From 131177e13cf50ecc39cd24edf96879d6132802eb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 28 Mar 2025 23:49:15 +0100 Subject: [PATCH] cviceni 1 + 2 --- Makefile | 7 +++++ cviceni.pdf | Bin 0 -> 68502 bytes cviceni.tex | 36 ++++++++++++++++++++++ cviceni1.tex | 19 ++++++++++++ cviceni2.tex | 82 +++++++++++++++++++++++++++++++++++++++++++++++++++ 5 files changed, 144 insertions(+) create mode 100644 Makefile create mode 100644 cviceni.pdf create mode 100644 cviceni.tex create mode 100644 cviceni1.tex create mode 100644 cviceni2.tex diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..e8e17b3 --- /dev/null +++ b/Makefile @@ -0,0 +1,7 @@ +.PHONY: clean + +cviceni.pdf: $(wildcard *.tex) + tectonic cviceni.tex + +clean: + rm cviceni.pdf diff --git a/cviceni.pdf b/cviceni.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dc76136acaf3c38f3ecc24da122f6b0bcb0358dd GIT binary patch literal 68502 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z)@S>Ny7<3~eGMg3GkQr8Z5}ozb`uU(GZRC0b{1A9BU2_0R+GP{riL6WX6(!?CVc<@ zSLo6!+1tB7{Z}Bn61}9InLWXO@!LxD%9fs{1dME)^y=D-1k40Xy8jn0frX9npPnYx zc}W{tLyVETlkR1!Gn3Nt74K z5JN+AOG2aw4NX@N# zPmCea4uEuGOWLMXUPwO^-oFim#p~|;>NpW?excb+L$tuYve&RQpM;h)?xd-f9>&$V zn7=oyFDav?Zj1N)W-RdyIl4F3U&(xz8gP1QroWI`g&t(OrQ%3O?pix4yO3?w3qE>g z)n`i&6Wi;@bZ)=BA?v5_*}8AY6F7Rykvl}!<-g?tscg1Zw19!At;O*FeH)!!44qs& VoJ`H2*#6E;VuK 0, c_2 > 0$ vztah $\log y = c_2 + t$, tedy $x - \log y$ je konstantní funkce. Z toho máme, že $u(x, y) = U(x - \log y)$ a dosazením počáteční podmínky dostáváme $\frac{1}{y} = u(0, y) = U(-\log y) \Rightarrow U(z) = e^z$. Máme tedy $u(x, y) = \frac{1}{y}e^x$, obdobná myšlenka funguje i pro případ $y < 0$ (dostaneme stejný výsledek). +\end{proof} diff --git a/cviceni2.tex b/cviceni2.tex new file mode 100644 index 0000000..5a6c80d --- /dev/null +++ b/cviceni2.tex @@ -0,0 +1,82 @@ +\section{Cvičení 2} + +\begin{example} + Nalezněte $u$ řešení $(z + y - x) \pdv{y}{x} + (z + x - y) \pdv{u}{y} + z \pdv{u}{z} = 0$ splňující $u(x, y, 1) = sin(x + y)$. + \begin{proof} + Máme soustavu ODR $\odv{x}{t} = z + y - x, \odv{y}{t} = z + x - y, \odv{z}{t} = z$. Platí $\odv{x}{t} + \odv{y}{t} - 2\odv{z}{t} = 0$, jinými slovy $\odv*{(x + y - 2z)}{t} = 0$. Tedy platí $u(x, y, z) = U(x + y - 2z)$. Dosazením počáteční podmínky dostáváme $u(x, y, 1) = sin(x + y) = U(x + y - 2)$, tedy $u(x, y, z) = sin(x + y - 2z + 2)$. + \end{proof} +\end{example} + +\begin{example} + Nalezněte $u$ řešení $x \pdv{u}{x} + (x + y) \pdv{u}{y} = 0$ takové, že $u(x, y) = u_0(x)$ na okolí bodu $(1, 0)$, případně na okolí bodu $(0, 0)$. +\end{example} + +\begin{proof} + Řešíme soustavu ODR $\odv{x}{t} = x, \odv{y}{t} = x + y$. Máme $x = c_1 e^t$, tedy $y = c_2 e_t + c_1 t e^t$. Všimneme si, že $\frac{y}{x} = \frac{c_2 + c_1 t}{c_1}$, tedy $xe^{-\frac{y}{x}} = c_1 e^t e^{-\frac{c_2+c_1 t}{c_1}}$ je konstantní funkce. Proto řešení hledáme ve tvaru $u(x, y) = U(x e^{-\frac{y}{x}})$. Má platit $u(x, 0) = u_0(x) = U(x)$, tedy $u(x, y) = u_0(xe^{-\frac{y}{x}})$. Tento výraz má smysl na okolí $(x, 0), x \neq 0$. Pro okolí bodu $(0, 0)$ má smysl jen řešení pro $u_0$ konstatní. + + Ukážeme si jiný postup. Vycházíme ze vztahů $x(t) = c_1 e^t, y(t) = c_2 e^t + c_1 t e^t$. Pro jaké hodnoty $c_1, c_2$ prochází bodem $(x_0, y_0)$ pro $t = 0$? Máme $x_0 = c_1, y_0 = c_2 + c_1 t = c_2$. Hledaná charakteristika tedy má tvar $x(t) = x_0 e^t, y(t) = y_0 e^t + x_0 t e^t$. Tyto charakteristiky protnou přímku $y = 0$ v bodě $t = -\frac{y_0}{x_0}$. Potom máme $x(t) = x_0 e^{-\frac{y_0}{x_0}}$. Máme tedy $u(x, y) = x e^{-\frac{y}{x}}$. +\end{proof} + +\begin{example} + Řešte úlohu $x^2 \pdv{u}{x} + xy \pdv{u}{y} = 0$, $u(x, 1) = u_0(x)$ na okolí bodu $(1, 1)$ (případně $(0,0)$). +\end{example} + +\begin{proof} + Soustava ODR $\odv{x}{t} = x^2, \odv{y}{t} = xy$. Platí $\odv*{\log x}{t} = \odv*{\log y}{t} = x$, tedy $\odv*{\log{\frac{x}{y}}}{t} = 0$. Řešení je tedy ve tvaru $u(x, y) = \tilde{U}(\ln(\frac{x}{y})) = U(\frac{y}{x})$. + Platí $U(\frac{1}{x}) = u(x, 1) = u_0(x)$, tedy $U(z) = u_0(\frac{1}{z})$, z čehož dostaneme $u(x, y) = u_0(\frac{x}{y})$. Na okolí počátku má smysl jen $u_0 = const$. +\end{proof} + +\begin{example} + Řešte úlohu $(3x + 4y) \pdv{u}{x} + (4x + 3y) \pdv{u}{y} = 0$ s počáteční podmínkou $u(x, 0) = u_0(x)$ na okolí $(1, 0)$. +\end{example} + +\begin{proof} + Řešíme soustavu ODR $\odv{x}{t} = 3x + 4y, \odv{y}{t} = 4x + 3y$. Máme lambda-matici $\begin{pmatrix} 3 - \lambda & 4 \\ 4 & 3 - \lambda\end{pmatrix}$. Tato matice je singulární pro $\lambda = 7$ a $\lambda = -1$ (výpočet determinantu). Příslušné vlastní vektory jsou $\begin{pmatrix}1\\1\end{pmatrix}$ a $\begin{pmatrix}1\\-1\end{pmatrix}$. Dostáváme, že $\begin{pmatrix}x(t)\\y(t)\end{pmatrix} = \begin{matrix}c_1 e^{-7t} + c_2 e^t \\ c_1 e^{-7t} - c_2 e^t \end{matrix}$ je obecné řešení této soustavy. + + Alternativní postup: víme, že $\odv{x}{t} + \odv{y}{t} = 7(x + y)$, tedy $\odv*{\log(x + y)}{t} = 7$, analogicky $\odv*{\log(x - y)}{t} = -1$. Můžeme tedy psát $\odv*{\log(x + y) + 7\log(x - y)}{t} = 0$. Potom $(x + y)(x - y)^7$ je konstantní podél řešení, tedy $u(x, y) = U((x + y)(x - y)^7)$. + + Aplikujeme počáteční podmínku, dostaneme $u_0(x) = u(x, 0) = U(x^8)$. Celkově dostáváme $u(x, y) = u_0((x + y)^{1/8} (x - y)^{7/8})$. +\end{proof} + +\begin{example} + Nalezněte fundamentální systém rovnice $\pdv{u}{x} + xz \pdv{u}{y} - xy \pdv {u}{z} = 0$. +\end{example} + +\begin{proof} + Řešíme soustavu ODR $\odv{x}{t} = 1, \odv{y}{t} = xz, \odv{z}{t} = -xy$. Vidíme, že $\odv{y^2}{t} + \odv{z^2}{t} = 2xyz - 2xyz = 0$. Tedy $\psi_1(x, y, z) = y^2 + z^2$ řeší danou soustavu. Hledejme druhé nezávislé řešení. Použijeme větu o redukci. Nechť tedy $\tilde{x} = x, \tilde{y} = y^2 + z^2, \tilde{z} = z$. + + Řešíme novou rovnici $\pdv{\tilde{u}}{\tilde{x}} - \tilde{x}\sqrt{\tilde{y} - \tilde{z}^2} \pdv{\tilde{u}}{\tilde{z}} = 0$. + Odpovídající soustava ODR je $\odv{\tilde{x}}{t} = 1, \odv{\tilde{z}}{t} = -\tilde{x} \sqrt{\tilde{y} - \tilde{z}^2}$. + Úpravami ji dostaneme do tvaru $\frac{1}{\sqrt{\tilde{y}}} \frac{\text{d}\tilde{z}}{\sqrt{1 - \frac{\tilde z^2}{\tilde{y}}}} = -\tilde{x} \text{d}\tilde{x}$. + Po zintegrování dostáváme $\arcsin (\frac{\tilde{z}}{\sqrt{\tilde{y}}}) + \frac{1}{2}x^2 = C$. + + Aplikujeme na tuto rovnost funkci sinus a pomocí příslušného součtového vzorce dostaneme rovnost + $$ \frac{\tilde{z}}{\sqrt{\tilde{y}}} \cos\left(\frac{\tilde{x}^2}{2}\right) + \sqrt{1 - \frac{\tilde{z}^2}{\tilde{y}}} \sin\left(\frac{1}{2}x^2\right) = C.$$ + Zpětným dosazením původních proměnných dostáváme + $$\frac{z}{\sqrt{y^2 + z^2}} \cos\left(\frac{x^2}{2}\right) + \frac{y}{\sqrt{y^2 + z^2}}\sin\left(\frac{x^2}{2}\right) = C.$$ + Můžeme tedy psát $\psi_2(x, y, z) = z \cos \frac{x^2}{2} + y\sin\frac{x^2}{2}$. + Tímto jsme nalezli fundamentální systém $\{\psi_1, \psi_2\}$. + + Na okolí $(1, 1, 0)$ aplikujeme počáteční podmínku $u(x, y, 0) = xy$. \\ Platí $\psi_1(x, y, 0) = y^2$, tedy $y = \sqrt{\psi_1}$. + Obdobně $\psi_2(x, y, 0) = y\sin\frac{x^2}{2}$, tedy $\frac{x^2}{2} = \arcsin\frac{\psi_2}{\sqrt{\psi_1}}$ a nakonec + $x = \sqrt{2\arcsin\frac{\psi_2}{\sqrt{\psi_1}}}$. + Z tohoto již dostáváme $u(x, y, z) = \sqrt{y^2 + z^2}\sqrt{2\arcsin\frac{z\cos\frac{x^2}{2} + y\sin\frac{x^2}{2}}{\sqrt{y^2 + z^2}}}$. +\end{proof} + +\begin{example} + Řešte úlohu $x\pdv{u}{x} + \log x \pdv{u}{y} = 0$ s počáteční podmínkou $u(x, 0) = u_0(x)$ na okolí $(2, 0)$. +\end{example} + +\begin{proof} + Máme soustavu ODR $\odv{x}{t} = x, \odv{y}{t} = \log x$. Triviálně dostáváme $x = c_1 e^t, y = \log c_1 t + \frac{1}{2}t^2 + c_2$. + + Fix $x_0, y_0$, hledáme konstanty tak, aby v čase $t=0$ jsme měli řešení $(x_0, y_0)$. Máme $x_0 = c_1$ a $y_0 = c_2$. Potom $x(t) = x_0 e^t, y(t) = \log x_0 t + \frac{1}{2}t^2 + y_0$. Najdeme průsečík s osou $x$ (položíme $y = 0$). Máme $t^2 + 2\log x_0 t + 2 y_0 = 0$. Toto je kvadratická rovnice, jejíž řešení jsou $t_{1,2} = -\log x_0 \pm \sqrt{\log^2x_0 - 2y_0}$. Zpětným dosazením dostáváme $x(t) = x_0 e^{-\log x_0 + \sqrt{\log^2 x_0 - 2 y_0}}$ a tedy $u(x, y) = u_0(xe^{-\log x + \sqrt{\log^2 x - 2y}})$. +\end{proof} + +\begin{example} + Řešte úlohu $\sqrt{x}\pdv{u}{x} + \sqrt{y}\pdv{u}{y} + \sqrt{z}\pdv{u}{z} = 0$ s počáteční podmínkou $u(1, y, z) = y - z$. +\end{example} + +\begin{proof} + Soustava ODR $\odv{x}{t} = \sqrt{x}, \odv{y}{t} = \sqrt{y}, \odv{z}{t} = \sqrt{z}$. Jejím řešením je $2\sqrt{x} = t + c_1, 2\sqrt{y} = t + c_2, 2\sqrt{z} = t + c_3$. Z toho dostáváme, že fundamentální systém tvoří například funkce $\sqrt{x} - \sqrt{y}$ a $\sqrt{x} - \sqrt{z}$. Máme $u(x, y, z) = U(\sqrt{x} - \sqrt{y}, \sqrt{x} - \sqrt{z})$, Aplikujeme počáteční podmínku: $u(1, y, z) = U(1 - \sqrt{y}, 1 - \sqrt{z}) = y - z$. Tedy $U(a, b) = (1 - a)^2 - (1 - b)^2$. Celkově dostáváme $u(x, y, z) = (1 - \sqrt{x} - \sqrt{y})^2 - (1 - \sqrt{x} - \sqrt{z})^2$. +\end{proof}