Green's function of an ODE boundary value problem

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Problem 12.2 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 12.1 "Examples of Problems with Solutions"

Find the spectrum, the eigenvalues, the eigenfunctions, and the norm of the operator $A$ in $L_{2}(\left[0,1\right])$ , defined by the formula $(A x)(t) = \int_{0}^{1} \min (t, s) x(s) d s$ .

Problem 12.25 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 12.2 "Problems to Solve"

Find the norm, the spectrum, the eigenvalues and the eigenfunctions of an operator $A$ in $L_{2}(\left[a, b\right])$ , defined by the formula $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in \left[a, b\right]$ , in the following cases:

(1)

$K(t, s) = \begin{cases} t(1-s), & t \leq s, \\ s(1-t), & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(2)

$K(t, s) = \begin{cases} \sin t \cos s, & t \leq s, \\ \cos t \sin s, & s \leq t, \end{cases} \quad a = 0, b = \pi$ ;

(3)

$K(t, s) = \begin{cases} \cos t \sin s, & t \leq s, \\ \sin t \cos s, & s \leq t, \end{cases} \quad a = 0, b = \pi$ ;

(4)

$K(t, s) = \begin{cases} (t+1) s, & t \leq s, \\ t(s+1), & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(5)

$K(t, s) = \begin{cases} \sin t \sin (1-s), & t \leq s, \\ \sin (1-t) \sin s, & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(6)

$K(t, s) = \begin{cases} (1+t)(1-s), & t \leq s, \\ (1-t)(1+s), & s \leq t, \end{cases} \quad a = -1, b = 1$ .

Problem 13.22 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 13.2 "Problems to Solve"

(1)(2)(3)(4)

Transform an integral equation $\mu x(t) = \int_{a}^{b} K(t, s) x(s) d s+y(t), t \in [a, b]$ , into a differential one and solve it:

(1)

$y(t) = \frac{t}{2}, K(t, s) = \begin{cases} \frac{\pi^{2}}{32} t(2-s), & t \leq s, \\ \frac{\pi^{2}}{32} s(2-t), & s < t, \end{cases} \quad [a, b] = \left[0,2\right], \mu = 1$ ;

(2)

$y(t) = t e^{t}, \quad K(t, s) = \begin{cases} -\frac{1}{\sinh 1} \sinh t \sinh (s-1), & t \leq s, \\ -\frac{1}{\sinh 1} \sinh s \sinh (t-1), & s < t, \end{cases} \quad [a, b] = \left[0,1\right], \mu = 1 ;$

(3)

$y(t) = \sin^{3} \pi t, K(t, s) = \begin{cases} (s-1) t, & 0 \leq t \leq s, \\ (t-1) s, & s < t \leq 1, \end{cases} \quad [a, b] = \left[0,1\right], \quad \mu = 0 ;$

(4)

$y(t) = 2 t^{2}-3 t+\frac{5}{2}, K(t, s) = \left|t-s\right|+t^{2},\left[a, b\right] = \left[1,2\right], \mu = 0$ .

Problem 11.96 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.5 "Integral equations"

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the following integral equations for all values $\lambda \in \mathbb {C}$ :

(a)

$x(t)-\lambda \int_{0}^{1} \min (t, s) x(s) d s = 0$ ;

(b)

$x(t)+\lambda \int_{0}^{1} \max (t, s) x(s) d s = 0$ .

Problem 11.97 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.5 "Integral equations"

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}(t s-\min (t, s)) x(s) d s = y(t)

(a)

$y(t) = t$ ;

(b)

$y(t) = \sin 2 \pi t$ .

Problem 11.98 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.5 "Integral equations"

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}\left(\frac{1}{2}-\left|t-s\right|\right) x(s) d s = y(t)

(a)

$y(t) = \cos \pi t$ ;

(b)

$y(t) = \cos 2 \pi t$ .

Problem 11.99 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 11.5 "Integral equations"

(a)(b)

In the space $L_{2}[0,1]$ , find the solution of the integral equation for all values $\lambda \in \mathbb {C}$ :

x(t)+\lambda \int _{0}^{1}(t s+\max (t, s)) x(s) d s = y(t)

(a)

$y(t) = \cos \pi t$ ;

(b)

$y(t) = \cos 2 \pi t$ .

Problem 12.73 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.2 "The spectrum of a compact operator"

In the space $L_{2}[0, \pi ]$ , find the eigenvalues and eigenfunctions of the integral operator

(A x)(t) = \int _{0}^{\pi } K(t, s) x(s) d s

K(t, s) = \begin{cases} \sin t \cos s & \text{ for } 0 \leqslant t \leqslant s \leqslant \pi , \\ \sin s \cos t & \text{ for } 0 \leqslant s \leqslant t \leqslant \pi . \end{cases}

Problem 12.74 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.2 "The spectrum of a compact operator"

(a)(b)

In the space $L_{2}[0,1]$ , find the eigenvalues and eigenfunctions of the integral operator

(A x)(t) = \int _{0}^{1} K(t, s) x(s) d s

(a)

$K(t, s) = t s+\max (t, s)$ ;

(b)

$K(t, s) = t s+\max (t, s)-2 \min (t, s)-$ -2.

Problem II.6.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section II.6 "An Application: Sturm-Liouville Systems"

(a)(b)(c)(d)

Consider the Sturm-Liouville operator $Lh = -h''$ with $a=0$ , $b=1$ , and for each of the following boundary conditions find the eigenvalues $\left\{ \lambda_{n}\right\}$ , the eigenvectors $\left\{ e_{n}\right\}$ , and the Green function $g(x,y)$ :

(a)

$h(0) = h(1) = 0$ ;

(b)

$h'(0) = h'(1) = 0$ ;

(c)

$h(0) = 0$ and $h'(1) = 0$ ;

(d)

$h(0) = h'(0)$ and $h(1) = -h'(1)$ .

Problem 3728 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.1 "Proper Integrals Depending on a Parameter"

Given the function

u(x)=\int _{0}^{1}K(x,y)v(y)\, dy ,

where

K(x,y)= \begin{cases} x(1-y) , & \text{if } x \leqslant y; \\ y(1-x) , & \text{if } x \geqslant y, \end{cases}

and $v(y)$ are both continuous. Prove that the given function satisfies the equation

u''(x)=-v(x) \quad (0 \leqslant x \leqslant 1) .

Problem 5900 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

At the point $x = \xi$ of a homogeneous string ( $\rho = 1$ ), subject to the action of a strong tension $T_{0}$ and fastened at the ends $x = 0$ and $x = 1$ , a weight $Q = 1$ is suspended (Fig. 15.22).

Fig. 15.22

Prove that the equation of the sag of the string has the form

u = K(x,\xi ) = \begin{cases} x(1-\xi ) \cdot \dfrac {1}{T_{0}} & \text{for } x \leqslant \xi , \\ \xi (1-x) \cdot \dfrac {1}{T_{0}} & \text{for } x \geqslant \xi . \end{cases}

Problem 5901 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

A heavy homogeneous string is fastened at the point $x = 1$ and is subject at the point $x = \xi$ to the action of a horizontal force $Q = 1$ (Fig. 15.23).

Fig. 15.23

Prove that the equilibrium equation of the string has the form

u = K(x,\xi ) = \begin{cases} -\ln \xi , & \text{for } x \leqslant \xi , \\ -\ln x, & \text{for } x \geqslant \xi . \end{cases}

Remark.

Hint. The tension of the string at the point $M(x)$ is taken equal to $-x$ .

Problem 5902 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

A homogeneous string of length $l = 1$ is fastened by the end $x = 0$ to a fixed axis and rotates about this axis with constant angular velocity. To the point $x = \xi$ of such a string a force $Q = 1$ is applied (Fig. 15.24).

Fig. 15.24

Prove that the equation of relative equilibrium of the string has the form

u = K(x,\xi ) = \begin{cases} \dfrac {1}{2}\ln \dfrac {1+x}{1-x} & \text{for } x \leqslant \xi , \\ \dfrac {1}{2}\ln \dfrac {1+\xi }{1-\xi } & \text{for } x \geqslant \xi . \end{cases}

Remark.

Hint. The tension of the string at the point $M(x)$ , caused by the centrifugal force, is taken equal to

R = 2\int _{x}^{1} x \, dx = 1-x^{2}.

Problem 5903 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Prove that for a continuously distributed load, the equation of the sag of the string has the form

u = \int _{0}^{1} K(x,\xi )p(\xi ) \, d\xi ,

where $p(\xi )$ denotes the magnitude of the load, computed per unit length of the string.

Remark.

Hint. Use the result of problem 5900 (problem 5900: $K(x,\xi ) = x(1-\xi )/T_{0}$ for $x \leqslant \xi$ , $K(x,\xi ) = \xi (1-x)/T_{0}$ for $x \geqslant \xi$ ).

Problem 5904 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Prove that the problem of small vibrations of a fastened string can be reduced to solving the homogeneous Fredholm equation

y(x) = \lambda \int _{0}^{1} K(x,\xi )y(\xi ) \, d\xi ,

where the symmetric kernel $K(x,\xi )$ has the form given in problem 5901 (problem 5901: $K(x,\xi ) = -\ln \xi$ for $x \leqslant \xi$ , $K(x,\xi ) = -\ln x$ for $x \geqslant \xi$ ).

Problem 5905 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

On a homogeneous string, fastened at the ends, a continuously distributed force acts

F(\xi ,t) = p_{0}(\xi )\sin \omega t.

Prove that the problem of forced vibrations of such a string reduces to solving the inhomogeneous Fredholm equation

y(x) = f(x) + \lambda \int _{0}^{1} K(x,\xi )y(\xi ) \, d\xi .

Problem 5941 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Reduce to an integral equation the problem of finding the solutions of the equation

(xy')'+\lambda xy = 0

under the boundary conditions: $y(0)$ bounded, $y(1) = 0$ .

Problem 5942 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Reduce to an integral equation the problem of finding the solutions of the equation

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

under the boundary conditions: $y(0) = 0$ , $y(1) = 0$ .

Problem 5943 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Construct the integral equation whose fundamental functions are the Legendre functions of order $m$ :

\left((1-x^{2})y'\right)'+\left(n(n+1)-\frac{m^{2}}{1-x^{2}}\right)y = 0;

boundary condition: $y$ bounded at $x = -1$ and $x = +1$ .

Problem 5944 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Solve problem 5943 for the case $m = 0$ of the Legendre polynomials.

Remark.

Hint. Consider the equation

\left((1-x^{2})y'\right)'+n(n+1)y = \frac{1}{2}.

Problem 5945 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Construct the integral equation whose fundamental functions are the Hermite polynomials:

y''+(1-x^{2})y+\lambda y = 0,

$y = 0$ at $x = \pm \infty$ .

Remark.

Hint. Set $\lambda = -2+\mu$ .

Problem 5946 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Construct the integral equation whose fundamental functions are the Laguerre polynomials:

xy''+(1-x)y'+\lambda y = 0;

$y$ bounded at $x = 0$ and $x = +\infty$ .

Remark.

Hint. Set $\lambda = -1+\mu$ .

Problem 5947 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Having found the corresponding Green's function, replace the equation

y''+k^{2}y = f(x), \quad y(0) = y(1) = 0

by an integral one.

Problem 5948 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Having found the corresponding Green's function, replace the equation

y''+k^{2}y = f(x), \quad y'(0) = hy(0), \quad y'(1) = -Hy(1)

by an integral equation.

Problem 5950 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 15.6 "Integral Equations"

Having found the corresponding Green's function $G(\xi ,x)$ , replace the equation

y''+k^{2}y = f(x), \quad y(0) = 0, \quad y'(1) = -hy(1)

by an integral one.

Problem 3.13.30 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.13.2 "Problems"

Let $K(x ; y) = \begin{cases} x(1-y), & \text{ if } x \leqslant y, \\ y(1-x), & \text{ if } x > y, \end{cases}$ and let $\varphi (y)$ be a function continuous on the segment $[0 ; 1]$ . Show that the function

u(x) = \int _{0}^{1} K(x ; y) \varphi (y) d y

satisfies on the segment $[0 ; 1]$ the equation $u^{\prime \prime }(x) = -\varphi (x)$ .

Problem 127 from Le A. "Functional Analysis Problems with Solutions" [Le13]More problems in Chapter 9 "Compact Operators and Their Spectra"

(a)(b)

Consider the operator $K: L^{2}([0,1]) \rightarrow L^{2}([0,1])$ defined by

(K f)(t) = \int _{0}^{1} k(t, s) f(s) d s

where $k(s, t) = \min \left\{ t, s: t, s \in [0,1]\right\}$ .

(a)

Prove that $K$ is a compact self-adjoint operator.

(b)

Find the spectrum $\sigma (K)$ and the norm $\left\| K\right\|$ .

Problem 3.4.16 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.4 "Duality in Banach Spaces"

(a)(b)(c)(d)

Define

K(s, t) = \begin{cases} (1-s) t & \text{ if } 0 \leq t \leq s \\ (1-t) s & \text{ if } s \leq t \leq 1\end{cases}

and define $T \in \mathscr {B}\left(L^{2}(0,1)\right)$ by

(T f)(s) = \int _{0}^{1} K(s, t) f(t) d t \quad (0 \leq s \leq 1)

(a)

Show that the eigenvalues of $T$ are $(n \pi )^{-2}, n = 1,2,3, \ldots$ , that the corresponding eigenfunctions are $\sin n \pi x$ , and that each eigenspace is onedimensional. Hint: If $\lambda \neq 0$ , the equation $T f = \lambda f$ implies that $f$ is infinitely differentiable, that $\lambda f^{\prime \prime }+f = 0$ , and that $f(0) = f(1) = 0$ . The case $\lambda = 0$ can be treated separately.

(b)

Show that the above eigenfunctions form an orthogonal basis for $L^{2}(0,1)$ .

(c)

Suppose $g(t) = \sum c_{n} \sin n \pi t$ . Discuss the equation $T f-\lambda f = g$ .

(d)

Show that $T$ is also a compact operator on $C$ , the space of all continuous functions on [0,1]. Hint: If $\left\{ f_{i}\right\}$ is uniformly bounded, then $\left\{ T f_{i}\right\}$ is equicontinuous.

Problem 3.13.7 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.13 "Unbounded Operators"

Consider the following three boundary value problems. The differential equation is

f^{\prime \prime }-f = g

where $g \in L^{2}([0,1])$ is given. The choices of boundary conditions are

(i) $f(0) = f(1) = 0$ .

(ii) $f^{\prime }(0) = f^{\prime }(1) = 0$ .

(iii) $f(0) = f(1)$ and $f^{\prime }(0) = f^{\prime }(1)$ .

Show that each of these problems has a unique solution $f$ such that $f^{\prime }$ is absolutely continuous and $f^{\prime \prime } \in L^{2}([0,1])$ . Hint: Combine Example 13.4 with Theorem 13.13.

Do this also by solving the problems explicitly.

Problem 41.3 from Bass R.F. "Stochastic Processes" [Bas11]More problems in Chapter 41 "One-dimensional diffusions"

Show that if

\int G_{a b}(x, y) m_{1}(d y)=\int G_{a b}(x, y) m_{2}(d y)

for all $x, a$ , and $b$ , then $m_{1}=m_{2}$ .

References

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[Le13] Le A. Functional Analysis Problems with Solutions. 2013.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.

[Bas11] Bass R.F. Stochastic Processes. Cambridge University Press, 2011.