. Animate $T(x, t)$ for nominal values of the parameters.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:28

Problem 23

In the text recipe, confirm that the solution becomes numerically unstable for $r>0.5$. Numerical instability is signaled by the appearance of increasingly wild oscillations in the solution as time increases.

Manish Jain
Manish Jain
Numerade Educator
03:04

Problem 24

Explore the change in the percentage error in the numerical mesh values at the end of the run compared with the exact values as $M$ is increased.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:05

Problem 25

Modify the text recipe to produce an animated numerical solution for the initial temperature profile $T(x, 0)=25 \sin (\pi x)$. Compare the numerical solution in the center of the rod with the exact solution as a function of time. At what time is the temperature in the middle equal to one-quarter of the initial value?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:59

Problem 26

For $a=0$, the KGE reduces to the linear wave equation. Run the code for $a=0$ and then explore how the results are affected by increasing $a$.

M S
M S
Numerade Educator
01:19

Problem 27

Modify the text recipe for the KGE to handle the initial conditions $f(x)=0$, $g(x)=\sin (\pi x)$. You may have to adjust the viewing box.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 28

Modify the text recipe to numerically simulate the nonlinear KGE (6.25). Take the same parameters as in text recipe and explore what happens when increasing positive values of $b$ are considered.

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05:07

Problem 29

A better approximation to the first time row to use for solving the KGE is
$$
\psi_{i, 1}=f\left(x_i\right)+k g\left(x_i\right)+\frac{r}{2}\left(f\left(x_{i+1}\right)-2 f\left(x_i\right)+f\left(x_{i-1}\right)\right)+\mathrm{O}\left(k^3\right) .
$$
Execute the text recipe with this improved approximation.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
01:09

Problem 30

The steady-state temperature distribution $T(x, y)$ in a thin square metal plate $0.5 \mathrm{~m}$ on a side satisfies Laplace's equation, $\nabla^2 T(x, y)=0$. The boundary conditions on the edges of the plate are
$$
T(0, y)=0, \quad T(x, 0)=0, \quad T(x, 0.5)=200 x, \quad T(0.5, y)=200 y .
$$
Using the standard CDA for the second derivatives, and choosing a suitable mesh spacing, numerically determine the temperature distribution in the plate and make a 3-dimensional plot.

Raj Bala
Raj Bala
Numerade Educator
07:20

Problem 31

A square inner conductor $3 \mathrm{~cm}$ on a side is held at a potential of $100 \mathrm{~V}$. A second square conductor, concentric with the first and $9 \mathrm{~cm}$ long on each of its inner sides, is held at $0 \mathrm{~V}$. The potential $\Phi(x, y)$ in the region between the two conductors satisfies Laplace's equation, $\nabla^2 \Phi(x, y)=0$.
(a) Taking the mesh spacing in both directions to be $1 \mathrm{~cm}$, make a mesh diagram showing all the interior mesh points for which $\Phi$ is to be found.
(b) Using CDAs for the second derivatives, write out the mesh equations for the interior points. Make use of symmetry arguments to show that only seven interior points need to be used in the calculation of $\Phi$.
(c) Solve the mesh equations and determine $\Phi$ at each interior point.
(d) Plot $\Phi$ in the region stretching from the inner to the outer conductor.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator