Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 19 Partial differential equations: separation of variables and other methods - all with Video Answers

Educators

Chapter Questions

Problem 1

Solve the following first-order partial differential equations by separating the variables:
(a) $\frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=0$;
(b) $x \frac{\partial u}{\partial x}-2 y \frac{\partial u}{\partial y}=0$.

Varsha Aggarwal

Numerade Educator

02:25

Problem 2

A conducting cube has as its six faces the planes $x=\pm a, y=\pm a$ and $z=\pm a$, and contains no internal heat sources. Verify that the temperature distribution
$$
u(x, y, z, t)=A \cos \frac{\pi x}{a} \sin \frac{\pi z}{a} \exp \left(-\frac{2 \kappa \pi^{2} t}{a^{2}}\right)
$$
obeys the appropriate diffusion equation. Across which faces is there heat flow? What is the direction and rate of heat flow at the point $(3 a / 4, a / 4, a)$ at time $t=a^{2} /\left(\kappa \pi^{2}\right) ?$

Joseph Liao

Numerade Educator

08:15

Problem 3

The wave equation describing the transverse vibrations of a stretched membrane under tension $T$ and having a uniform surface density $\rho$ is
$$
T\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)=\rho \frac{\partial^{2} u}{\partial t^{2}}
$$ Find a separable solution appropriate to a membrane stretched on a frame of length $a$ and width $b$, showing that the natural angular frequencies of such a membrane are
$$
\omega^{2}=\frac{\pi^{2} T}{\rho}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{b^{2}}\right)
$$
where $n$ and $m$ are any positive integers.

Shoukat Ali

Other Schools

09:39

Problem 4

Schrödinger's equation for a non-relativistic particle in a constant potential region can be taken as
$$
-\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}\right)=i \hbar \frac{\partial u}{\partial t}
$$
(a) Find a solution, separable in the four independent variables, that can be written in the form of a plane wave,
$$
\psi(x, y, z, t)=A \exp [i(\mathbf{k} \cdot \mathbf{r}-\omega t)]
$$
Using the relationships associated with de Broglie $(\mathbf{p}=\hbar \mathbf{k})$ and Einstein $(E=\hbar \omega)$, show that the separation constants must be such that
$$
p_{x}^{2}+p_{y}^{2}+p_{z}^{2}=2 m E
$$
(b) Obtain a different separable solution describing a particle confined to a box of side $a(\psi$ must vanish at the walls of the box). Show that the energy of the particle can only take the quantised values
$$
E=\frac{\hbar^{2} \pi^{2}}{2 m a^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)
$$
where $n_{x}, n_{y}, n_{z}$ are integers.

Khoobchandra Agrawal

Numerade Educator

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Problem 5

Denoting the three terms of $\nabla^{2}$ in spherical polars by $\nabla_{r}^{2}, \nabla_{\theta}^{2}, \nabla_{\phi}^{2}$ in an obvious way, evaluate $\nabla_{r}^{2} u$, etc. for the two functions given below and verify that, in each case, although the individual terms are not necessarily zero their sum $\nabla^{2} u$ is zero. Identify the corresponding values of $\ell$ and $m$.
(a) $u(r, \theta, \phi)=\left(A r^{2}+\frac{B}{r^{3}}\right) \frac{3 \cos ^{2} \theta-1}{2}$.
(b) $u(r, \theta, \phi)=\left(A r+\frac{B}{r^{2}}\right) \sin \theta \exp i \phi$.

Victor Salazar

Numerade Educator

03:54

Problem 6

Prove that the expression given in equation (19.47) for the associated Legendre function $P_{\ell}^{m}(\mu)$ satisfies the appropriate equation, (19.45), as follows.
(a) Evaluate $d P_{\ell}^{m}(\mu) / d \mu$ and $d^{2} P_{\ell}^{m}(\mu) / d \mu^{2}$ using the forms given in (19.47) and substitute them into (19.45).
(b) Differentiate Legendre's equation $m$ times using Leibniz' theorem.
(c) Show that the equations obtained in (a) and (b) are multiples of each other, and hence that the validity of (b) implies that of (a).

Zulfiqar Ali

Numerade Educator

05:53

Problem 7

Use the expressions at the end of subsection $19.3 .2$ to verify for $\ell=0,1,2$ that
$$
\sum_{m=-l}^{\prime}\left|Y_{\ell}^{m}(\theta, \phi)\right|^{2}=\frac{2 \ell+1}{4 \pi}
$$
and so is independent of the values of $\theta$ and $\phi .$ This is true for any $\ell$, but a general proof is more involved. This result helps to reconcile intuition with the apparently arbitrary choice of polar axis in a general quantum mechanical system.

Andrija Isakov

Numerade Educator

05:56

Problem 8

Express the function
$$
f(\theta, \phi)=\sin \theta\left[\sin ^{2}(\theta / 2) \cos \phi+i \cos ^{2}(\theta / 2) \sin \phi\right]+\sin ^{2}(\theta / 2)
$$
as a sum of spherical harmonics.

Eduard Sanchez

Numerade Educator

07:11

Problem 9

Continue the analysis of exercise $10.20$, concerned with the flow of a very viscous fluid past a sphere, to find the full expression for the stream function $\psi(r, \theta)$. At the surface of the sphere $r=a$ the velocity field $\mathbf{u}=\mathbf{0}$, whilst far from the sphere $\psi \simeq\left(U r^{2} \sin ^{2} \theta\right) / 2$

Show that $f(r)$ can be expressed as a superposition of powers of $r$, and determine which powers give acceptable solutions. Hence show that
$$
\psi(r, \theta)=\frac{U}{4}\left(2 r^{2}-3 a r+\frac{a^{3}}{r}\right) \sin ^{2} \theta
$$.

Khoobchandra Agrawal

Numerade Educator

22:53

Problem 10

The motion of a very viscous fluid in the two-dimensional (wedge) region $-\alpha<$ $\phi<\alpha$ can be described in ( $\rho, \phi$ ) coordinates by the (biharmonic) equation
$$
\nabla^{2} \nabla^{2} \psi \equiv \nabla^{4} \psi=0
$$
together with the boundary conditions $\partial \psi / \partial \phi=0$ at $\phi=\pm \alpha$, which represents the fact that there is no radial fluid velocity close to either of the bounding walls because of the viscosity, and $\partial \psi / \partial \rho=\pm \rho$ at $\phi=\pm \alpha$, which imposes the condition that azimuthal flow increases linearly with $r$ along any radial line. Assuming a solution in separated-variable form, show that the full expression for $\psi$ is
$$
\psi(\rho, \phi)=\frac{\rho^{2}}{2} \frac{\sin 2 \phi-2 \phi \cos 2 \alpha}{\sin 2 \alpha-2 \alpha \cos 2 \alpha}
$$.

Sheh Lit Chang

University of Washington

01:28

Problem 11

A circular disk of radius $a$ is such a way that its perimeter $\rho=a$ is maintained with a temperature distribution $A+B \cos ^{2} \phi$, where $\rho$ and $\phi$ are plane polar coordinates and $A$ and $B$ are constants. Find the temperature $T(\rho, \phi)$ everywhere in the region $\rho<a$.

Hast Aggarwal

Numerade Educator

05:12

Problem 12

(a) Find the form of the solution of Laplace's equation in plane polar coordinates $\rho, \phi$ that takes the value $+1$ for $0<\phi<\pi$ and the value $-1$ for $-\pi<\phi<0$ when $\rho=a$
(b) For a point $(x, y)$ on or inside the circle $x^{2}+y^{2}=a^{2}$, identify the angles $\alpha$ and $\beta$ defined by
$$
\alpha=\tan ^{-1} \frac{y}{a+x} \quad \text { and } \quad \beta=\tan ^{-1} \frac{y}{a-x}
$$
Show that $u(x, y)=(2 / \pi)(\alpha+\beta)$ is a solution of Laplace's equation that satisfies the boundary conditions given in (a).
(c) Deduce a Fourier series expansion for the function
$$
\tan ^{-1} \frac{\sin \phi}{1+\cos \phi}+\tan ^{-1} \frac{\sin \phi}{1-\cos \phi}
$$.

Aman Gupta

Numerade Educator

01:26

Problem 13

The free transverse vibrations of a thick rod satisfy the equation
$$
a^{4} \frac{\partial^{4} u}{\partial x^{4}}+\frac{\partial^{2} u}{\partial t^{2}}=0
$$
Obtain a solution in separated-variable form and, for a rod clamped at one end, $x=0$, and free at the other, $x=L$, show that the angular frequency of vibration $\omega$ satisfies
$$
\cosh \left(\frac{\omega^{1 / 2} L}{a}\right)=-\sec \left(\frac{\omega^{1 / 2} L}{a}\right)
$$. (At a clamped end both $u$ and $\partial u / \partial x$ vanish, whilst at a free end, where there is no bending moment, $\partial^{2} u / \partial x^{2}$ and $\partial^{3} u / \partial x^{3}$ are both zero.)

Manik Pulyani

Numerade Educator

01:07

Problem 14

A membrane is stretched between two concentric rings of radii $a$ and $b(b>a)$. If the smaller ring is transversely distorted from the planar configuration by an amount $c|\phi|,-\pi \leq \phi \leq \pi$, show that the membrane then has a shape given by
$$
u(\rho, \phi)=\frac{c \pi}{2} \frac{\ln (b / \rho)}{\ln (b / a)}-\frac{4 c}{\pi} \sum_{m \text { odd }} \frac{a^{m}}{m^{2}\left(b^{2 m}-a^{2 m}\right)}\left(\frac{b^{2 m}}{\rho^{m}}-\rho^{m}\right) \cos m \phi
$$.

Carson Merrill

Numerade Educator

03:16

Problem 15

A string of length $L$, fixed at its two ends, is plucked at its mid-point by an amount $A$ and then released. Prove that the subsequent displacement is given by
$$
u(x, t)=\sum_{n=0}^{\infty} \frac{8 A}{\pi^{2}(2 n+1)^{2}} \sin \left[\frac{(2 n+1) \pi x}{L}\right] \cos \left[\frac{(2 n+1) \pi c t}{L}\right]
$$
where, in the usual notation, $c^{2}=T / \rho$.
Find the total kinetic energy of the string when it passes through its unplucked position, by calculating it in each mode (each $n$ ) and summing, using the result
$$
\sum_{0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8}
$$
Confirm that the total energy is equal to the work done in plucking the string initially.

Khoobchandra Agrawal

Numerade Educator

01:42

Problem 16

Prove that the potential for $\rho<a$ associated with a vertical split cylinder of radius $a$, the two halves of which $(\cos \phi>0$ and $\cos \phi<0$ ) are maintained at equal and opposite potentials $\pm V$, is given by
$$
u(\rho, \phi)=\frac{4 V}{\pi} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1}\left(\frac{\rho}{a}\right)^{2 n+1} \cos (2 n+1) \phi
$$.

Manik Pulyani

Numerade Educator

01:55

Problem 17

A conducting spherical shell of radius $a$ is cut round its equator and the two halves connected to voltages of $+V$ and $-V .$ Show that an expression for the potential at the point $(r, \theta, \phi)$ anywhere inside the two hemispheres is
$$
u(r, \theta, \phi)=V \sum_{n=0}^{\infty} \frac{(-1)^{n}(2 n) !(4 n+3)}{2^{2 n+1} n !(n+1) !}\left(\frac{r}{a}\right)^{2 n+1} P_{2 n+1}(\cos \theta)
$$
(This is the spherical polar analogue of the previous question.)

Mahipal Kumawat

Numerade Educator

09:40

Problem 18

A slice of biological material of thickness $L$ is placed into a solution of a radioactive isotope of constant concentration $C_{0}$ at time $t=0 .$ For a later time $t$ find the concentration of radioactive ions at a depth $x$ inside one of its surfaces if the diffusion constant is $\kappa$.

Dorcas Attuabea Addo

Numerade Educator

03:34

Problem 19

Two identical copper bars are each of length $a .$ Initially, one is at $0^{\circ} \mathrm{C}$ and the other at $100^{\circ} \mathrm{C}$; they are then joined together end to end and thermally isolated. Obtain in the form of a Fourier series an expression $u(x, t)$ for the temperature at any point a distance $x$ from the join at a later time $t$. (Bear in mind the heat flow conditions at the free ends of the bars.)

Taking $a=0.5 \mathrm{~m}$ estimate the time it takes for one of the free ends to attain a temperature of $55^{\circ} \mathrm{C}$. The thermal conductivity of copper is $3.8 \times$ $10^{2} \mathrm{~J} \mathrm{~m}^{-1} \mathrm{~K}^{-1} \mathrm{~s}^{-1}$, and its specific heat capacity is $3.4 \times 10^{6} \mathrm{~J} \mathrm{~m}^{-3} \mathrm{~K}^{-1}$.

Khoobchandra Agrawal

Numerade Educator

02:34

Problem 20

A sphere of radius $a$ and thermal conductivity $k_{1}$ is surrounded by an infinite medium of conductivity $k_{2}$ in which, far away, the temperature tends to $T_{\infty}$. A distribution of heat sources $q(\theta)$ embedded in the sphere's surface establish steady temperature fields $T_{1}(r, \theta)$ inside the sphere and $T_{2}(r, \theta)$ outside it. It can be shown, by considering the heat flow through a small volume that includes part of the sphere's surface, that
$$
k_{1} \frac{\partial T_{1}}{\partial r}-k_{2} \frac{\partial T_{2}}{\partial r}=q(\theta) \quad \text { on } \quad r=a
$$
Given that
$$
q(\theta)=\frac{1}{a} \sum_{n=0}^{\infty} q_{n} P_{n}(\cos \theta)
$$
find complete expressions for $T_{1}(r, \theta)$ and $T_{2}(r, \theta)$. What is the temperature at the centre of the sphere?

Mahendra K

Numerade Educator

01:02

Problem 21

Using result (19.77) from the worked example in the text, find the general expression for the temperature $u(x, t)$ in the bar, given that the temperature distribution at time $t=0$ is $u(x, 0)=\exp \left(-x^{2} / a^{2}\right)$.

Raj Bala

Numerade Educator

21:00

Problem 22

(a) Show that the gravitational potential due to a uniform disc of radius $a$ and mass $M$, centred at the origin, is given for $r<a$ by
$$
\frac{2 G M}{a}\left[1-\frac{r}{a} P_{1}(\cos \theta)+\frac{1}{2}\left(\frac{r}{a}\right)^{2} P_{2}(\cos \theta)-\frac{1}{8}\left(\frac{r}{a}\right)^{4} P_{4}(\cos \theta)+\cdots\right]
$$
and for $r>a$ by
$$
\frac{G M}{r}\left[1-\frac{1}{4}\left(\frac{a}{r}\right)^{2} P_{2}(\cos \theta)+\frac{1}{8}\left(\frac{a}{r}\right)^{4} P_{4}(\cos \theta)-\cdots\right]
$$
where the polar axis is normal to the plane of the disc.
(b) Reconcile the presence of a term $P_{1}(\cos \theta)$, which is odd under $\theta \rightarrow \pi-\theta$, with the symmetry with respect to the plane of the disc of the physical system.
(c) Deduce that the gravitational field near an infinite sheet of matter of constant density $\rho$ per unit area is $2 \pi G \rho$.

Chris Trentman

Numerade Educator

08:06

Problem 23

In the region $-\infty<x, y<\infty$ and $-t \leq z \leq t$, a charge-density wave $\rho(\mathbf{r})=$ $A \cos q x$, in the $x$-direction, is represented by
$$
\rho(\mathbf{r})=\frac{e^{i q x}}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \tilde{\rho}(\alpha) e^{i x z} d \alpha
$$
The resulting potential is represented by
$$
V(\mathbf{r})=\frac{e^{i q x}}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \tilde{V}(\alpha) e^{i x z} d \alpha
$$
Determine the relationship between $\tilde{V}(\alpha)$ and $\tilde{\rho}(\alpha)$, and hence show that the potential at the point $(x, 0,0)$ is
$$
\frac{A}{\pi} \int_{-\infty}^{\infty} \frac{\sin k t}{k\left(k^{2}+q^{2}\right)} d k
$$.

Bruce Edelman

Numerade Educator

01:37

Problem 24

Point charges $q$ and $-q a / b$ (with $a<b$ ) are placed respectively at a point $P$, a distance $b$ from the origin $O$, and a point $Q$ between $O$ and $P$, a distance $a^{2} / b$ from $O$. Show, by considering similar triangles $Q O S$ and $S O P$, where $S$ is any point on the surface of the sphere centred at $O$ and of radius $a$, that the net potential anywhere on the sphere due to the two charges is zero.

Use this result (backed up by the uniqueness theorem) to find the force with which a point charge $q$ placed a distance $b$ from the centre of a spherical conductor of radius $a(<b)$ is attracted to the sphere (i) if the sphere is earthed, and (ii) if the sphere is uncharged and insulated.

Dominador Tan

Numerade Educator

02:25

Problem 25

Find the Green's function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$ in the half-space $z>0$ for the solution of $\nabla^{2} \Phi=0$ with $\Phi$ specified in cylindrical polar coordinates $(\rho, \phi, z)$ on the plane $z=0$ by
$$
\Phi(\rho, \phi, z)= \begin{cases}1 & \text { for } \rho \leq 1 \\ 1 / \rho & \text { for } \rho>1\end{cases}
$$
Determine the variation of $\Phi(0,0, z)$ along the $z$-axis.

John Nicolle

Numerade Educator

12:14

Problem 26

Electrostatic charge is distributed in a sphere of radius $R$ centred on the origin. Determine the form of the resultant potential $\phi(\mathbf{r})$ at distances much greater than $R$, as follows.
(a) express in the form of an integral over all space the solution of
$$
\nabla^{2} \phi=-\frac{\rho(\mathbf{r})}{\epsilon_{0}}
$$
(b) show that, for $r \gg r^{\prime}$,
$$
\left|\mathbf{r}-\mathbf{r}^{\prime}\right|=r-\frac{\mathbf{r} \cdot \mathbf{r}^{\prime}}{r}+\mathrm{O}\left(\frac{1}{r}\right)
$$
(c) use results (a) and (b) to show that $\phi(\mathbf{r})$ has the form
$$
\phi(\mathbf{r})=\frac{M}{r}+\frac{\mathbf{d} \cdot \mathbf{r}}{r^{3}}+\mathrm{O}\left(\frac{1}{r^{3}}\right)
$$
Find expressions for $M$ and $\mathbf{d}$, and identify them physically.

Laszlo Zalavari

Numerade Educator

01:18

Problem 27

Find, in the form of an infinite series the Green's function of the $\nabla^{2}$ operator for the Dirichlet problem in the region $-\infty<x<\infty,-\infty<y<\infty,-c \leq z \leq c$.

Hast Aggarwal

Numerade Educator

01:42

Problem 28

Find the Green's function for the three-dimensional Neumann problem
$$
\nabla^{2} \phi=0 \quad \text { for } z>0 \quad \text { and } \quad \frac{\partial \phi}{\partial z}=f(x, y) \quad \text { on } z=0
$$
Determine $\phi(x, y, z)$ if
$$
f(x, y)= \begin{cases}\delta(y) & \text { for }|x|<a. \\ 0 & \text { for }|x| \geq a.\end{cases}
$$

Sriram Soundarrajan

Numerade Educator

06:00

Problem 29

(a) By applying the divergence theorem to the volume integral
$$
\int_{V}\left[\phi\left(\nabla^{2}-m^{2}\right) \psi-\psi\left(\nabla^{2}-m^{2}\right) \phi\right] d V
$$
obtain a Green's function expression, as the sum of a volume integral and a surface integral, for $\phi\left(\mathbf{r}^{\prime}\right)$ that satisfies
$$
\nabla^{2} \phi-m^{2} \phi=\rho
$$
in $V$ and takes the specified form $\phi=f$ on $S$, the boundary of $V .$ The Green's function $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ to be used satisfies
$$
\nabla^{2} G-m^{2} G=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)
$$
and vanishes when $\mathbf{r}$ is on $S .$
(b) When $V$ is all space, $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ can be written as $G(t)=g(t) / t$ where $t=\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$ and $g(t)$ is bounded as $t \rightarrow \infty .$ Find the form of $G(t)$.
(c) Find $\phi(\mathbf{r})$ in the half space $x>0$ if $\rho(\mathbf{r})=\delta\left(\mathbf{r}-\mathbf{r}_{1}\right)$ and $\phi=0$ both on $x=0$ and as $r \rightarrow \infty$ .

Arwa Ali

Numerade Educator

01:41

Problem 30

Consider the PDE $\mathcal{L} u(\mathbf{r})=\rho(\mathbf{r})$, for which the differential operator $\mathcal{L}$ is given by
$$
\mathcal{L}=\nabla \cdot[p(\mathbf{r}) \nabla]+q(\mathbf{r})
$$
where $p(\mathbf{r})$ and $q(\mathbf{r})$ are functions of position. By proving the generalised form of Green's theorem,
$$
\int_{V}(\phi \mathcal{L} \psi-\psi \mathcal{L} \phi) d V=\oint_{S} p(\phi \nabla \psi-\psi \nabla \phi) \cdot \hat{\mathbf{n}} d S
$$
show that the solution of the PDE is given by
$$
u\left(\mathbf{r}_{0}\right)=\int_{V} G\left(\mathbf{r}, \mathbf{r}_{0}\right) \rho(\mathbf{r}) d V(\mathbf{r})+\oint_{S} p(\mathbf{r})\left[u(\mathbf{r}) \frac{\partial G\left(\mathbf{r}, \mathbf{r}_{0}\right)}{\partial n}-G\left(\mathbf{r}, \mathbf{r}_{0}\right) \frac{\partial u(\mathbf{r})}{\partial n}\right] d S(\mathbf{r})
$$
where $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$ is the Green's function satisfying $\mathcal{L} G\left(\mathbf{r}, \mathbf{r}_{0}\right)=\delta\left(\mathbf{r}-\mathbf{r}_{0}\right)$.

Manik Pulyani

Numerade Educator