• Paper 1, Section II, C

(a) By introducing the variables $\xi=x+c t$ and $\eta=x-c t$ (where $c$ is a constant), derive d'Alembert's solution of the initial value problem for the wave equation:

$u_{t t}-c^{2} u_{x x}=0, \quad u(x, 0)=\phi(x), \quad u_{t}(x, 0)=\psi(x)$

where $-\infty and $\phi$ and $\psi$ are given functions (and subscripts denote partial derivatives).

(b) Consider the forced wave equation with homogeneous initial conditions:

$u_{t t}-c^{2} u_{x x}=f(x, t), \quad u(x, 0)=0, \quad u_{t}(x, 0)=0$

where $-\infty and $f$ is a given function. You may assume that the solution is given by

$u(x, t)=\frac{1}{2 c} \int_{0}^{t} \int_{x-c(t-s)}^{x+c(t-s)} f(y, s) d y d s$

For the forced wave equation $u_{t t}-c^{2} u_{x x}=f(x, t)$, now in the half space $x \geqslant 0$ (and with $t \geqslant 0$ as before), find (in terms of $f$ ) the solution for $u(x, t)$ that satisfies the (inhomogeneous) initial conditions

$u(x, 0)=\sin x, \quad u_{t}(x, 0)=0, \quad \text { for } x \geqslant 0$

and the boundary condition $u(0, t)=0$ for $t \geqslant 0$.

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• Paper 2, Section I, C

Consider the differential operator

$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}$

acting on real functions $y(x)$ with $0 \leqslant x \leqslant 1$.

(i) Recast the eigenvalue equation $\mathcal{L} y=-\lambda y$ in Sturm-Liouville form $\tilde{\mathcal{L}} y=-\lambda w y$, identifying $\tilde{\mathcal{L}}$ and $w$.

(ii) If boundary conditions $y(0)=y(1)=0$ are imposed, show that the eigenvalues form an infinite discrete set $\lambda_{1}<\lambda_{2}<\ldots$ and find the corresponding eigenfunctions $y_{n}(x)$ for $n=1,2, \ldots$. If $f(x)=x-x^{2}$ on $0 \leqslant x \leqslant 1$ is expanded in terms of your eigenfunctions i.e. $f(x)=\sum_{n=1}^{\infty} A_{n} y_{n}(x)$, give an expression for $A_{n}$. The expression can be given in terms of integrals that you need not evaluate.

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• Paper 2, Section II, A

The Fourier transform $\tilde{f}(k)$ of a function $f(x)$ and its inverse are given by

$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x, \quad f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) e^{i k x} d k$

(a) Calculate the Fourier transform of the function $f(x)$ defined by:

$f(x)= \begin{cases}1 & \text { for } 0

(b) Show that the inverse Fourier transform of $\tilde{g}(k)=e^{-\lambda|k|}$, for $\lambda$ a positive real constant, is given by

$g(x)=\frac{\lambda}{\pi\left(x^{2}+\lambda^{2}\right)}$

(c) Consider the problem in the quarter plane $0 \leqslant x, 0 \leqslant y$ :

\begin{aligned} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}} &=0 ; \\ u(x, 0) &= \begin{cases}1 & \text { for } 0

Use the answers from parts (a) and (b) to show that

$u(x, y)=\frac{4 x y}{\pi} \int_{0}^{1} \frac{v d v}{\left[(x-v)^{2}+y^{2}\right]\left[(x+v)^{2}+y^{2}\right]}$

(d) Hence solve the problem in the quarter plane $0 \leqslant x, 0 \leqslant y$ :

\begin{aligned} \frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}} &=0 ; \\ w(x, 0) &= \begin{cases}1 & \text { for } 0

[You may quote without proof any property of Fourier transforms.]

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• Paper 3, Section I, A

Let $f(\theta)$ be a $2 \pi$-periodic function with Fourier expansion

$f(\theta)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)$

Find the Fourier coefficients $a_{n}$ and $b_{n}$ for

f(\theta)=\left\{\begin{aligned} 1, & 0<\theta<\pi \\ -1, & \pi<\theta<2 \pi \end{aligned}\right.

Hence, or otherwise, find the Fourier coefficients $A_{n}$ and $B_{n}$ for the $2 \pi$-periodic function $F$ defined by

$F(\theta)=\left\{\begin{array}{cc} \theta, & 0<\theta<\pi \\ 2 \pi-\theta, & \pi<\theta<2 \pi \end{array}\right.$

$\sum_{r=0}^{\infty} \frac{(-1)^{r}}{2 r+1} \quad \text { and } \quad \sum_{r=0}^{\infty} \frac{1}{(2 r+1)^{2}}$

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• Paper 3, Section II, A

Let $P(x)$ be a solution of Legendre's equation with eigenvalue $\lambda$,

$\left(1-x^{2}\right) \frac{d^{2} P}{d x^{2}}-2 x \frac{d P}{d x}+\lambda P=0$

such that $P$ and its derivatives $P^{(k)}(x)=d^{k} P / d x^{k}, k=0,1,2, \ldots$, are regular at all points $x$ with $-1 \leqslant x \leqslant 1$.

(a) Show by induction that

$\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}\left[P^{(k)}\right]-2(k+1) x \frac{d}{d x}\left[P^{(k)}\right]+\lambda_{k} P^{(k)}=0$

for some constant $\lambda_{k}$. Find $\lambda_{k}$ explicitly and show that its value is negative when $k$ is sufficiently large, for a fixed value of $\lambda$.

(b) Write the equation for $P^{(k)}(x)$ in part (a) in self-adjoint form. Hence deduce that if $P^{(k)}(x)$ is not identically zero, then $\lambda_{k} \geqslant 0$.

[Hint: Establish a relation between integrals of the form $\int_{-1}^{1}\left[P^{(k+1)}(x)\right]^{2} f(x) d x$ and $\int_{-1}^{1}\left[P^{(k)}(x)\right]^{2} g(x) d x$ for certain functions $f(x)$ and $\left.g(x) .\right]$

(c) Use the results of parts (a) and (b) to show that if $P(x)$ is a non-zero, regular solution of Legendre's equation on $-1 \leqslant x \leqslant 1$, then $P(x)$ is a polynomial of degree $n$ and $\lambda=n(n+1)$ for some integer $n=0,1,2, \ldots$

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• Paper 4, Section II, C

The function $\theta(x, t)$ obeys the diffusion equation

$\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}}$

Verify that

$\theta(x, t)=\frac{1}{\sqrt{t}} e^{-x^{2} / 4 D t}$

is a solution of $(*)$, and by considering $\int_{-\infty}^{\infty} \theta(x, t) d x$, find the solution having the initial form $\theta(x, 0)=\delta(x)$ at $t=0$.

Find, in terms of the error function, the solution of $(*)$ having the initial form

$\theta(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases}$

Sketch a graph of this solution at various times $t \geqslant 0$.

[The error function is

$\left.\operatorname{Erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-y^{2}} d y .\right]$

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• Paper 1, Section II, B

Consider the equation

$\nabla^{2} \phi=\delta(x) g(y)$

on the two-dimensional strip $-\infty, where $\delta(x)$ is the delta function and $g(y)$ is a smooth function satisfying $g(0)=g(a)=0 . \phi(x, y)$ satisfies the boundary conditions $\phi(x, 0)=\phi(x, a)=0$ and $\lim _{x \rightarrow \pm \infty} \phi(x, y)=0$. By using solutions of Laplace's equation for $x<0$ and $x>0$, matched suitably at $x=0$, find the solution of $(*)$ in terms of Fourier coefficients of $g(y)$.

Find the solution of $(*)$ in the limiting case $g(y)=\delta(y-c)$, where $0, and hence determine the Green's function $\phi(x, y)$ in the strip, satisfying

$\nabla^{2} \phi=\delta(x-b) \delta(y-c)$

and the same boundary conditions as before.

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• Paper 2, Section I, B

Find the Fourier transform of the function

$f(x)= \begin{cases}A, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases}$

Determine the convolution of the function $f(x)$ with itself.

State the convolution theorem for Fourier transforms. Using it, or otherwise, determine the Fourier transform of the function

$g(x)= \begin{cases}B(2-|x|), & |x| \leqslant 2 \\ 0, & |x|>2\end{cases}$

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• Paper 2, Section II, A

(i) The solution to the equation

$\frac{d}{d x}\left(x \frac{d F}{d x}\right)+\alpha^{2} x F=0$

that is regular at the origin is $F(x)=C J_{0}(\alpha x)$, where $\alpha$ is a real, positive parameter, $J_{0}$ is a Bessel function, and $C$ is an arbitrary constant. The Bessel function has infinitely many zeros: $J_{0}\left(\gamma_{k}\right)=0$ with $\gamma_{k}>0$, for $k=1,2, \ldots$. Show that

$\int_{0}^{1} J_{0}(\alpha x) J_{0}(\beta x) x d x=\frac{\beta J_{0}(\alpha) J_{0}^{\prime}(\beta)-\alpha J_{0}(\beta) J_{0}^{\prime}(\alpha)}{\alpha^{2}-\beta^{2}}, \quad \alpha \neq \beta$

(where $\alpha$ and $\beta$ are real and positive) and deduce that

$\int_{0}^{1} J_{0}\left(\gamma_{k} x\right) J_{0}\left(\gamma_{\ell} x\right) x d x=0, \quad k \neq \ell ; \quad \int_{0}^{1}\left(J_{0}\left(\gamma_{k} x\right)\right)^{2} x d x=\frac{1}{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}$

[Hint: For the second identity, consider $\alpha=\gamma_{k}$ and $\beta=\gamma_{k}+\epsilon$ with $\epsilon$ small.]

(ii) The displacement $z(r, t)$ of the membrane of a circular drum of unit radius obeys

$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial r}\right)=\frac{\partial^{2} z}{\partial t^{2}}, \quad z(1, t)=0$

where $r$ is the radial coordinate on the membrane surface, $t$ is time (in certain units), and the displacement is assumed to have no angular dependence. At $t=0$ the drum is struck, so that

$z(r, 0)=0, \quad \frac{\partial z}{\partial t}(r, 0)=\left\{\begin{array}{cc} U, & rb \end{array}\right.$

where $U$ and $b<1$ are constants. Show that the subsequent motion is given by

$z(r, t)=\sum_{k=1}^{\infty} C_{k} J_{0}\left(\gamma_{k} r\right) \sin \left(\gamma_{k} t\right) \quad \text { where } \quad C_{k}=-2 b U \frac{J_{0}^{\prime}\left(\gamma_{k} b\right)}{\gamma_{k}^{2}\left(J_{0}^{\prime}\left(\gamma_{k}\right)\right)^{2}}$

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• Paper 1, Section II, B

The Bessel functions $J_{n}(r)(n \geqslant 0)$ can be defined by the expansion

$e^{i r \cos \theta}=J_{0}(r)+2 \sum_{n=1}^{\infty} i^{n} J_{n}(r) \cos n \theta$

By using Cartesian coordinates $x=r \cos \theta, y=r \sin \theta$, or otherwise, show that

$\left(\nabla^{2}+1\right) e^{i r \cos \theta}=0$

Deduce that $J_{n}(r)$ satisfies Bessel's equation

$\left(r^{2} \frac{d^{2}}{d r^{2}}+r \frac{d}{d r}-\left(n^{2}-r^{2}\right)\right) J_{n}(r)=0$

By expanding the left-hand side of $(*)$ up to cubic order in $r$, derive the series expansions of $J_{0}(r), J_{1}(r), J_{2}(r)$ and $J_{3}(r)$ up to this order.

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• Paper 2, Section I, B

Let $r, \theta, \phi$ be spherical polar coordinates, and let $P_{n}$ denote the $n$th Legendre polynomial. Write down the most general solution for $r>0$ of Laplace's equation $\nabla^{2} \Phi=0$ that takes the form $\Phi(r, \theta, \phi)=f(r) P_{n}(\cos \theta)$.

Solve Laplace's equation in the spherical shell $1 \leqslant r \leqslant 2$ subject to the boundary conditions

\begin{aligned} &\Phi=3 \cos 2 \theta \text { at } r=1 \\ &\Phi=0 \quad \text { at } r=2 \end{aligned}

[The first three Legendre polynomials are

$\left.P_{0}(x)=1, \quad P_{1}(x)=x \quad \text { and } \quad P_{2}(x)=\frac{3}{2} x^{2}-\frac{1}{2} .\right]$

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• Paper 2, Section II, D

For $n=0,1,2, \ldots$, the degree $n$ polynomial $C_{n}^{\alpha}(x)$ satisfies the differential equation

$\left(1-x^{2}\right) y^{\prime \prime}-(2 \alpha+1) x y^{\prime}+n(n+2 \alpha) y=0$

where $\alpha$ is a real, positive parameter. Show that, when $m \neq n$,

$\int_{a}^{b} C_{m}^{\alpha}(x) C_{n}^{\alpha}(x) w(x) d x=0$

for a weight function $w(x)$ and values $a that you should determine.

Suppose that the roots of $C_{n}^{\alpha}(x)$ that lie inside the domain $(a, b)$ are $\left\{x_{1}, x_{2}, \ldots, x_{k}\right\}$, with $k \leqslant n$. By considering the integral

$\int_{a}^{b} C_{n}^{\alpha}(x) \prod_{i=1}^{k}\left(x-x_{i}\right) w(x) d x$

show that in fact all $n$ roots of $C_{n}^{\alpha}(x)$ lie in $(a, b)$.

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• Paper 3, Section I, D

Define the discrete Fourier transform of a sequence $\left\{x_{0}, x_{1}, \ldots, x_{N-1}\right\}$ of $N$ complex numbers.

Compute the discrete Fourier transform of the sequence

$x_{n}=\frac{1}{N}\left(1+e^{2 \pi i n / N}\right)^{N-1} \quad \text { for } n=0, \ldots, N-1 .$

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• Paper 3, Section II, D

By differentiating the expression $\psi(t)=H(t) \sin (\alpha t) / \alpha$, where $\alpha$ is a constant and $H(t)$ is the Heaviside step function, show that

$\frac{d^{2} \psi}{d t^{2}}+\alpha^{2} \psi=\delta(t)$

where $\delta(t)$ is the Dirac $\delta$-function.

Hence, by taking a Fourier transform with respect to the spatial variables only, derive the retarded Green's function for the wave operator $\partial_{t}^{2}-c^{2} \nabla^{2}$ in three spatial dimensions.

[You may use that

$\frac{1}{2 \pi} \int_{\mathbb{R}^{3}} e^{i \mathbf{k} \cdot(\mathbf{x}-\mathbf{y})} \frac{\sin (k c t)}{k c} d^{3} k=-\frac{i}{c|\mathbf{x}-\mathbf{y}|} \int_{-\infty}^{\infty} e^{i k|\mathbf{x}-\mathbf{y}|} \sin (k c t) d k$

without proof.]

Thus show that the solution to the homogeneous wave equation $\partial_{t}^{2} u-c^{2} \nabla^{2} u=0$, subject to the initial conditions $u(\mathbf{x}, 0)=0$ and $\partial_{t} u(\mathbf{x}, 0)=f(\mathbf{x})$, may be expressed as

$u(\mathbf{x}, t)=\langle f\rangle t$

where $\langle f\rangle$ is the average value of $f$ on a sphere of radius $c t$ centred on $\mathbf{x}$. Interpret this result.

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• Paper 4, Section I, D

Let

$g_{\epsilon}(x)=\frac{-2 \epsilon x}{\pi\left(\epsilon^{2}+x^{2}\right)^{2}} .$

By considering the integral $\int_{-\infty}^{\infty} \phi(x) g_{\epsilon}(x) d x$, where $\phi$ is a smooth, bounded function that vanishes sufficiently rapidly as $|x| \rightarrow \infty$, identify $\lim _{\epsilon \rightarrow 0} g_{\epsilon}(x)$ in terms of a generalized function.

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• Paper 4, Section II, B

(a) Show that the operator

$\frac{d^{4}}{d x^{4}}+p \frac{d^{2}}{d x^{2}}+q \frac{d}{d x}+r$

where $p(x), q(x)$ and $r(x)$ are real functions, is self-adjoint (for suitable boundary conditions which you need not state) if and only if

$q=\frac{d p}{d x}$

(b) Consider the eigenvalue problem

$\frac{d^{4} y}{d x^{4}}+p \frac{d^{2} y}{d x^{2}}+\frac{d p}{d x} \frac{d y}{d x}=\lambda y$

on the interval $[a, b]$ with boundary conditions

$y(a)=\frac{d y}{d x}(a)=y(b)=\frac{d y}{d x}(b)=0$

Assuming that $p(x)$ is everywhere negative, show that all eigenvalues $\lambda$ are positive.

(c) Assume now that $p \equiv 0$ and that the eigenvalue problem (*) is on the interval $[-c, c]$ with $c>0$. Show that $\lambda=1$ is an eigenvalue provided that

$\cos c \sinh c \pm \sin c \cosh c=0$

and show graphically that this condition has just one solution in the range $0.

[You may assume that all eigenfunctions are either symmetric or antisymmetric about $x=0 .]$

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• Paper 1, Section II, 14C

Define the convolution $f * g$ of two functions $f$ and $g$. Defining the Fourier transform $\tilde{f}$ of $f$ by

$\tilde{f}(k)=\int_{-\infty}^{\infty} \mathrm{e}^{-\mathrm{i} k x} f(x) \mathrm{d} x$

show that

$\widehat{f * g}(k)=\tilde{f}(k) \tilde{g}(k) .$

Given that the Fourier transform of $f(x)=1 / x$ is

$\tilde{f}(k)=-\mathrm{i} \pi \operatorname{sgn}(k),$

find the Fourier transform of $\sin (x) / x^{2}$.

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• Paper 2, Section I, $5 \mathrm{C}$

Show that

$a(x, y)\left(\frac{d y}{d s}\right)^{2}-2 b(x, y) \frac{d x}{d s} \frac{d y}{d s}+c(x, y)\left(\frac{d x}{d s}\right)^{2}=0$

along a characteristic curve $(x(s), y(s))$ of the $2^{\text {nd }}$-order pde

$a(x, y) u_{x x}+2 b(x, y) u_{x y}+c(x, y) u_{y y}=f(x, y)$

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• Paper 2, Section II, A

(a) Let $f(x)$ be a $2 \pi$-periodic function (i.e. $f(x)=f(x+2 \pi)$ for all $x$ ) defined on $[-\pi, \pi]$ by

$f(x)=\left\{\begin{array}{cl} x & x \in[0, \pi] \\ -x & x \in[-\pi, 0] \end{array}\right.$

Find the Fourier series of $f(x)$ in the form

$f(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos (n x)+\sum_{n=1}^{\infty} b_{n} \sin (n x)$

(b) Find the general solution to

$y^{\prime \prime}+2 y^{\prime}+y=f(x)$

where $f(x)$ is as given in part (a) and $y(x)$ is $2 \pi$-periodic.

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• Paper 3, Section I, A

(a) Determine the Green's function $G(x ; \xi)$ satisfying

$G^{\prime \prime}-4 G^{\prime}+4 G=\delta(x-\xi),$

with $G(0 ; \xi)=G(1 ; \xi)=0$. Here ' denotes differentiation with respect to $x$.

(b) Using the Green's function, solve

$y^{\prime \prime}-4 y^{\prime}+4 y=e^{2 x}$

with $y(0)=y(1)=0$.

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• Paper 3, Section II, A

Consider the Dirac delta function, $\delta(x)$, defined by the sampling property

$\int_{-\infty}^{\infty} f(x) \delta\left(x-x_{0}\right) d x=f\left(x_{0}\right)$

for any suitable function $f(x)$ and real constant $x_{0}$.

(a) Show that $\delta(\alpha x)=|\alpha|^{-1} \delta(x)$ for any non-zero $\alpha \in \mathbb{R}$.

(b) Show that $x \delta^{\prime}(x)=-\delta(x)$, where ${ }^{\prime}$ denotes differentiation with respect to $x$.

(c) Calculate

$\int_{-\infty}^{\infty} f(x) \delta^{(m)}(x) d x$

where $\delta^{(m)}(x)$ is the $m^{\text {th }}$derivative of the delta function.

(d) For

$\gamma_{n}(x)=\frac{1}{\pi} \frac{n}{(n x)^{2}+1}$

show that $\lim _{n \rightarrow \infty} \gamma_{n}(x)=\delta(x)$.

(e) Find expressions in terms of the delta function and its derivatives for

(i)

$\lim _{n \rightarrow \infty} n^{3} x e^{-x^{2} n^{2}}$

(ii)

$\lim _{n \rightarrow \infty} \frac{1}{\pi} \int_{0}^{n} \cos (k x) d k .$

(f) Hence deduce that

$\lim _{n \rightarrow \infty} \frac{1}{2 \pi} \int_{-n}^{n} e^{i k x} d k=\delta(x)$

[You may assume that

$\left.\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi} \quad \text { and } \quad \int_{-\infty}^{\infty} \frac{\sin y}{y} d y=\pi .\right]$

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• Paper 4, Section I, A

By using separation of variables, solve Laplace's equation

$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \quad 0

subject to

$\begin{array}{ll} u(0, y)=0 & 0 \leqslant y \leqslant 1 \\ u(1, y)=0 & 0 \leqslant y \leqslant 1 \\ u(x, 0)=0 & 0 \leqslant x \leqslant 1 \\ u(x, 1)=2 \sin (3 \pi x) & 0 \leqslant x \leqslant 1 \end{array}$

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• Paper 4, Section II, 17C

Let $\Omega$ be a bounded region in the plane, with smooth boundary $\partial \Omega$. Green's second identity states that for any smooth functions $u, v$ on $\Omega$

$\int_{\Omega}\left(u \nabla^{2} v-v \nabla^{2} u\right) \mathrm{d} x \mathrm{~d} y=\oint_{\partial \Omega} u(\mathbf{n} \cdot \nabla v)-v(\mathbf{n} \cdot \nabla u) \mathrm{d} s$

where $\mathbf{n}$ is the outward pointing normal to $\partial \Omega$. Using this identity with $v$ replaced by

$G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)=\frac{1}{2 \pi} \ln \left(\left\|\mathbf{x}-\mathbf{x}_{0}\right\|\right)=\frac{1}{4 \pi} \ln \left(\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right)$

and taking care of the singular point $(x, y)=\left(x_{0}, y_{0}\right)$, show that if $u$ solves the Poisson equation $\nabla^{2} u=-\rho$ then

\begin{aligned} u(\mathbf{x})=-\int_{\Omega} G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \rho\left(\mathbf{x}_{0}\right) \mathrm{d} x_{0} \mathrm{~d} y_{0} \\ &+\oint_{\partial \Omega}\left(u\left(\mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right)-G_{0}\left(\mathbf{x} ; \mathbf{x}_{0}\right) \mathbf{n} \cdot \nabla u\left(\mathbf{x}_{0}\right)\right) \mathrm{d} s \end{aligned}

at any $\mathbf{x}=(x, y) \in \Omega$, where all derivatives are taken with respect to $\mathbf{x}_{0}=\left(x_{0}, y_{0}\right)$.

In the case that $\Omega$ is the unit disc $\|\mathbf{x}\| \leqslant 1$, use the method of images to show that the solution to Laplace's equation $\nabla^{2} u=0$ inside $\Omega$, subject to the boundary condition

$u(1, \theta)=\delta(\theta-\alpha),$

is

$u(r, \theta)=\frac{1}{2 \pi} \frac{1-r^{2}}{1+r^{2}-2 r \cos (\theta-\alpha)}$

where $(r, \theta)$ are polar coordinates in the disc and $\alpha$ is a constant.

[Hint: The image of a point $\mathbf{x}_{0} \in \Omega$ is the point $\mathbf{y}_{0}=\mathbf{x}_{0} /\left\|\mathbf{x}_{0}\right\|^{2}$, and then

$\left\|\mathbf{x}-\mathbf{x}_{0}\right\|=\left\|\mathbf{x}_{0}\right\|\left\|\mathbf{x}-\mathbf{y}_{0}\right\|$

for all $\mathbf{x} \in \partial \Omega .]$

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• Paper 1, Section II, 14B

(a)

(i) Compute the Fourier transform $\tilde{h}(k)$ of $h(x)=e^{-a|x|}$, where $a$ is a real positive constant.

(ii) Consider the boundary value problem

$-\frac{d^{2} u}{d x^{2}}+\omega^{2} u=e^{-|x|} \quad \text { on }-\infty

with real constant $\omega \neq \pm 1$ and boundary condition $u(x) \rightarrow 0$ as $|x| \rightarrow \infty$.

Find the Fourier transform $\tilde{u}(k)$ of $u(x)$ and hence solve the boundary value problem. You should clearly state any properties of the Fourier transform that you use.

(b) Consider the wave equation

$v_{t t}=v_{x x} \quad \text { on } \quad-\infty0$

with initial conditions

$v(x, 0)=f(x) \quad v_{t}(x, 0)=g(x) .$

Show that the Fourier transform $\tilde{v}(k, t)$ of the solution $v(x, t)$ with respect to the variable $x$ is given by

$\tilde{v}(k, t)=\tilde{f}(k) \cos k t+\frac{\tilde{g}(k)}{k} \sin k t$

where $\tilde{f}(k)$ and $\tilde{g}(k)$ are the Fourier transforms of the initial conditions. Starting from $\tilde{v}(k, t)$ derive d'Alembert's solution for the wave equation:

$v(x, t)=\frac{1}{2}(f(x-t)+f(x+t))+\frac{1}{2} \int_{x-t}^{x+t} g(\xi) d \xi$

You should state clearly any properties of the Fourier transform that you use.

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• Paper 2, Section I, B

Expand $f(x)=x$ as a Fourier series on $-\pi.

By integrating the series show that $x^{2}$ on $-\pi can be written as

$x^{2}=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos n x$

where $a_{n}, n=1,2, \ldots$, should be determined and

$a_{0}=8 \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} .$

By evaluating $a_{0}$ another way show that

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$

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• Paper 2, Section II, A

Laplace's equation for $\phi$ in cylindrical coordinates $(r, \theta, z)$, is

$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0$

Use separation of variables to find an expression for the general solution to Laplace's equation in cylindrical coordinates that is $2 \pi$-periodic in $\theta$.

Find the bounded solution $\phi(r, \theta, z)$ that satisfies

\begin{aligned} \nabla^{2} \phi &=0 \quad z \geqslant 0, \quad 0 \leqslant r \leqslant 1 \\ \phi(1, \theta, z) &=e^{-4 z}(\cos \theta+\sin 2 \theta)+2 e^{-z} \sin 2 \theta \end{aligned}

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• Paper 3, Section I, A

Using the substitution $u(x, y)=v(x, y) e^{-x^{2}}$, find $u(x, y)$ that satisfies

$u_{x}+x u_{y}+2 x u=e^{-x^{2}}$

with boundary data $u(0, y)=y e^{-y^{2}}$.

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• Paper 3, Section II, A

Let $\mathcal{L}$ be the linear differential operator

$\mathcal{L} y=y^{\prime \prime \prime}-y^{\prime \prime}-2 y^{\prime}$

where $^{\prime}$ denotes differentiation with respect to $x$.

Find the Green's function, $G(x ; \xi)$, for $\mathcal{L}$ satisfying the homogeneous boundary conditions $G(0 ; \xi)=0, G^{\prime}(0 ; \xi)=0, G^{\prime \prime}(0 ; \xi)=0$.

Using the Green's function, solve

$\mathcal{L} y=e^{x} \Theta(x-1)$

with boundary conditions $y(0)=1, y^{\prime}(0)=-1, y^{\prime \prime}(0)=0$. Here $\Theta(x)$ is the Heaviside step function having value 0 for $x<0$ and 1 for $x>0$.

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• Paper 4, Section I, A

The Legendre polynomials, $P_{n}(x)$ for integers $n \geqslant 0$, satisfy the Sturm-Liouville equation

$\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d}{d x} P_{n}(x)\right]+n(n+1) P_{n}(x)=0$

and the recursion formula

$(n+1) P_{n+1}(x)=(2 n+1) x P_{n}(x)-n P_{n-1}(x), \quad P_{0}(x)=1, \quad P_{1}(x)=x$

(i) For all $n \geqslant 0$, show that $P_{n}(x)$ is a polynomial of degree $n$ with $P_{n}(1)=1$.

(ii) For all $m, n \geqslant 0$, show that $P_{n}(x)$ and $P_{m}(x)$ are orthogonal over the range $x \in[-1,1]$ when $m \neq n$.

(iii) For each $n \geqslant 0$ let

$R_{n}(x)=\frac{d^{n}}{d x^{n}}\left[\left(x^{2}-1\right)^{n}\right]$

Assume that for each $n$ there is a constant $\alpha_{n}$ such that $P_{n}(x)=\alpha_{n} R_{n}(x)$ for all $x$. Determine $\alpha_{n}$ for each $n$.

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• Paper 4, Section II, B

(a)

(i) For the diffusion equation

$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0 \quad \text { on }-\infty

with diffusion constant $K$, state the properties (in terms of the Dirac delta function) that define the fundamental solution $F(x, t)$ and the Green's function $G(x, t ; y, \tau)$.

You are not required to give expressions for these functions.

(ii) Consider the initial value problem for the homogeneous equation:

$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=0, \quad \phi\left(x, t_{0}\right)=\alpha(x) \quad \text { on }-\infty

and the forced equation with homogeneous initial condition (and given forcing term $h(x, t))$ :

$\frac{\partial \phi}{\partial t}-K \frac{\partial^{2} \phi}{\partial x^{2}}=h(x, t), \quad \phi(x, 0)=0 \quad \text { on }-\infty

Given that $F$ and $G$ in part (i) are related by

$G(x, t ; y, \tau)=\Theta(t-\tau) F(x-y, t-\tau)$

(where $\Theta(t)$ is the Heaviside step function having value 0 for $t<0$ and 1 for $t>0$, show how the solution of (B) can be expressed in terms of solutions of (A) with suitable initial conditions. Briefly interpret your expression.

(b) A semi-infinite conducting plate lies in the $\left(x_{1}, x_{2}\right)$ plane in the region $x_{1} \geqslant 0$. The boundary along the $x_{2}$ axis is perfectly insulated. Let $(r, \theta)$ denote standard polar coordinates on the plane. At time $t=0$ the entire plate is at temperature zero except for the region defined by $-\pi / 4<\theta<\pi / 4$ and $1 which has constant initial temperature $T_{0}>0$. Subsequently the temperature of the plate obeys the two-dimensional heat equation with diffusion constant $K$. Given that the fundamental solution of the twodimensional heat equation on $\mathbb{R}^{2}$ is

$F\left(x_{1}, x_{2}, t\right)=\frac{1}{4 \pi K t} e^{-\left(x_{1}^{2}+x_{2}^{2}\right) /(4 K t)}$

show that the origin $(0,0)$ on the plate reaches its maximum temperature at time $t=3 /(8 K \log 2)$.

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• Paper 1, Section II, A

(a) Consider the general self-adjoint problem for $y(x)$ on $[a, b]$ :

$-\frac{d}{d x}\left[p(x) \frac{d}{d x} y\right]+q(x) y=\lambda w(x) y ; \quad y(a)=y(b)=0$

where $\lambda$ is the eigenvalue, and $w(x)>0$. Prove that eigenfunctions associated with distinct eigenvalues are orthogonal with respect to a particular inner product which you should define carefully.

(b) Consider the problem for $y(x)$ given by

$x y^{\prime \prime}+3 y^{\prime}+\left(\frac{1+\lambda}{x}\right) y=0 ; \quad y(1)=y(e)=0 .$

(i) Recast this problem into self-adjoint form.

(ii) Calculate the complete set of eigenfunctions and associated eigenvalues for this problem. [Hint: You may find it useful to make the substitution $\left.x=e^{s} .\right]$

(iii) Verify that the eigenfunctions associated with distinct eigenvalues are indeed orthogonal.

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• Paper 2, Section I, A

Use the method of characteristics to find $u(x, y)$ in the first quadrant $x \geqslant 0, y \geqslant 0$, where $u(x, y)$ satisfies

$\frac{\partial u}{\partial x}-2 x \frac{\partial u}{\partial y}=\cos x$

with boundary data $u(x, 0)=\cos x$.

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• Paper 2, Section II, A

Consider a bar of length $\pi$ with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$ :

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$

for $x \in(0, \pi)$ and $t>0$ with boundary conditions:

$\frac{\partial y}{\partial x}(0, t)=\frac{\partial y}{\partial x}(\pi, t)=0$

for $t>0$. The bar is initially at rest so that

$\frac{\partial y}{\partial t}(x, 0)=0$

for $x \in(0, \pi)$, with a spatially varying initial longitudinal displacement given by

$y(x, 0)=b x$

for $x \in(0, \pi)$, where $b$ is a real constant.

(a) Using separation of variables, show that

$y(x, t)=\frac{b \pi}{2}-\frac{4 b}{\pi} \sum_{n=1}^{\infty} \frac{\cos [(2 n-1) x] \cos [(2 n-1) c t]}{(2 n-1)^{2}}$

(b) Determine a periodic function $P(x)$ such that this solution may be expressed as

$y(x, t)=\frac{1}{2}[P(x+c t)+P(x-c t)]$

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• Paper 3, Section $\mathbf{I}$, A

Calculate the Green's function $G(x ; \xi)$ given by the solution to

$\frac{d^{2} G}{d x^{2}}=\delta(x-\xi) ; \quad G(0 ; \xi)=\frac{d G}{d x}(1 ; \xi)=0$

where $\xi \in(0,1), x \in(0,1)$ and $\delta(x)$ is the Dirac $\delta$-function. Use this Green's function to calculate an explicit solution $y(x)$ to the boundary value problem

$\frac{d^{2} y}{d x^{2}}=x e^{-x}$

where $x \in(0,1)$, and $y(0)=y^{\prime}(1)=0$.

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• Paper 3, Section II, B

(a) Show that the Fourier transform of $f(x)=e^{-a^{2} x^{2}}$, for $a>0$, is

$\tilde{f}(k)=\frac{\sqrt{\pi}}{a} e^{-\frac{k^{2}}{4 a^{2}}},$

stating clearly any properties of the Fourier transform that you use.

[Hint: You may assume that $\int_{0}^{\infty} e^{-t^{2}} d t=\sqrt{\pi} / 2$.]

(b) Consider now the Cauchy problem for the diffusion equation in one space dimension, i.e. solving for $\theta(x, t)$ satisfying:

$\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}} \quad \text { with } \theta(x, 0)=g(x)$

where $D$ is a positive constant and $g(x)$ is specified. Consider the following property of a solution:

Property P: If the initial data $g(x)$ is positive and it is non-zero only within a bounded region (i.e. there is a constant $\alpha$ such that $\theta(x, 0)=0$ for all $|x|>\alpha)$, then for any $\epsilon>0$ (however small) and $\beta$ (however large) the solution $\theta(\beta, \epsilon)$ can be non-zero, i.e. the solution can become non-zero arbitrarily far away after an arbitrarily short time.

Does Property P hold for solutions of the diffusion equation? Justify your answer (deriving any expression for the solution $\theta(x, t)$ that you use).

(c) Consider now the wave equation in one space dimension:

$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$

with given initial data $u(x, 0)=\phi(x)$ and $\frac{\partial u}{\partial t}(x, 0)=0$ (and $c$ is a constant).

Does Property $\mathrm{P}$ (with $g(x)$ and $\theta(\beta, \epsilon)$ now replaced by $\phi(x)$ and $u(\beta, \epsilon)$ respectively) hold for solutions of the wave equation? Justify your answer again as above.

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• Paper 4, Section I, A

Consider the function $f(x)$ defined by

$f(x)=x^{2}, \text { for }-\pi

Calculate the Fourier series representation for the $2 \pi$-periodic extension of this function. Hence establish that

$\frac{\pi^{2}}{6}=\sum_{n=1}^{\infty} \frac{1}{n^{2}}$

and that

$\frac{\pi^{2}}{12}=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}}$

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• Paper 4, Section II, B

Let $\mathcal{D}$ be a 2-dimensional region in $\mathbb{R}^{2}$ with boundary $\partial \mathcal{D}$. In this question you may assume Green's second identity:

$\int_{\mathcal{D}}\left(f \nabla^{2} g-g \nabla^{2} f\right) d A=\int_{\partial \mathcal{D}}\left(f \frac{\partial g}{\partial n}-g \frac{\partial f}{\partial n}\right) d l$

where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial \mathcal{D}$, and $f$ and $g$ are suitably regular functions that include the free space Green's function in two dimensions. You may also assume that the free space Green's function for the Laplace equation in two dimensions is given by

$G_{0}\left(\boldsymbol{r}, \boldsymbol{r}_{0}\right)=\frac{1}{2 \pi} \log \left|\boldsymbol{r}-\boldsymbol{r}_{0}\right|$

(a) State the conditions required on a function $G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right)$ for it to be a Dirichlet Green's function for the Laplace operator on $\mathcal{D}$. Suppose that $\nabla^{2} \psi=0$ on $\mathcal{D}$. Show that if $G$ is a Dirichlet Green's function for $\mathcal{D}$ then

$\psi\left(\boldsymbol{r}_{\mathbf{0}}\right)=\int_{\partial \mathcal{D}} \psi(\boldsymbol{r}) \frac{\partial}{\partial n} G\left(\boldsymbol{r}, \boldsymbol{r}_{\mathbf{0}}\right) d l \quad \text { for } \boldsymbol{r}_{\mathbf{0}} \in \mathcal{D}$