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Integral equations: a practical treatment, from spectral theory to applications

David Porter, David S. G. Stirling

Chapter 3

Integral equations of the second kind - all with Video Answers

Educators

Chapter Questions

04:07

Problem 1

(i) Solve the integral equation
$$
\phi(x)-\int_0^1(x+t) \phi(t) \mathrm{d} t=1 \quad(0 \leqslant x \leqslant 1) .
$$
(ii) Solve
$$
\phi(x)-\int_0^1(x+t) \phi(t) \mathrm{d} t=x^2 \quad(0 \leqslant x \leqslant 1) .
$$

John Connell
John Connell
Numerade Educator
01:12

Problem 2

Show that if $f$ is a given continuous function and $i \neq 2 \sqrt{3} /(\sqrt{3} \pm 2)$ the integral equation
$$
\phi(x)-i \int_0^1(x+t) \phi(t) \mathrm{d} t=f(x) \quad(0 \leqslant x \leqslant 1)
$$
has a unique solution, and find that solution.

Carson Merrill
Carson Merrill
Numerade Educator

Problem 3

Solve the integral equation
$$
\phi(x)=\hat{\lambda} \int_0^1 \cos \{\alpha(x-t)\} \phi(t) \mathrm{d} t+f(x) \quad(0 \leqslant x \leqslant 1)
$$
in the cases
(i) $f(x)=0(0 \leqslant x \leqslant 1), \alpha \in \mathbb{R}$;
(ii) $f(x)=1(0 \leqslant x \leqslant 1), \alpha=n \pi, n \in \mathbb{N}$.

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02:54

Problem 4

Suppose that $f$ is a real-valued continuous function on $[0,1]$. Show that the integral equation
$$
\phi(x)-\int_0^1(12 x t+6(t-x)-2) \phi(t) \mathrm{d} t=f(x) \quad(0 \leqslant x \leqslant 1)
$$
has a solution if and only if $\int_0^1(1-2 t) f(t) \mathrm{d} t=0$. Show also that if $f$ satisfies the stated condition the solution of the integral equation is not unique. Find the solution in the case $f=1$.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 5

Define the integral operator $K$ on $L_2(0,1)$ by
$$
(K \phi)(x)=\int_0^1(1-\max (x, t)) \phi(t) \mathrm{d} t .
$$

By observing that the kernel is an $L_2$-kernel, show that $\|K\| \leqslant 1 / 6^{ \pm}$and therefore that the homogeneous equation
$$
\phi(x)-i \int_0^1(1-\max (x, t)) \phi(t) \mathrm{d} t=0 \quad(0 \leqslant x \leqslant 1)
$$
has no non-trivial solution in $L_2(0,1)$ if $|\lambda|<6^{\frac{1}{4}}$. Deduce that if $|\lambda|<6^{\frac{1}{2}}$ and $f$ is a square-integrable function, the equation
$$
\phi(x)-i \int_0^1(1-\max (x, t)) \phi(t) \mathrm{d} t=f(x) \quad(0 \leqslant x \leqslant 1)
$$
has a unique square-integrable solution $\phi$.

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Problem 6

Suppose that $k:[-1,1] \rightarrow \mathbb{R}$ is a measurable function and that $\int_{-1}^1|k(s)| \mathrm{d} s<\infty$. Define $K$ by
$$
(K \phi)(x)=\int_0^1 k(x-t) \phi(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1)
$$
and show that $K$ is a bounded linear map from $L_2(0,1)$ to itself with $\|K\| \leqslant \int_{-1}^1|k(s)|$ ds. (Hint: the kernel is a Schur kernel.) Now define $k_n$ by $k_n(s)=k(s)$ if $|k(s)| \leqslant n$ and $k_n(s)=0$ if $|k(s)|>n$, so that $k_n$ is a bounded function and the operator $K_n$ defined by $\left(K_n \phi\right)(x)=\int_0^1 k_n(x-t) \phi(t) \mathrm{d} t(0 \leqslant x \leqslant 1)$ is compact. Deduce that $\left\|K_n-K\right\| \rightarrow 0$ and thus that $K$ is compact.

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Problem 7

Suppose that $k:[0, \infty) \rightarrow \mathbb{C}$ is a measurable function and that $\int_0^{\infty}|k(u)| u^{- \pm} \mathrm{d} u$ exists. Use the technique of Example 3.6 to show that the operator $K$ defined by
$$
(K \phi)(x)=\frac{1}{x} \int_0^1 k\left(\frac{t}{x}\right) \phi(t) \mathrm{d} t \quad(0<x \leqslant 1)
$$
is a bounded linear map from $L_2(0,1)$ to itself and that $\|K\| \leqslant \int_0^{\infty}|k(u)| u^{-\frac{1}{2}} \mathrm{~d} u$.
By considering the function $\phi$ where $\phi(t)=t^{-\frac{1}{2}}(a \leqslant t \leqslant 1)$ and $\phi(t)=0$ for $t<a$, or otherwise, show that if for all $u>0 k(u) \geqslant 0$ then $\|K\|=\int_0^{\infty} k(u) u^{-\frac{1}{2}} \mathrm{~d} u$.

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Problem 8

Example 3.6 shows that not all operators of the form considered in Problem 3.7 are compact. Show that if $\int_0^{\infty}|k(u)|^2 \mathrm{~d} u<\infty$ and $0<\alpha<1$, then the operator defined by
$$
(K \phi)(x)=\frac{1}{x^\alpha} \int_0^1 k\left(\frac{t}{x}\right) \phi(t) \mathrm{d} t \quad(0<x \leqslant 1)
$$
is compact. (It has an $L_2$-kernel.) Deduce that for $0<\alpha<1$ the operator $K_\alpha$ given by
$$
\left(K_\alpha \phi\right)(x)=\frac{1}{x^\alpha} \int_0^x \phi(t) \mathrm{d} t \quad(0<x \leqslant 1)
$$
is compact.

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04:07

Problem 9

$ \quad$ Let $(T \phi)(x)=\int_0^1 \phi(t) /(x+t) \mathrm{d} t(0<x \leqslant 1)$. Prove that $T$ is a bounded operator from $L_2(0,1)$ to itself and that $\|T\| \leqslant \pi$.

John Connell
John Connell
Numerade Educator

Problem 10

Suppose that $U$ is a bounded linear map from $L_2(0,1)$ to itself whose inverse is also bounded, and let $K: L_2(0,1) \rightarrow L_2(0,1)$ be compact and linear. By considering $I+U^{-1} K$, or otherwise, show that for $f \in L_2(0,1)$ either the equation $U \phi+K \phi=f$ has a unique solution $\phi \in L_2(0,1)$ or the homogeneous equation $U \phi+K \phi=0$ has a non-trivial solution.
Use this to show that the integral equations
$$
(1+x) \phi(x)-\int_0^x \phi(t) \mathrm{d} t=f(x) \quad(0 \leqslant x \leqslant 1)
$$
and
$$
\phi(1-x)-\int_0^x \phi(t) \mathrm{d} t=f(x) \quad(0 \leqslant x \leqslant 1)
$$
both possess a unique solution $\phi$ in $L_2(0,1)$ provided $f \in L_2(0,1)$.

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Problem 11

Let $p$ and $f$ be continuous functions and $\alpha$ be constant. Show that the initial value problem
$$
\left.\begin{array}{l}
\phi^{\prime}(x)-p(x) \phi(x)=f(x) \\
\phi(0)=\alpha
\end{array}\right\} \quad(0<x<1)
$$
is equivalent to the integral equation
$$
\phi(x)-\int_0^x p(t) \phi(t) \mathrm{d} t=\alpha+F(x) \quad(0 \leqslant x \leqslant 1),
$$
where $F(x)=\int_0^x f(t) \mathrm{d} t$. Let $K$ be the operator given by $(K \phi)(x)=\int_0^x p(t) \phi(t) \mathrm{d} t$ and show that
$$
\left(K^n \phi\right)(x)=\int_0^x \frac{(P(x)-P(t))^{n-1}}{(n-1) !} p(t) \phi(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1, n \in \mathbb{N}),
$$
where $P(x)=\int_0^x p(t) \mathrm{d} t$. Obtain the Neumann series for the solution $\phi$ of (3.27) and, by observing that the uniform convergence of $\Sigma(P(x)-P(t))^{n-1} p(t) f(t) /$ $(n-1)$ ! allows us to interchange the integration and summation, show that
$$
\begin{aligned}
\phi(x) & =\alpha+F(x)+\int_0^x(\alpha+F(t)) \mathrm{e}^{P(x)-P(t)} p(t) \mathrm{d} t \\
& =\alpha \mathrm{e}^{P(x)}+\int_0^x f(t) \mathrm{e}^{P(x)-P(t)} \mathrm{d} t \quad(0 \leqslant x \leqslant 1) .
\end{aligned}
$$

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01:31

Problem 12

Show that the boundary value problem
$$
\left.\begin{array}{l}
\phi^{\prime \prime}(x)-\lambda x \phi(x)=0 \quad(0<x<1) \\
\phi(0)=\phi^{\prime}(1)=0
\end{array}\right\}
$$
has a non-trivial solution $\phi$ if and only if the integral equation
$$
\phi(x)+\lambda \int_0^1 t \min (x, t) \phi(t) \mathrm{d} t=0 \quad(0 \leqslant x \leqslant 1)
$$
possesses a non-trivial solution. By showing that the operator $K$ defined by $(K \phi)(x)=\int_0^1 t \min (x, t) \phi(t) \mathrm{d} t$ has $\|K\| \leqslant 2 /(3 \sqrt{ } 5)$ deduce that if $(3.28)$ has a non-trivial solution then $|\lambda|>3 \sqrt{ } 5 / 2$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 13

Let $(K \phi)(x)=\int_0^x(x t)^{ \pm} \phi(t) \mathrm{d} t(0 \leqslant x \leqslant 1)$. Find the kernel $k_n$ for which
$$
\left(K^n \phi\right)(x)=\int_0^x k_n(x, t) \phi(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1)
$$
and deduce that the (unique) solution of the equation
$$
\phi-K \phi=f
$$

satisfies
$$
\phi(x)=f(x)+\int_0^x(x t)^{\frac{1}{2}} \mathrm{e}^{\left(x^2-t^2\right) / 2} f(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1) .
$$

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02:44

Problem 14

Let $(K \phi)(x)=\int_0^x x \phi(t) \mathrm{d} t(0 \leqslant x \leqslant 1)$ and deduce that the integral equation
$$
\phi(x)-\int_0^x x \phi(t) \mathrm{d} t=1 \quad(0 \leqslant x \leqslant 1)
$$
has a unique solution $\phi$. By letting $f=1$ and finding $K^n f$ obtain an expression for $\phi$ in the form of a convergent series.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:15

Problem 15

Show that the initial value problem $\phi^{\prime \prime}(x)-(\sin x) \phi(x)=0, \phi(0)=0, \phi^{\prime}(0)=1$ is equivalent to
$$
\phi(x)-\int_0^x(x-t)(\sin t) \phi(t) \mathrm{d} t=x \quad(0 \leqslant x \leqslant 1) .
$$

Show that $\phi(x)=x(1-\sin x)+2(1-\cos x)+\epsilon(x) \quad(0 \leqslant x \leqslant 1) \quad$ where $\left(\int_0^1|\epsilon(x)|^2 \mathrm{~d} x\right)^{\frac{1}{2}}<0.0028$.

Katelyn Chen
Katelyn Chen
Numerade Educator

Problem 16

(i) Let $\phi$ satisfy $\phi=f+\lambda K \phi$ in $[0,1]$ where $f(x)=1(0 \leqslant x \leqslant 1)$ and $(K \phi)(x)=\int_0^x t \log (t / x) \phi(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1)$. Show by induction that $\left(K^n f\right)(x)=\left(-\frac{1}{4} x^2\right)^n /(n !)^2(n \in \mathbb{N})$ and hence prove that
$$
\phi(x)=1+\sum_{n=1}^{\infty}\left(-\frac{1}{4} i x^2\right)^n /(n !)^2 \quad(0 \leqslant x \leqslant 1)
$$
satisfies the given integral equation. Show also that, with $\lambda=1$, $\phi(x)=1-\frac{1}{4} x^2+\frac{1}{64} x^4+\epsilon(x)(0 \leqslant x \leqslant 1)$ where $\int_0^1|\epsilon(x)|^2 \mathrm{~d} x<1.4 \times 10^{-4}$.
(ii) Let $\phi=f+\lambda K \phi$ in $(0,1]$ where $f(x)=\log x \quad(0<x \leqslant 1)$ and $K$ is as defined in (i). Show that
$$
\left(K^n f\right)(x)=\left(-\frac{1}{4} x^2\right)^n\left\{\log x-1-\frac{1}{2}-\cdots-\frac{1}{n}\right\} /(n !)^2 \quad(n \in \mathbb{N})
$$
and deduce the solution of the integral equation. Show that, with $\lambda=1$,
$$
\phi(x)=\left(1-\frac{x^2}{4}+\frac{x^4}{64}\right) \log x+\frac{x^2}{4}-\frac{3 x^4}{128}+\epsilon(x) \quad(0<x \leqslant 1),
$$
where $\int_0^1|\epsilon(x)|^2 \mathrm{~d} x<2.7 \times 10^{-4}$.

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Problem 17

Show that the integral equation
$$
\phi(x, y)+\int_x^y \mathrm{~d} t \int_x^t \phi(s, t) \mathrm{d} s=\frac{1}{2}(f(x)+f(y)) \quad(x \in \mathbb{R}, x \leqslant y)
$$
is equivalent to the initial value problem
$$
\begin{gathered}
\phi_{x y}(x, y)-\phi(x, y)=0 \quad(x \in \mathbb{R}, x \leqslant y) \\
\phi(x, x)=f(x), \quad \phi_x(x, x)-\phi_y(x, x)=0 \quad(x \in \mathbb{R}) .
\end{gathered}
$$

Let $(K \phi)(x, y)=-\int_x^y \mathrm{~d} t \int_x^t \phi(s, t) \mathrm{d} s(x \in \mathbb{R}, x \leqslant y \leqslant x+a)$ and show that
$$
\left(K^n \phi\right)(x, y)=(-1)^n \int_x^y \mathrm{~d} t \int_x^t \frac{(y-t)^{n-1}(s-x)^{n-1}}{((n-1) !)^2} \phi(s, t) \mathrm{d} s .
$$

By showing that, as a map from $L_2(\mathbb{R} \times[0, a])$ to itself, $\left\|K^n\right\| \leqslant a^n / n$ ! deduce that (3.29) has a unique solution $\phi$. Deduce that the solution of (3.29) is
$$
\phi(x, y)=\frac{1}{2}(f(x)+f(y))-\frac{1}{2} \int_x^y \mathrm{~d} t \int_x^t J_0\left(2(y-t)^{\frac{1}{2}}(s-x)^{\frac{1}{2}}\right)(f(s)+f(t)) \mathrm{d} s
$$
where $J_0(z)=\sum_{n=0}^{\infty}(-1)^n \frac{z^{2 n}}{4^n(n !)^2}$ (the Bessel function of the first kind of order zero).

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Problem 18

The following simple transformations give rise to bounded linear maps from $L_2(0,1)$ to itself; check the details.
(i) If $a:[0,1] \rightarrow \mathbb{R}$ is a bounded measurable function and $(A \phi)(x)=a(x) \phi(x)$ $(x \in[0,1])$, then $\|\boldsymbol{A}\|=$ ess $\sup \{|a(x)|: 0 \leqslant x \leqslant 1\}$. Here ess sup denotes the essential supremum, the essential supremum of a set $S$ being $\sup \{\lambda: \mu\{x \in S: x \geqslant \lambda\} \neq 0\}$ where $\mu$ denotes Lebesgue measure. (In fact, in these terms, $a$ need only be an essentially bounded function.)
(ii) Let $h:[0,1] \rightarrow[0,1]$ be a bijective function such that $h$ and $h^{-1}$ are differentiable and have continuous derivatives. Then
$$
(U \phi)(x)=\phi(h(x))\left(\left|h^{\prime}(x)\right|\right)^{ \pm} \quad(0 \leqslant x \leqslant 1)
$$
defines a linear map from $L_2(0,1)$ to itself with the property that for all $\phi \in L_2(0,1)\|U \phi\|=\|\phi\| . U$ is invertible.

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Problem 19

Let $H$ be a Hilbert space and $S, T \in B(H)$. Show that if $S T=T S$ then $\rho(S T) \leqslant \rho(S) \rho(T)$.

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Problem 20

Let $V \in B\left(L_2(0,1)\right)$ be defined by $(V \phi)(x)=\int_0^x \phi$. From our work on Volterra operators we know $\rho(V)=0$. Let $U$ be defined by $(U \phi)(x)=\phi(1-x)(c f$. Problem 3.18). By observing that if $\phi(x)=\cos \left(\frac{1}{2} \pi x\right)(0 \leqslant x \leqslant 1)$ then $U V \phi=\frac{2}{\pi} \phi$, show that $\rho(U V) \geqslant \frac{2}{\pi}$, and therefore that the requirement that $S T=T S$ in Problem 3.19 cannot be omitted. Show also that $\rho(U V U)=0$.

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Problem 21

Suppose that $K \in B\left(L_2(a, b)\right)$ has rank $N$, and let $\left\{\phi_1, \ldots, \phi_N\right\}$ be an orthonormal basis for $\mathscr{R}(K)$. By observing that for all $\phi \in L_2(a, b) K \phi=$ $\Sigma_{n=1}^N\left(K \phi, \phi_n\right) \phi_n$, or otherwise, show that there are functions $\psi_1, \ldots, \psi_N$ in $L_2(a, b)$ for which, for all $\phi \in L_2(a, b)$,
$$
(K \phi)(x)=\int_a^b\left(\sum_{n=1}^N \phi_n(x) \psi_n(t)\right) \phi(t) \mathrm{d} t
$$
almost everywhere in $(a, b)$. (Thus every operator of finite rank in $L_2(a, b)$ arises from a degenerate kernel.)

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