Paper 2, Section I, B

with $y(0)=1$ , and show that $y$ is a polynomial if and only if $\lambda$ is a non-negative integer $n$ . Let $y_{n}$ be the solution for $\lambda=n$ . Establish an orthogonality relation between $y_{m}$ and $y_{n}(m \neq n)$ .

Show that $y_{m} y_{n}$ is a polynomial of degree $m+n$ , and hence that

y_{m} y_{n}=\sum_{p=0}^{m+n} a_{p} y_{p}

for appropriate choices of the coefficients $a_{p}$ and with $a_{m+n} \neq 0$ .

For given $n>0$ , show that the functions

\left\{y_{m}, y_{m} y_{n}: m=0,1,2, \ldots, n-1\right\}

are linearly independent.

Let $f(x)$ be a polynomial of degree 3. Explain why the expansion

f(x)=a_{0} y_{0}(x)+a_{1} y_{1}(x)+a_{2} y_{2}(x)+a_{3} y_{1}(x) y_{2}(x)

holds for appropriate choices of $a_{p}, p=0,1,2,3$ . Hence show that

\int_{0}^{\infty} e^{-x} f(x) d x=w_{1} f\left(\alpha_{1}\right)+w_{2} f\left(\alpha_{2}\right)

where

w_{1}=\frac{y_{1}\left(\alpha_{2}\right)}{y_{1}\left(\alpha_{2}\right)-y_{1}\left(\alpha_{1}\right)}, \quad w_{2}=\frac{-y_{1}\left(\alpha_{1}\right)}{y_{1}\left(\alpha_{2}\right)-y_{1}\left(\alpha_{1}\right)},

and $\alpha_{1}, \alpha_{2}$ are the zeros of $y_{2}$ . You need not construct the polynomials $y_{1}(x), y_{2}(x)$ explicitly.

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Paper 2, Section II, B

Mathematical Methods

Part IB, 2009

A string of uniform density $\rho$ is stretched under tension along the $x$ -axis and undergoes small transverse oscillations in the $(x, y)$ plane with amplitude $y(x, t)$ . Given that waves in the string travel at velocity $c$ , write down the equation of motion satisfied by $y(x, t)$ .

The string is now fixed at $x=0$ and $x=L$ . Derive the general separable solution for the amplitude $y(x, t)$ .

For $t<0$ the string is at rest. At time $t=0$ the string is struck by a hammer in the interval $[l-a / 2, l+a / 2]$ , distance being measured from one end. The effect of the hammer is to impart a constant velocity $v$ to the string inside the interval and zero velocity outside it. Calculate the proportion of the total energy given to the string in each mode.

If $l=L / 3$ and $a=L / 10$ , find all the modes of the string which are not excited in the motion.

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A2.17

Mathematical Methods

Part II, 2001

(i) A certain physical quantity $q(x)$ can be represented by the series $\sum_{n=0}^{\infty} c_{n} x^{n}$ in $0 \leqslant x<x_{0}$ , but the series diverges for $x>x_{0}$ . Describe the Euler transformation to a new series which may enable $q(x)$ to be computed for $x>x_{0}$ . Give the first four terms of the new series.

Describe briefly the disadvantages of the method.

(ii) The series $\sum_{1}^{\infty} c_{r}$ has partial sums $S_{n}=\sum_{1}^{n} c_{r}$ . Describe Shanks' method to approximate $S_{n}$ by

S_{n}=A+B C^{n}

giving expressions for $A, B$ and $C$ .

Denote by $B_{N}$ and $C_{N}$ the values of $B$ and $C$ respectively derived from these expressions using $S_{N-1}, S_{N}$ and $S_{N+1}$ for some fixed $N$ . Now let $A^{(n)}$ be the value of $A$ obtained from $(*)$ with $B=B_{N}, C=C_{N}$ . Show that, if $\left|C_{N}\right|<1$ ,

\sum_{1}^{\infty} c_{r}=\lim _{n \rightarrow \infty} A^{(n)}

If, in fact, the partial sums satisfy

S_{n}=a+\alpha c^{n}+\beta d^{n}

with $1>|c|>|d|$ , show that

A^{(n)}=A+\gamma d^{n}+o\left(d^{n}\right)

where $\gamma$ is to be found.

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A3.17

Mathematical Methods

Part II, 2001

(i) The function $y(x)$ satisfies the differential equation

y^{\prime \prime}+b y^{\prime}+c y=0, \quad 0<x<1,

where $b$ and $c$ are constants, with boundary conditions $y(0)=0, y^{\prime}(0)=1$ . By integrating this equation or otherwise, show that $y$ must also satisfy the integral equation

y(x)=g(x)+\int_{0}^{1} K(x, t) y(t) d t

and find the functions $g(x)$ and $K(x, t)$ .

(ii) Solve the integral equation

\varphi(x)=1+\lambda^{2} \int_{0}^{x}(x-t) \varphi(t) d t, \quad x>0, \quad \lambda \text { real }

by finding an ordinary differential equation satisfied by $\varphi(x)$ together with boundary conditions.

Now solve the integral equation by the method of successive approximations and show that the solutions are the same.

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A4.21

Mathematical Methods

Part II, 2001

The equation

\mathbf{A x}=\lambda \mathbf{x}

where $\mathbf{A}$ is a real square matrix and $\mathbf{x}$ a column vector, has a simple eigenvalue $\lambda=\mu$ with corresponding right-eigenvector $\mathbf{x}=\mathbf{X}$ . Show how to find expressions for the perturbed eigenvalue and right-eigenvector solutions of

\mathbf{A} \mathbf{x}+\epsilon \mathbf{b}(\mathbf{x})=\lambda \mathbf{x}, \quad|\epsilon| \ll 1

to first order in $\epsilon$ , where $\mathbf{b}$ is a vector function of $\mathbf{x}$ . State clearly any assumptions you make.

If $\mathbf{A}$ is $(n \times n)$ and has a complete set of right-eigenvectors $\mathbf{X}^{(j)}, j=1,2, \ldots n$ , which span $\mathbb{R}^{n}$ and correspond to separate eigenvalues $\mu^{(j)}, j=1,2, \ldots n$ , find an expression for the first-order perturbation to $\mathbf{X}^{(1)}$ in terms of the $\left\{\mathbf{X}^{(j)}\right\}$ and the corresponding lefteigenvectors of $\mathbf{A}$ .

Find the normalised eigenfunctions and eigenvalues of the equation

\frac{d^{2} y}{d x^{2}}+\lambda y=0,0<x<1

with $y(0)=y(1)=0$ . Let these be the zeroth order approximations to the eigenfunctions of

\frac{d^{2} y}{d x^{2}}+\lambda y+\epsilon b(y)=0,0<x<1

with $y(0)=y(1)=0$ and where $b$ is a function of $y$ . Show that the first-order perturbations of the eigenvalues are given by

\lambda_{n}^{(1)}=-\epsilon \sqrt{2} \int_{0}^{1} \sin (n \pi x) \quad b(\sqrt{2} \sin n \pi x) d x

Part II

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A2.17

Mathematical Methods

Part II, 2002

(i) Show that the equation

\epsilon x^{4}-x^{2}+5 x-6=0, \quad|\epsilon| \ll 1

has roots in the neighbourhood of $x=2$ and $x=3$ . Find the first two terms of an expansion in $\epsilon$ for each of these roots.

Find a suitable series expansion for the other two roots and calculate the first two terms in each case.

(ii) Describe, giving reasons for the steps taken, how the leading-order approximation for $\lambda \gg 1$ to an integral of the form

I(\lambda) \equiv \int_{A}^{B} f(t) e^{i \lambda g(t)} d t

where $\lambda$ and $g$ are real, may be found by the method of stationary phase. Consider the cases where (a) $g^{\prime}(t)$ has one simple zero at $t=t_{0}$ with $A<t_{0}<B$ ; (b) $g^{\prime}(t)$ has more than one simple zero in $A<t<B$ ; and (c) $g^{\prime}(t)$ has only a simple zero at $t=B$ . What is the order of magnitude of $I(\lambda)$ if $g^{\prime}(t)$ is non-zero for $A \leq t \leq B$ ?

Use the method of stationary phase to find the leading-order approximation to

J(\lambda) \equiv \int_{0}^{1} \sin \left[\lambda\left(2 t^{4}-t\right)\right] d t

for $\lambda \gg 1$ .

[You may use the fact that $\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4}$ .]

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A3.17

Mathematical Methods

Part II, 2002

(i) State the Fredholm alternative for Fredholm integral equations of the second kind.

Show that the integral equation

\phi(x)-\lambda \int_{0}^{1}(x+t) \phi(t) d t=f(x), \quad 0 \leqslant x \leqslant 1

where $f$ is a continuous function, has a unique solution for $\phi$ if $\lambda \neq-6 \pm 4 \sqrt{3}$ . Derive this solution.

(ii) Describe the WKB method for finding approximate solutions $f(x)$ of the equation

\frac{d^{2} f(x)}{d x^{2}}+q(\epsilon x) f(x)=0

where $q$ is an arbitrary non-zero, differentiable function and $\epsilon$ is a small parameter. Obtain these solutions in terms of an exponential with slowly varying exponent and slowly varying amplitude.

Hence, by means of a suitable change of independent variable, find approximate solutions $w(t)$ of the equation

\frac{d^{2} w}{d t^{2}}+\lambda^{2} t w=0

in $t>0$ , where $\lambda$ is a large parameter.

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A4.21

Mathematical Methods

Part II, 2002

State Watson's lemma giving an asymptotic expansion as $\lambda \rightarrow \infty$ for an integral of the form

I_{1}=\int_{0}^{A} f(t) e^{-\lambda t} d t, \quad A>0

Show how this result may be used to find an asymptotic expansion as $\lambda \rightarrow \infty$ for an integral of the form

I_{2}=\int_{-A}^{B} f(t) e^{-\lambda t^{2}} d t, \quad A>0, B>0

Hence derive Laplace's method for obtaining an asymptotic expansion as $\lambda \rightarrow \infty$ for an integral of the form

I_{3}=\int_{a}^{b} f(t) e^{\lambda \phi(t)} d t

where $\phi(t)$ is differentiable, for the cases: (i) $\phi^{\prime}(t)<0$ in $a \leq t \leq b$ ; and (ii) $\phi^{\prime}(t)$ has a simple zero at $t=c$ with $a<c<b$ and $\phi^{\prime \prime}(c)<0$ .

Find the first two terms in the asymptotic expansion as $x \rightarrow \infty$ of

I_{4}=\int_{-\infty}^{\infty} \log \left(1+t^{2}\right) e^{-x t^{2}} d t

[You may leave your answer expressed in terms of $\Gamma$ -functions.]

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A2.17

Mathematical Methods

Part II, 2003

(i) Explain how to solve the Fredholm integral equation of the second kind,

f(x)=\mu \int_{a}^{b} K(x, t) f(t) d t+g(x)

in the case where $K(x, t)$ is of the separable (degenerate) form

K(x, t)=a_{1}(x) b_{1}(t)+a_{2}(x) b_{2}(t)

(ii) For what values of the real constants $\lambda$ and $A$ does the equation

u(x)=\lambda \sin x+A \int_{0}^{\pi}(\cos x \cos t+\cos 2 x \cos 2 t) u(t) d t

have (a) a unique solution, (b) no solution?

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A3.17

Mathematical Methods

Part II, 2003

(i) Explain what is meant by the assertion: "the series $\sum_{0}^{\infty} b_{n} x^{n}$ is asymptotic to $f(x)$ as $x \rightarrow 0 "$ .

Consider the integral

I(\lambda)=\int_{0}^{A} e^{-\lambda x} g(x) d x

where $A>0, \lambda$ is real and $g$ has the asymptotic expansion

g(x) \sim a_{0} x^{\alpha}+a_{1} x^{\alpha+1}+a_{2} x^{\alpha+2}+\ldots

as $x \rightarrow+0$ , with $\alpha>-1$ . State Watson's lemma describing the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$ , and determine an expression for the general term in the asymptotic series.

(ii) Let

h(t)=\pi^{-1 / 2} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x

for $t \geqslant 0$ . Show that

h(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 . \cdots \cdot(2 k-1) t^{k}

as $t \rightarrow+0$ .

Suggest, for the case that $t$ is smaller than unity, the point at which this asymptotic series should be truncated so as to produce optimal numerical accuracy.

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A4.21

Mathematical Methods

Part II, 2003

Let $y(x, \lambda)$ denote the solution for $0 \leqslant x<\infty$ of

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

subject to the conditions that $y(0, \lambda)=a$ and $y(x, \lambda) \rightarrow 0$ as $x \rightarrow \infty$ , where $a>0$ ; it may be assumed that $y(x, \lambda)>0$ for $x>0$ . Write $y(x, \lambda)$ in the form

y(x, \lambda)=\exp (z(x, \lambda))

and consider an asymptotic expansion of the form

z(x, \lambda) \sim \sum_{n=0}^{\infty} \lambda^{1-n} \phi_{n}(x),

valid in the limit $\lambda \rightarrow \infty$ with $x=O(1)$ . Find $\phi_{0}(x), \phi_{1}(x), \phi_{2}(x)$ and $\phi_{3}(x)$ .

It is known that the solution $y(x, \lambda)$ is of the form

y(x, \lambda)=c Y(X)

where

X=x+\lambda^{2}

and the constant factor $c$ depends on $\lambda$ . By letting $Y(X)=\exp (Z(X))$ , show that the expression

Z(X)=-\frac{2}{3} X^{3 / 2}-\frac{1}{4} \ln X

satisfies the relevant differential equation with an error of $O\left(1 / X^{3 / 2}\right)$ as $X \rightarrow \infty$ . Comment on the relationship between your answers for $z(x, \lambda)$ and $Z(X)$ .

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A2.17

Mathematical Methods

Part II, 2004

(i) Consider the integral equation

\phi(x)=-\lambda \int_{a}^{b} K(x, t) \phi(t) d t+g(x)

for $\phi$ in the interval $a \leq x \leq b$ , where $\lambda$ is a real parameter and $g(x)$ is given. Describe the method of successive approximations for solving ( $\dagger$ ).

Suppose that

|K(x, t)| \leq M, \quad \forall x, t \in[a, b]

By using the Cauchy-Schwarz inequality, or otherwise, show that the successive-approximation series for $\phi(x)$ converges absolutely provided

|\lambda|<\frac{1}{M(b-a)} .

(ii) The real function $\psi(x)$ satisfies the differential equation

-\psi^{\prime \prime}(x)+\lambda \psi(x)=h(x), \quad 0<x<1

where $h(x)$ is a given smooth function on $[0,1]$ , subject to the boundary conditions

\psi^{\prime}(0)=\psi(0), \quad \psi(1)=0 .

By integrating $(\star)$ , or otherwise, show that $\psi(x)$ obeys

\psi(0)=\frac{1}{2} \int_{0}^{1}(1-t) h(t) d t-\frac{1}{2} \lambda \int_{0}^{1}(1-t) \psi(t) d t

Hence, or otherwise, deduce that $\psi(x)$ obeys an equation of the form ( $\dagger$ ), with

\begin{gathered} K(x, t)= \begin{cases}\frac{1}{2}(1-x)(1+t), & 0 \leq t \leq x \leq 1 \\ \frac{1}{2}(1+x)(1-t), & 0 \leq x \leq t \leq 1\end{cases} \\ \text { and } g(x)=\int_{0}^{1} K(x, t) h(t) d t \end{gathered}

Deduce that the series solution for $\psi(x)$ converges provided $|\lambda|<2$ .

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A3.17

Mathematical Methods

Part II, 2004

(i) Give a brief description of the method of matched asymptotic expansions, as applied to a differential equation of the type

\epsilon y^{\prime \prime}+K y^{\prime}+f(y)=0, \quad 0<x<1,

where $0<\epsilon \ll 1, K$ is a non-zero constant, $f$ is a suitable smooth function and the boundary values $y(0), y(1)$ are specified. An outline of Van Dyke's asymptotic matching principle should be included.

(ii) Consider the boundary-value problem

\epsilon y^{\prime \prime}+y^{\prime}-(2 x+1) y=0, \quad y(0)=0, \quad y(1)=e^{2}

with $0<\epsilon \ll 1$ . Find the integrating factor for the leading-order outer problem. Hence obtain the first two terms in the outer expansion.

Rewrite the problem using an appropriate stretched inner variable. Hence obtain the first two terms of the inner exansion.

Use van Dyke's matching principle to determine all the constants. Hence show that $y^{\prime}(0)=\epsilon^{-1}+\frac{25}{3}+O(\epsilon) .$

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A4.21

Mathematical Methods

Part II, 2004

State Watson's lemma, describing the asymptotic behaviour of the integral

I(\lambda)=\int_{0}^{A} e^{-\lambda t} f(t) d t, \quad A>0

as $\lambda \rightarrow \infty$ , given that $f(t)$ has the asymptotic expansion

f(t) \sim \sum_{n=0}^{\infty} a_{n} t^{n \beta}

as $t \rightarrow 0_{+}$ , where $\beta>0$ .

Consider the integral

J(\lambda)=\int_{a}^{b} e^{\lambda \phi(t)} F(t) d t,

where $\lambda \gg 1$ and $\phi(t)$ has a unique maximum in the interval $[a, b]$ at $c$ , with $a<c<b$ , such that

\phi^{\prime}(c)=0, \quad \phi^{\prime \prime}(c)<0 .

By using the change of variable from $t$ to $\zeta$ , defined by

\phi(t)-\phi(c)=-\zeta^{2}

deduce an asymptotic expansion for $J(\lambda)$ as $\lambda \rightarrow \infty$ . Show that the leading-order term gives

J(\lambda) \sim e^{\lambda \phi(c)} F(c)\left(\frac{2 \pi}{\lambda\left|\phi^{\prime \prime}(c)\right|}\right)^{\frac{1}{2}}

The gamma function $\Gamma(x)$ is defined for $x>0$ by

\Gamma(x)=\int_{0}^{\infty} e^{(x-1) \log t-t} d t

By means of the substitution $t=(x-1) s$ , or otherwise, deduce that

\Gamma(x+1) \sim x^{\left(x+\frac{1}{2}\right)} e^{-x} \sqrt{2 \pi}\left(1+\frac{1}{12 x}+\ldots\right)

as $x \rightarrow \infty$

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