From 96525643447829f55960be1e2e91ece28fdd4a02 Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 1 Apr 2021 00:38:45 +0200 Subject: [PATCH] Added Stieltjes --- chapters/measures_integrals.tex | 1 + chapters/ode.tex | 10 ++ chapters/sections/solution_methods.tex | 137 +++++++++++++++++++++++++ chapters/sections/stieltjes.tex | 35 +++++++ script.pdf | Bin 1096751 -> 1112109 bytes script.tex | 1 + 6 files changed, 184 insertions(+) create mode 100644 chapters/ode.tex create mode 100644 chapters/sections/solution_methods.tex create mode 100644 chapters/sections/stieltjes.tex diff --git a/chapters/measures_integrals.tex b/chapters/measures_integrals.tex index 6ec57eb..92f89d7 100644 --- a/chapters/measures_integrals.tex +++ b/chapters/measures_integrals.tex @@ -11,4 +11,5 @@ \subfile{sections/int_real_nums.tex} \subfile{sections/product_measures.tex} \subfile{sections/trans_thm.tex} + \subfile{sections/stieltjes.tex} \end{document} \ No newline at end of file diff --git a/chapters/ode.tex b/chapters/ode.tex new file mode 100644 index 0000000..f9a157b --- /dev/null +++ b/chapters/ode.tex @@ -0,0 +1,10 @@ +% !TeX root = ../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} + \chapter{Ordinary Differential Equations} + \vspace*{\fill}\par + \pagebreak + + \subfile{sections/solution_methods.tex} +\end{document} \ No newline at end of file diff --git a/chapters/sections/solution_methods.tex b/chapters/sections/solution_methods.tex new file mode 100644 index 0000000..dcac273 --- /dev/null +++ b/chapters/sections/solution_methods.tex @@ -0,0 +1,137 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Solution Methods} + +\begin{defi} + An ordinary differential equation (ODE) is an equation of the form + \[ + F(x, y, y', \cdots, y^{(n)}) = 0 + \] + with $F: \realn^{n+2} \rightarrow \realn$. $n$ is the order of the ODE. + Let $I$ be an open interval. A function $y: I \rightarrow \realn$ is a solution of the ODE if $y \in C^n(\realn)$ and + \[ + F(x, y(x), y'(x), \cdots, y^{(n)}(x)) = 0 ~~\forall x \in I + \] +\end{defi} + +\begin{eg} + \begin{align*} + y'' = -\frac{1}{y^2} && \text{Gravitational field} \\ + y'' = -\sin y && \text{Pendulum} + \end{align*} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Often times $F$ is only defined on subsets of $\realn^{n+2}$ + \item ODEs are not simple to solve + \item Even if we can't calculate explicit solutions, we can inspect the following properties + \begin{itemize} + \item Existence of solutions + \item Uniqueness of solutions + \item Dependency of solutions from initial conditions + \item Sability + \end{itemize} + \end{enumerate} +\end{rem} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $I$ be an open interval and $f: I \rightarrow \realn$ continuous. Then the solution of + \[ + y' = f(x) + \] + is the antiderivative of $f$. Let $x_0 \in I$, then + \[ + y(x) = \int_{x_0}^x f(t) \dd{t} + c ~~c \in \realn + \] + + \item Consider the ODE + \[ + y' = y + \] + The functions $x \mapsto c e^x$ are solutions $\forall c \in \realn$. Are those all the solutions that exist? + Let $y: I \rightarrow \realn$ be any solution, and consider + \[ + u(x) = y(x)e^{-x} + \] + Then + \begin{align*} + u'(x) &= y'(x) e^{-x} - y(x)e^{-x} \\ + &= \left(y'(x) - y(x)\right) e^{-x} = 0 ~~\forall x \in I + \end{align*} + So $u(x) = c$. + \end{enumerate} +\end{eg} + +\begin{defi}[Initial Value Problem] + Let $y_0, \cdots, y_{n-1} \in \realn$ and also $F: \realn^{n+2} \rightarrow \realn$. The system of equations + \begin{align*} + F(x, y, y', \cdots, y^{(n)}) = 0 && \begin{cases} + y(0) = y_0 \\ + y'(0) = y_1\\ + \cdots \\ + y^{(n-1)}(0) = y_{n-1} + \end{cases} + \end{align*} + is said to be an initial value problem (IVP). +\end{defi} + +\begin{eg} + Consider the problem + \begin{align*} + y'' = -\rec{y^2} && \begin{cases} + y(0) = y_0 \\ + y'(0) = y_1 + \end{cases} + \end{align*} + This describes the movement of a point mass in the gravitational field of the earth along a straight line + through the center of the earth with the initial position $y_0$ and the initial velocity $y_1$. +\end{eg} + +\begin{eg} + Consider the problem + \begin{align*} + y' = -y^2 && y(0) = 1 + \end{align*} + Assume $y: I \rightarrow \realn$ is a solution and $y(x) > 0 ~~\forall x \in I$. Then + \[ + 1 = -\frac{1}{y(t)^2} ~y'(t) ~~\forall t \in I + \] + By integrating we get + \begin{align*} + x = -\int_0^x \frac{1}{y(t)^2} y'(t) \dd{t} &\equalexpl{Substitution} -\int_1^{y(x)} \rec{y^2} \dd{y} \\ + &= \left. \rec{y} \right\vert_1^{y(x)} = \rec{y(x)} - 1 ~~\forall x \in I + \end{align*} + So a solution is + \[ + y(x) = \frac{1}{1+x} + \] + The biggest domain that makes sense is $(-1, \infty)$. Analogously one can approach equations with "separated variables", so of the form + \begin{align*} + y' = f(y)g(x) && y(x_0) = y_0 + \end{align*} +\end{eg} + +\begin{thm}[Separation of Variables] + Let $I, J$ be open intervals, and let + \begin{align*} + f: I \longrightarrow \realn && g: J \longrightarrow \realn + \end{align*} + be continuous with $0 \ne f(I)$. Let $x_0 \in J, ~y_0 \in I$. + Then there exists an open interval $I_2 \subset J$ and $x_0 \in I_2$ such that the IVP + \begin{align*} + y' = f(y)g(x) && y(x_0) = y_0 + \end{align*} + has exactly one solution on $I_2$. Set + \[ + F(y) = \int_{y_0}^y \rec{f(t)} \dd{t} + \] + Then $y: I_2 \rightarrow I$ is uniquely defined by + \[ + F(y(x)) = \int_{x_0}^x g(t) \dd{t} + \] +\end{thm} +\end{document} \ No newline at end of file diff --git a/chapters/sections/stieltjes.tex b/chapters/sections/stieltjes.tex new file mode 100644 index 0000000..1566823 --- /dev/null +++ b/chapters/sections/stieltjes.tex @@ -0,0 +1,35 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Lebesgue-Stieltjes Integral} + +\begin{defi} + Let $F: \realn \rightarrow \realn$ be a monotonically increasing, continuous function. Then we set + \begin{align*} + \lambda_F(\varnothing) := 0 && \lambda_F((a, b]) = F(b) - F(a), ~~(a, b] \in \intervals + \end{align*} +\end{defi} + +\begin{thm} + $\lambda_F$ is a measure on $H$. +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{defi} + The integral + \[ + \int_A f \dd{\lambda_F} + \] + is called the Lebesgue-Stieltjes integral on $\realn$ and is denoted by + \[ + \int_A f(x) \dd{F(x)} := \int_A f \dd{\lambda_F} + \] + If $A = [a, b]$, then we write + \[ + \int_a^b f(x) \dd{F(x)} + \] +\end{defi} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 284e110474e97b3c4d9b8f2e786bf99684dd1c76..14619d566c281b677c1319773712b080ed131ae8 100644 GIT binary patch delta 285833 zcmcG%cVJXS*9ZLFyFCq(-A#5QB?L$U39!8&E%aWMCL&cu5fbc5^-)1OTwstQMG=uM zWr0Ve7nP16BGry`3o7;d&D^`2JG*Q?Uf=ioCz&&I@7;TU=ggUN&YYQ9`FF$R*BY&w zX`gb ze_o(}21FVLM(UoCw&~NQI@T%))gnJ4_14>^_C8}ufctG}iatHD9ldi1{nJl6Y%uFt 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