Chapter 23, Integral equations Video Solutions, Mathematical Methods for Physics and Engineering: A Comprehensive Guide

Problem 1

Solve the integral equation
$$
\int_{0}^{\infty} \cos (x v) y(v) d v=\exp \left(-x^{2} / 2\right)
$$
for the function $y=y(x)$ for $x>0$. Note that for $x<0, y(x)$ can be chosen as is most convenient.

Tyler Moulton

Numerade Educator

00:35

Problem 2

Solve
$$
\int_{0}^{\infty} f(t) \exp (-s t) d t=\frac{a}{a^{2}+s^{2}}
$$

Sajin Shajee

Numerade Educator

01:44

Problem 3

Use the fact that its kernel is separable to solve for $y(x)$ the integral equation
$$
y(x)=A \cos (x+a)+\lambda \int_{0}^{\pi} \sin (x+z) y(z) d z
$$
(This equation is an inhomogeneous extension of the homogeneous Fredholm equation (23.13), and is similar to equation (23.57).)

Gabrielle Lee

Numerade Educator

03:06

Problem 4

Convert
$$
f(x)=\exp x+\int_{0}^{x}(x-y) f(y) d y
$$
into a differential equation, and hence show that its solution is
$$
(\alpha+\beta x) \exp x+\gamma \exp (-x)
$$
where $\alpha, \beta, \gamma$ are constants that should be determined.

Kyle Findley

Numerade Educator

04:02

Problem 5

Solve for $\phi(x)$ the integral equation
$$
\phi(x)=f(x)+\lambda \int_{0}^{1}\left[\left(\frac{x}{y}\right)^{n}+\left(\frac{y}{x}\right)^{n}\right] \phi(y) d y
$$
where $f(x)$ is bounded for $0<x<1$ and $-\frac{1}{2}<n<\frac{1}{2}$, expressing your answer in terms of the quantities $F_{\mathrm{m}}=\int_{0}^{1} f(y) y^{m} d y .$
(a) Give the explicit solution when $\lambda=1$.
(b) For what values of $\lambda$ are there no solutions unless $F_{\pm n}$ take particular values? What are these values?

Farhana Sharmin

Numerade Educator

25:32

Problem 6

(a) Consider the inhomogeneous integral equation
$$
f(x)=g(x)+\lambda \int_{a}^{b} K(x, y) f(y) d y
$$
its kernel $K(x, y)$ is real, symmetric and continuous in $a \leq x \leq b, a \leq y \leq b$. If $\lambda$ is one of the eigenvalues $\lambda_{i}$ of the homogeneous equation
$$
f_{i}(x)=\lambda_{l} \int_{a}^{b} K(x, y) f_{i}(y) d y
$$
prove that the inhomogeneous equation can only a have non-trivial solution if $g(x)$ is orthogonal to the corresponding eigenfunction $f_{i}(x)$.
(b) Show that the only values of $\lambda$. for which
$$
f(x)=\lambda \int_{0}^{1} x y(x+y) f(y) d y
$$
has a non-trivial solution are the roots of the equation.
$$
\lambda^{2}+120 \lambda-240=0
$$
(c) Solve
$$
f(x)=\mu x^{2}+\int_{0}^{1} 2 x y(x+y) f(y) d y
$$

Anas Venkitta

Numerade Educator

21:28

Problem 7

(a) If the kernel of the integral equation
$$
\psi(x)=\lambda \int_{a}^{b} K(x, y) \psi(y) d y
$$
has the form
$$
K(x, y)=\sum_{n=0}^{\infty} h_{n}(x) g_{n}(y)
$$
where the $h_{n}(x)$ form a complete orthonormal set of functions over the interval $[a, b]$, show that the eigenvalues $\lambda_{1}$ are given by
$$
\mid \mathrm{M}-\lambda^{-1} \|=0
$$ where $M$ is the matrix with elements
$$
M_{k j}=\int_{a}^{b} g_{k}(u) h_{j}(u) d u
$$
If the corresponding solutions are $\varphi^{(9}(x)=\sum_{n=0}^{\infty} a_{n}^{(9} h_{n}(x)$, find an expression for $a_{n}^{(i)}$.
(b) Obtain the eigenvalues and eigenfunctions over the interval $[0,2 \pi]$ if
$$
K(x, y)=\sum_{n=1}^{\infty} \frac{1}{n} \cos n x \cos n y
$$

R M

Numerade Educator

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Problem 8

By taking its Laplace transform, and that of $x^{n} e^{-a x}$, obtain the explicit solution of
$$
f(x)=e^{-x}\left[x+\int_{0}^{x}(x-u) e^{w} f(u) d u\right]
$$
Verify your answer by substitution.

Philip Olivier

Numerade Educator

06:38

Problem 9

For $f(t)=\exp \left(-\frac{r^{2}}{2}\right)$, use the relationships of the Fourier transforms of $f^{\prime}(t)$ and $t f(t)$ to that of $f(t)$ itself to find a simple differential equation satisfied by $\tilde{f}(\omega)$, the Fourier transform of $f(t)$ and hence determine $\tilde{f}(\omega)$ to within a constant. Use this result to solve the integral equation
$$
\int_{-x}^{\infty} e^{-t(t-2 x) / 2} h(t) d t=e^{3 x^{2} / 8}
$$
for $h(t)$

Amit Srivastava

Numerade Educator

03:53

Problem 10

Show that the equation
$$
f(x)=x^{-1 / 3}+\lambda \int_{0}^{\infty} f(y) \exp (-x y) d y
$$
has a solution of the form $A x^{\alpha}+B x^{\beta}$. Determine the values of $\alpha$ and $\beta$ and show that those of $A$ and $B$ are
$$
\frac{1}{1-\lambda^{2} \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)} \quad \text { and } \quad \frac{\lambda \Gamma\left(\frac{2}{3}\right)}{1-\lambda^{2} \Gamma\left(\frac{1}{3}\right) \Gamma\left(\frac{2}{3}\right)}
$$
where $\Gamma(z)$ is the gamma function, discussed in the appendix.

Christopher Stanley

Numerade Educator

00:58

Problem 11

At an international "peace' conference a large number of delegates are seated around a circular table with each delegation sitting near its allies and diametrically opposite the delegation most bitterly opposed to it. The position of a delegate is denoted by $\theta$, with $0 \leq \theta \leq 2 \pi$. The fury $f(\theta)$ felt by the delegate at $\theta$ is the sum of his own natural hostility $h(\theta)$ and the influences on him of each of the other delegates; a delegate at position $\phi$ contributes an amount $K(\theta-\phi) f(\phi)$. Thus
$$
f(\theta)=h(\theta)+\int_{0}^{2 \pi} K(\theta-\phi) f(\phi) d \phi
$$
Show that if $K(\varphi)$ takes the form $K(\varphi)=k_{0}+k_{1} \cos \psi$ then
$$
f(\theta)=h(\theta)+p+q \cos \theta+r \sin \theta
$$
and evaluate $p, q$ and $r$. A positive value for $k_{1}$ implies that delegates tend to placate their opponents but upset their allies, whilst negative values imply that they calm their allies but infuriate their opponents. A walkout will occur if $f(\theta)$ exceeds a certain threshold value for some $\theta .$ Is this more likely to happen for positive or for negative values of $k_{1} ?$

Robert Leedy

Numerade Educator

01:46

Problem 12

By considering functions of the form $h(x)=\int_{0}^{x}(x-y) f(y) d y$, show that the solution $f(x)$ of the integral equation
$$
f(x)=x+\frac{1}{2} \int_{0}^{1}|x-y| f(y) d y
$$
satisfies the equation $f^{\prime \prime}(x)=f(x)$
By examining the special cases $x=0$ and $x=1$, show that
$$
f(x)=\frac{2}{(e+3)(e+1)}\left[(e+2) e^{x}-e e^{-x}\right]
$$

Hoan Nguyen

Numerade Educator

08:26

Problem 13

The operator $\mathscr{H}$ is defined by
$$
M f(x)=\int_{-\infty}^{\infty} K(x, y) f(y) d y
$$
where $K(x, y)=1$ inside the square $|x|<a,|y|<a$, and is equal to 0 elsewhere. Consider the possible eigenvalues of $\mathscr{H}$ and the eigenfunctions that correspond to them; show that the only possible eigenvalues are 0 and $2 a$ and determine the corresponding eigenfunctions. Hence find the general solution of
$$
f(x)=g(x)+\lambda \int_{-x}^{x} K(x, y) f(y) d y
$$

Hafiz Shahzaib

Numerade Educator

02:45

Problem 14

For the integral equation
$$
y(x)=x^{-3}+\lambda \int_{a}^{b} x^{2} z^{2} y(z) d z
$$
show that the resolvent kernel is $5 x^{2} z^{2} /\left[5-\lambda\left(b^{5}-a^{5}\right)\right]$ and hence solve the equation. For what range of $\lambda$ is the solution valid?

Gaurav Kalra

Numerade Educator

04:18

Problem 15

Use Fredholm theory to show that, for the kernel
$$
K(x, z)=(x+z) \exp (x-z)
$$
over the interval $[0,1]$, the resolvent kernel is
$$
R(x, z ; \lambda)=\frac{\exp (x-z)\left[(x+z)-\lambda\left(\frac{1}{2} x+\frac{1}{2} z-x z-\frac{1}{3}\right)\right]}{1-\lambda-\frac{1}{12} \lambda^{2}}
$$
and hence solve
$$
y(x)=x^{2}+2 \int_{0}^{1}(x+z) \exp (x-z) y(z) d z
$$
expressing your answer in terms of $I_{n}$, where $I_{n}=\int_{0}^{1} u^{n} \exp (-u) d u$.

Sriram Soundarrajan

Numerade Educator

01:19

Problem 16

(a) Determine the eigenvalues $\lambda_{\pm}$of the kernel $K(x, z)=(x z)^{1 / 2}\left(x^{1 / 2}+z^{1 / 2}\right)$ and show that the corresponding eigenfunctions have the forms
$$
y_{\pm}(x)=A_{\pm}\left(\sqrt{2} x^{1 / 2} \pm \sqrt{3} x\right)
$$
where $A_{\pm}^{2}=5 /(10 \pm 4 \sqrt{6})$.
(b) Use Schmidt-Hilbert theory to solve,
$$
y(x)=1+\frac{5}{2} \int_{0}^{1} K(x, z) y(z) d z
$$
(c) As may be apparent, the algebra involved in the formal method used in (b) is long and error-prone, and it is in fact much more straightforward to use a trial function $1+\alpha x^{1 / 2}+\beta x$. Check your answer by doing so.

Victor Salazar

Numerade Educator