From 528772aa7dd5d48086334f3bc025a8322e71f784 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Thu, 26 Oct 2023 14:18:42 +0200 Subject: [PATCH] changed pde->PDE and ode->ODE --- .../Differential_equations.tex | 122 +++++++-------- .../Differential_geometry.tex | 4 +- .../Dynamical_systems/Dynamical_systems.tex | 6 +- .../Harmonic_analysis/Harmonic_analysis.tex | 6 +- .../Numerical_calculus/Numerical_calculus.tex | 2 +- ...tion_of_partial_differential_equations.tex | 26 ++-- .../Partial_differential_equations.tex | 10 +- .../Stochastic_processes.tex | 2 +- .../Advanced_dynamical_systems.tex | 144 ++++++++++++++++-- .../Continuous_optimization.tex | 30 ++++ .../Montecarlo_methods/Montecarlo_methods.tex | 11 +- .../Numerical_methods_for_PDEs.tex | 6 +- .../Stochastic_calculus.tex | 6 +- .../Classical_mechanics.tex | 10 +- preamble_formulas.sty | 2 +- 15 files changed, 271 insertions(+), 116 deletions(-) diff --git a/Mathematics/3rd/Differential_equations/Differential_equations.tex b/Mathematics/3rd/Differential_equations/Differential_equations.tex index 915c50c..d5e44bb 100644 --- a/Mathematics/3rd/Differential_equations/Differential_equations.tex +++ b/Mathematics/3rd/Differential_equations/Differential_equations.tex @@ -33,21 +33,21 @@ where $\vf{g}:\Omega\subseteq\RR\times\RR^{m\cdot n}\rightarrow\RR^m$\footnote{Sometimes we will write $\vf{x}^{(n)}=\vf{g}\left(t,\vf{x},\vf{x}',\ldots,\vf{x}^{(n-1)}\right)$ instead of $\vf{x}^{(n)}(t)=\vf{g}\left(t,\vf{x}(t),\vf{x}'(t),\ldots,\vf{x}^{(n-1)}(t)\right)$ in order to simplify the notation.}. \end{definition} \begin{definition} - Consider the following ode of $m$ unknowns and of order $n$: + Consider the followingODE of $m$ unknowns and of order $n$: \begin{equation}\label{DE:ode1} \vf{x}^{(n)}(t)=\vf{f}\left(t,\vf{x}(t),\vf{x}'(t),\ldots,\vf{x}^{(n-1)}(t)\right) \end{equation} - We say that $\vf{\varphi}:I\subseteq\RR\rightarrow\RR^m$ is a \emph{solution of the ode} if: + We say that $\vf{\varphi}:I\subseteq\RR\rightarrow\RR^m$ is a \emph{solution of theODE} if: \begin{itemize} \item $\vf{\varphi}$ is $n$ times differentiable on $I$. \item $\displaystyle\left\{\left(t,\vf{\varphi}(t),\vf{\varphi}'(t),\ldots,\vf{\varphi}^{(n-1)}(t)\right):t\in I\right\}\subseteq\domain \vf{f}$ \item For all $t\in I$ we have: $$\vf{\varphi}^{(n)}(t)=\vf{f}\left(t,\vf{\varphi}(t),\vf{\varphi}'(t),\ldots,\vf{\varphi}^{(n-1)}(t)\right)$$ \end{itemize} - The set of all solutions of an ode is called \emph{general solution of the ode}. + The set of all solutions of anODE is called \emph{general solution of theODE}. \end{definition} \begin{proposition} - Consider an ode of $m$ unknowns and order $n$ of the form of \mcref{DE:ode1}. Then, we can transform this ode to an ode of $m\cdot n$ unknowns and order 1 in the following way\footnote{Therefore, we will mainly study the odes of order 1.}. Define $\vf{y}_i=\vf{x}^{(i-1)}$ for $i=1,\ldots,n$. Therefore, the functions $\vf{y}_i$ must satisfy: + Consider anODE of $m$ unknowns and order $n$ of the form of \mcref{DE:ode1}. Then, we can transform thisODE to anODE of $m\cdot n$ unknowns and order 1 in the following way\footnote{Therefore, we will mainly study theODEs of order 1.}. Define $\vf{y}_i=\vf{x}^{(i-1)}$ for $i=1,\ldots,n$. Therefore, the functions $\vf{y}_i$ must satisfy: \begin{equation*} \left\{ \begin{aligned} @@ -62,25 +62,25 @@ This is called a \emph{system of ordinary differential equations} (of order 1) or a \emph{differential system}. \end{proposition} \begin{definition} - We say that an ode is \emph{autonomous} if it doesn't depend on the independent variable, that is, if it is of the form: $$\vf{x}'=\vf{f}(\vf{x})$$ Otherwise, we say that an ode is \emph{non-autonomous}. + We say that anODE is \emph{autonomous} if it doesn't depend on the independent variable, that is, if it is of the form: $$\vf{x}'=\vf{f}(\vf{x})$$ Otherwise, we say that anODE is \emph{non-autonomous}. \end{definition} \begin{definition} - We say that an ode of order $n$ is \emph{linear} if it is of the form: + We say that anODE of order $n$ is \emph{linear} if it is of the form: \begin{equation}\label{DE:linear0} a_0(t)\vf{x}+a_1(t)\vf{x}'+\cdots+a_n(t)\vf{x}^{(n)}=\vf{b}(t) \end{equation} - where $a_i\in\mathcal{C}(I,\RR)$ for $i=0,\ldots,n$ and $\vf{b}\in\mathcal{C}(I,\RR^m)$ are arbitrary functions which do not need to be linear. We say that the linear ode of \mcref{DE:linear0} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linear ode of \mcref{DE:linear0} is of \emph{constant coefficients} if $a_i(t):=a_{i0}\in\RR$ $\forall t\in I$ and $\forall i=0,\ldots,n$. + where $a_i\in\mathcal{C}(I,\RR)$ for $i=0,\ldots,n$ and $\vf{b}\in\mathcal{C}(I,\RR^m)$ are arbitrary functions which do not need to be linear. We say that the linearODE of \mcref{DE:linear0} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linearODE of \mcref{DE:linear0} is of \emph{constant coefficients} if $a_i(t):=a_{i0}\in\RR$ $\forall t\in I$ and $\forall i=0,\ldots,n$. \end{definition} \begin{definition}[Initial value problem] - Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a function. Given $(t_0,\vf{x}_0)\in U$, the \emph{initial value problem} (\emph{ivp}) (or \emph{Cauchy problem}) consists in finding a solution of the ode $$\vf{x}'=\vf{f}(t,\vf{x})$$ with \emph{initial conditions} $\vf{x}(t_0)=\vf{x}_0$. + Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a function. Given $(t_0,\vf{x}_0)\in U$, the \emph{initial value problem} (\emph{ivp}) (or \emph{Cauchy problem}) consists in finding a solution of theODE $$\vf{x}'=\vf{f}(t,\vf{x})$$ with \emph{initial conditions} $\vf{x}(t_0)=\vf{x}_0$. \end{definition} - \subsubsection{Methods for solving odes} + \subsubsection{Methods for solvingODEs} \begin{proposition}[Separation of variables] - Let $f:(a,b)\rightarrow\RR$, $g:(c,d)\rightarrow\RR$ be continuous functions such that $f(x)\ne 0$ $\forall x\in (a,b)$. Consider the ode $x'=f(x)g(t)$. To find the solution of this ode, proceed as follows: - $$x'=f(x)g(t)\iff \int\frac{\dd{x}}{f(x)}=C+\int g(t)\dd{t}$$ where the constant $C$ is determined with the initial conditions of the ode. + Let $f:(a,b)\rightarrow\RR$, $g:(c,d)\rightarrow\RR$ be continuous functions such that $f(x)\ne 0$ $\forall x\in (a,b)$. Consider theODE $x'=f(x)g(t)$. To find the solution of thisODE, proceed as follows: + $$x'=f(x)g(t)\iff \int\frac{\dd{x}}{f(x)}=C+\int g(t)\dd{t}$$ where the constant $C$ is determined with the initial conditions of theODE. \end{proposition} \begin{proposition}[Variation of constants] - Let $I\subset \RR$ be an interval, $a,b\in\mathcal{C}(I,\RR)$. Consider the ode $x'=a(t)x+b(t)$. To find the solution of this ode, proceed as follows: + Let $I\subset \RR$ be an interval, $a,b\in\mathcal{C}(I,\RR)$. Consider theODE $x'=a(t)x+b(t)$. To find the solution of thisODE, proceed as follows: \begin{enumerate} \item Find the solution of the associated homogeneous system with the separation of variables method. Let's say that is $\varphi(t)c$, where $c\in\RR$. \item Try to find a general solution of the form $\varphi(t)c(t)$: @@ -91,12 +91,12 @@ \end{enumerate} \end{proposition} \begin{proposition}[Characteristic equation] - Consider the following ode of order $n$ of constant coefficients: + Consider the followingODE of order $n$ of constant coefficients: \begin{equation}\label{DE:char} x^{(n)} + a_{n-1}x^{(n-1)} + \cdots + a_1 x' + a_0 x = 0 \end{equation} We define the \emph{characteristic equation} of that system as the equation: $$p(r):=r^n + a_{n-1}r^{n-1} + \cdots + a_1 r + a_0 = 0$$ - In order to find the solution of this ode, we need to find the solutions to $p(r)=0$. So suppose $p$ has $s$ distinct real roots and $2(m-s)$ distinct complex roots. + In order to find the solution of thisODE, we need to find the solutions to $p(r)=0$. So suppose $p$ has $s$ distinct real roots and $2(m-s)$ distinct complex roots. $$\lambda_1,\ldots,\lambda_s,\lambda_{s+1},\overline{\lambda_{s+1}},\ldots,\lambda_{m},\overline{\lambda_m}$$ Here, $\lambda_i\in\RR$ $\forall i=1,\ldots,s$ and $\lambda_{i}=\alpha_i+\ii\beta_i\in\CC$ $\forall i=s+1,\ldots,m$. Assume, each of these roots have multiplicity $k_i\in\NN$. Then, the general solution to \mcref{DE:char} is: \begin{multline*} @@ -118,7 +118,7 @@ Then, the characteristic equation is precisely the characteristic polynomial of $\vf{A}$. \end{proposition} \begin{corollary} - Consider the following ode of order $n$ of constant coefficients: + Consider the followingODE of order $n$ of constant coefficients: \begin{equation}\label{DE:char2} x'' + px' + q = 0 \end{equation} @@ -130,8 +130,8 @@ \item If $p^2-4q<0$, then $\lambda_1=\alpha+\ii\beta\in\CC$ and the solution is: $$\varphi(t)=\exp{\alpha t}\left[c_1\cos(\beta t)+c_2\sin(\beta t)\right]$$ \end{itemize} \end{corollary} - \begin{proposition}[Reducible linear ode of second order] - Let $I\subset \RR$ be an interval, $a,b,c,d\in\mathcal{C}(I,\RR)$. Consider the system of odes: + \begin{proposition}[Reducible linearODE of second order] + Let $I\subset \RR$ be an interval, $a,b,c,d\in\mathcal{C}(I,\RR)$. Consider the system ofODEs: \begin{equation}\label{DE:ode-complex} \left\{ \begin{aligned} @@ -140,27 +140,27 @@ \end{aligned} \right. \end{equation} - In order to find the solution of this ode, consider the change of variable $z=x+\ii y$. Then, \mcref{DE:ode-complex} becomes: - $$z'=[a(t)+\ii b(t)]z+c(t)+\ii d(t)$$ which is a linear ode of order 1 and can be easily solved. + In order to find the solution of thisODE, consider the change of variable $z=x+\ii y$. Then, \mcref{DE:ode-complex} becomes: + $$z'=[a(t)+\ii b(t)]z+c(t)+\ii d(t)$$ which is a linearODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Bernoulli differential equation] Let $p,q\in\mathcal{C}((a,b),\RR)$ and $\alpha\in\RR$. Consider the \emph{Bernoulli differential equation}: \begin{equation}\label{DE:bernoulli} x'+p(t)x=q(t)x^\alpha \end{equation} - If $\alpha=0,1$ the ode is linear. So suppose $\alpha\ne 0,1$. In order to solve it, consider the change of variable $y=x^{1-\alpha}$. Then, \mcref{DE:bernoulli} becomes: - $$y'+(1-\alpha)p(t)y=(1-\alpha)q(t)$$ which is a linear ode of order 1 and can be easily solved. + If $\alpha=0,1$ theODE is linear. So suppose $\alpha\ne 0,1$. In order to solve it, consider the change of variable $y=x^{1-\alpha}$. Then, \mcref{DE:bernoulli} becomes: + $$y'+(1-\alpha)p(t)y=(1-\alpha)q(t)$$ which is a linearODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Riccati differential equation] Let $q_0,q_1,q_2\in\mathcal{C}((a,b),\RR)$. Consider the \emph{Riccati differential equation}: \begin{equation}\label{DE:riccati} x'=q_0(t)+q_1(t)x+q_2(t)x^2 \end{equation} - Suppose we have found a particular solution $x_1(t)$ of the ode of \mcref{DE:riccati}. In order to find the general solution, consider the change of variable $x=x_1(t)+\frac{1}{y}$. Then, \mcref{DE:riccati} becomes: - $$y'+[q_1(t)+2q_2(t)x_1(t)]y=-q_2(t)$$ which is a linear ode of order 1 and can be easily solved. + Suppose we have found a particular solution $x_1(t)$ of theODE of \mcref{DE:riccati}. In order to find the general solution, consider the change of variable $x=x_1(t)+\frac{1}{y}$. Then, \mcref{DE:riccati} becomes: + $$y'+[q_1(t)+2q_2(t)x_1(t)]y=-q_2(t)$$ which is a linearODE of order 1 and can be easily solved. \end{proposition} \begin{proposition}[Integrating factor] - Consider the ode: $$p(t,x)+q(t,x)x'=0\iff p(t,x)\dd{t}+q(t,x)\dd{x}=0$$ where $p,q\in\mathcal{C}^1(U,\RR)$ and $U\subseteq\RR^2$ is an open set. + Consider theODE: $$p(t,x)+q(t,x)x'=0\iff p(t,x)\dd{t}+q(t,x)\dd{x}=0$$ where $p,q\in\mathcal{C}^1(U,\RR)$ and $U\subseteq\RR^2$ is an open set. An \emph{integrating factor} $\mu(t,x)\in\mathcal{C}^1(U)$, $\mu(t,x)\ne 0$, is a function so that $$\mu(t,x)p(t,x)\dd{t}+\mu(t,x)q(t,x)\dd{x}$$ is an exact differential ($\dd{\Phi(t,x)}$) of a function $\Phi(t,x)$, that is: \begin{gather} \label{DE:ifactor1}\frac{\partial\Phi}{\partial t}(t,x)=\mu(t,x)p(t,x)\\ @@ -363,7 +363,7 @@ We say that linear equation of \mcref{DE:linear} is \emph{homogeneous} if $\vf{b}(t)=\vf{0}$ $\forall t\in I$. We say that linear equation of \mcref{DE:linear} is of \emph{constant coefficients} if $\vf{A}(t)=\vf{A}$ $\forall t\in I$, where $\vf{A}\in\mathcal{M}_n(\RR)$. \end{definition} \begin{definition} - Let $I\subseteq\RR$ be an interval, $t_0\in I$, $\vf{x}_0\in\RR^n$ and consider the ode of \mcref{DE:linear}. We define the \emph{flow of the linear ode} as the function: + Let $I\subseteq\RR$ be an interval, $t_0\in I$, $\vf{x}_0\in\RR^n$ and consider theODE of \mcref{DE:linear}. We define the \emph{flow of the linearODE} as the function: $$ \function{\vf{\phi}}{I\times I\times \RR^n}{\RR^n}{(t,t_0,\vf{x}_0)}{\vf{\varphi}_{(t_0,\vf{x}_0)}(t)} $$ @@ -383,7 +383,7 @@ \end{proposition} \subsubsection{Homogeneous systems} \begin{theorem} - Let $I\subseteq\RR$ be an interval and $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$. We define $\mathcal{A}_n$ as the set of all solutions of the linear ode: + Let $I\subseteq\RR$ be an interval and $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$. We define $\mathcal{A}_n$ as the set of all solutions of the linearODE: \begin{equation}\label{DE:homo} \vf{x}'=\vf{A}(t)\vf{x} \end{equation} Then, $\mathcal{A}_n$ is a vector space of dimension $n$ and for each $t_0\in I$, the function @@ -415,37 +415,37 @@ \end{enumerate} \end{corollary} \begin{definition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)=(m_{ij}(t))\in\mathcal{M}_n(\RR)$. We say that $\vf{M}(t)$ is a \emph{matrix solution} of the ode of \mcref{DE:homo} if $\vf{\varphi}_j={(m_{1j}(t),\ldots,m_{nj}(t))}^\mathrm{T}\in\mathcal{A}_n$ for $j=1,\ldots,n$. We say that $\vf{M}(t)$ is a \emph{fundamental matrix solution} of the ode of \mcref{DE:homo} if $\vf{M}(t)$ is a matrix solution and $\vf{\varphi}_1,\ldots,\vf{\varphi}_n$ are linearly independent. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)=(m_{ij}(t))\in\mathcal{M}_n(\RR)$. We say that $\vf{M}(t)$ is a \emph{matrix solution} of theODE of \mcref{DE:homo} if $\vf{\varphi}_j={(m_{1j}(t),\ldots,m_{nj}(t))}^\mathrm{T}\in\mathcal{A}_n$ for $j=1,\ldots,n$. We say that $\vf{M}(t)$ is a \emph{fundamental matrix solution} of theODE of \mcref{DE:homo} if $\vf{M}(t)$ is a matrix solution and $\vf{\varphi}_1,\ldots,\vf{\varphi}_n$ are linearly independent. \end{definition} \begin{proposition} Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{M}(t)\in\mathcal{M}_n(\RR)$. Then: \begin{enumerate} - \item $\vf{M}(t)$ is a matrix solution of the ode of \mcref{DE:homo} $\iff\vf{M}'(t)=\vf{A}(t)\vf{M}(t)$\footnote{By definition, if $\vf{M}(t)=(m_{ij}(t))$, then $\vf{M}'(t):=({m_{ij}}'(t))$.}. - \item $\vf{M}(t)$ is a matrix solution of the ode of \mcref{DE:homo} $\iff\forall \vf{c}\in\RR^n$, $\vf{M}(t)\vf{c}\in\mathcal{A}_n$. - \item If $\vf{M}(t)$ is a matrix solution of the ode of \mcref{DE:homo}, then $\forall \vf{C}\in\mathcal{M}_n(\RR)$, $\vf{M}(t)\vf{C}$ is a matrix solution of the ode of \mcref{DE:homo}. - \item If $\vf{M}(t)$ is a fundamental matrix solution of the ode of \mcref{DE:homo}, then $\det\vf{M}(t)\ne 0$ $\forall t\in I$. - \item $\vf{M}(t)$ is a fundamental matrix solution of the ode of \mcref{DE:homo} $\iff\vf{M}(t)$ is a matrix solution of the ode of \mcref{DE:homo} and $\exists t_0\in I$ such that $\det\vf{M}(t_0)\ne 0$. + \item $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} $\iff\vf{M}'(t)=\vf{A}(t)\vf{M}(t)$\footnote{By definition, if $\vf{M}(t)=(m_{ij}(t))$, then $\vf{M}'(t):=({m_{ij}}'(t))$.}. + \item $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} $\iff\forall \vf{c}\in\RR^n$, $\vf{M}(t)\vf{c}\in\mathcal{A}_n$. + \item If $\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo}, then $\forall \vf{C}\in\mathcal{M}_n(\RR)$, $\vf{M}(t)\vf{C}$ is a matrix solution of theODE of \mcref{DE:homo}. + \item If $\vf{M}(t)$ is a fundamental matrix solution of theODE of \mcref{DE:homo}, then $\det\vf{M}(t)\ne 0$ $\forall t\in I$. + \item $\vf{M}(t)$ is a fundamental matrix solution of theODE of \mcref{DE:homo} $\iff\vf{M}(t)$ is a matrix solution of theODE of \mcref{DE:homo} and $\exists t_0\in I$ such that $\det\vf{M}(t_0)\ne 0$. \end{enumerate} \end{proposition} \begin{proposition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{\Phi}(t),\vf{\psi}(t)\in\mathcal{M}_n(\RR)$ be matrix solutions of the ode of \mcref{DE:homo} such that $\vf{\Phi}(t)$ is fundamental. Then, $\exists! \vf{C}\in\mathcal{M}_n(\RR)$ such that $\vf{\psi}(t)=\vf{\Phi}(t)\vf{C}$. Moreover, $\vf{\psi}(t)$ is fundamental if and only if $\det \vf{C}\ne 0$. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{\Phi}(t),\vf{\psi}(t)\in\mathcal{M}_n(\RR)$ be matrix solutions of theODE of \mcref{DE:homo} such that $\vf{\Phi}(t)$ is fundamental. Then, $\exists! \vf{C}\in\mathcal{M}_n(\RR)$ such that $\vf{\psi}(t)=\vf{\Phi}(t)\vf{C}$. Moreover, $\vf{\psi}(t)$ is fundamental if and only if $\det \vf{C}\ne 0$. \end{proposition} \subsubsection{Non-homogeneous linear systems} \begin{proposition} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Suppose $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of the ode of \mcref{DE:linear}. Then, $$\vf{\phi}(t,t_0,\vf{x}_0)=\Phi(t)\left[{\Phi(t_0)}^{-1}\vf{x}_0+\int_{t_0}^t{\Phi(s)}^{-1}\vf{b}(s)\dd{s}\right]$$ where $\Phi(t)$ is a fundamental matrix of the associated homogeneous system. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Suppose $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of theODE of \mcref{DE:linear}. Then, $$\vf{\phi}(t,t_0,\vf{x}_0)=\Phi(t)\left[{\Phi(t_0)}^{-1}\vf{x}_0+\int_{t_0}^t{\Phi(s)}^{-1}\vf{b}(s)\dd{s}\right]$$ where $\Phi(t)$ is a fundamental matrix of the associated homogeneous system. \end{proposition} \begin{corollary} - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Then, the general solution $\vf{\varphi}(t)$ of the ode of \mcref{DE:homo} can be written as: $$\vf{\varphi}(t)=\vf{\varphi}_\mathrm{h}(t)+\vf{\varphi}_\mathrm{p}(t)$$ where $\vf{\varphi}_\mathrm{h}(t)$ is the general solution to the associated homogeneous system and $\vf{\varphi}_\mathrm{p}(t)$ is a particular solution of \mcref{DE:homo}. + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$ and $\vf{b}\in\mathcal{C}(I,\RR^n)$. Then, the general solution $\vf{\varphi}(t)$ of theODE of \mcref{DE:homo} can be written as: $$\vf{\varphi}(t)=\vf{\varphi}_\mathrm{h}(t)+\vf{\varphi}_\mathrm{p}(t)$$ where $\vf{\varphi}_\mathrm{h}(t)$ is the general solution to the associated homogeneous system and $\vf{\varphi}_\mathrm{p}(t)$ is a particular solution of \mcref{DE:homo}. \end{corollary} \begin{proposition}[Liouville's formula] - Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$, $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of the ode of \mcref{DE:homo} and $t_0\in I$. Then, for all $t\in I$ we have: $$\det(\Phi(t))=\det (\Phi(t_0))\exp{\int_{t_0}^t\trace(\vf{A}(s))\dd{s}}$$ + Let $I\subseteq \RR$ be an interval, $\vf{A}\in\mathcal{C}(I,\mathcal{L}(\RR^n))$, $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of theODE of \mcref{DE:homo} and $t_0\in I$. Then, for all $t\in I$ we have: $$\det(\Phi(t))=\det (\Phi(t_0))\exp{\int_{t_0}^t\trace(\vf{A}(s))\dd{s}}$$ \end{proposition} \subsubsection{Constant coefficients linear systems} \begin{lemma} Let $I\subseteq\RR$ be a compact interval and $\vf{f}:I\times\RR^n\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable. Let $\vf{\varphi}:I\rightarrow\RR^n$ be the solution of the ivp of \mcref{DE:ivp}. Then, $\forall\vf{\psi}\in\mathcal{C}(I,\RR^n)$ the sequence $(\vf{T}^m\vf{\psi})$ converges uniformly to $\vf{\varphi}$ on $I$. \end{lemma} \begin{theorem} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of the ode + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\vf{\Phi}(t)\in\mathcal{M}_n(\RR)$ be a matrix solution of theODE \begin{equation}\label{DE:coef-constants} \vf{x}'=\vf{A}\vf{x} \end{equation} @@ -463,13 +463,13 @@ \end{equation} \end{definition} \begin{proposition} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t,s\in \RR$. Then, the matrix exponential $\exp{\vf{A}t}$ is a fundamental matrix of the ode of \mcref{DE:coef-constants} and has the following properties: + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t,s\in \RR$. Then, the matrix exponential $\exp{\vf{A}t}$ is a fundamental matrix of theODE of \mcref{DE:coef-constants} and has the following properties: \begin{enumerate} \item $\exp{\vf{A}\cdot 0}=\vf{I}_n$ \item $\exp{\vf{A}(t+s)}=\exp{\vf{A}t}\exp{\vf{A}s}$ \item ${\left(\exp{\vf{A}t}\right)}^{-1}=\exp{-\vf{A}t}$ \item $\left(\exp{\vf{A}t}\right)'=\vf{A}\exp{\vf{A}t}=\exp{\vf{A}t}\vf{A}$ - \item If $\vf{\Phi}(t)$ is an arbitrary fundamental matrix of the ode of \mcref{DE:coef-constants}, then: $$\exp{\vf{A}t}=\vf{\Phi}(t){\vf{\Phi}(0)}^{-1}$$ + \item If $\vf{\Phi}(t)$ is an arbitrary fundamental matrix of theODE of \mcref{DE:coef-constants}, then: $$\exp{\vf{A}t}=\vf{\Phi}(t){\vf{\Phi}(0)}^{-1}$$ \end{enumerate} \end{proposition} \begin{lemma} @@ -492,7 +492,7 @@ $$ \end{proposition} \begin{corollary} - Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t\in\RR$ and consider the linear ode of \mcref{DE:coef-constants}. If $(\vf{v}_1,\ldots,\vf{v}_n)$ is a basis of eigenvectors with associated eigenvalues $\lambda_1,\ldots,\lambda_n$, respectively, then $(\vf{\varphi}_1,\ldots,\vf{\varphi}_n)$, where $\vf{\varphi}_i=\exp{\lambda_it}\vf{v}_i$ for $i=1,\ldots,n$, is a basis of $\mathcal{A}_n$. + Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $t\in\RR$ and consider the linearODE of \mcref{DE:coef-constants}. If $(\vf{v}_1,\ldots,\vf{v}_n)$ is a basis of eigenvectors with associated eigenvalues $\lambda_1,\ldots,\lambda_n$, respectively, then $(\vf{\varphi}_1,\ldots,\vf{\varphi}_n)$, where $\vf{\varphi}_i=\exp{\lambda_it}\vf{v}_i$ for $i=1,\ldots,n$, is a basis of $\mathcal{A}_n$. \end{corollary} \begin{lemma} Let $\vf{A}=\diag(\lambda_1,\ldots,\lambda_n)\in\mathcal{M}_n(\RR)$ and $t\in\RR$. Then: @@ -503,7 +503,7 @@ \begin{multline*} \exp{\vf{A}t}\vf{v}=\exp{\vf{A}t}\vf{u}+\ii\exp{\vf{A}t}\vf{w}=\exp{\alpha t}\left[\cos(\beta t)\vf{u}-\sin(\beta t)\vf{w}\right]+\\+\ii\exp{\alpha t}\left[\sin(\beta t)\vf{u}+\cos(\beta t)\vf{w}\right] \end{multline*} - and $\exp{\vf{A}t}\vf{u}$, $\exp{\vf{A}t}\vf{w}$ are linearly independent solutions of the ode of \mcref{DE:coef-constants} with initial conditions $\vf{x}(0)=\vf{u}$ and $\vf{x}(0)=\vf{w}$, respectively. + and $\exp{\vf{A}t}\vf{u}$, $\exp{\vf{A}t}\vf{w}$ are linearly independent solutions of theODE of \mcref{DE:coef-constants} with initial conditions $\vf{x}(0)=\vf{u}$ and $\vf{x}(0)=\vf{w}$, respectively. \end{proposition} \begin{definition} Let $\vf{A}\in\mathcal{M}_n(\RR)$. A vector $\vf{w}\in\RR^n$ is a \emph{generalized eigenvector} of rank $m$ of $\vf{A}$ corresponding to the eigenvalue $\lambda\in\RR$ if: $${(\vf{A}-\lambda\vf{I}_n)}^m\vf{w}=0\quad\text{but}\quad{(\vf{A}-\lambda\vf{I}_n)}^{m-1}\vf{w}\ne 0$$ @@ -526,7 +526,7 @@ \end{aligned} \right. $$ - are solutions of the ode of \mcref{DE:coef-constants}. Furthermore, if $\vf{v}_1,\ldots,\vf{v}_k$ are linearly independent, then so are $\vf{\varphi}_1,\ldots,\vf{\varphi}_k$. + are solutions of theODE of \mcref{DE:coef-constants}. Furthermore, if $\vf{v}_1,\ldots,\vf{v}_k$ are linearly independent, then so are $\vf{\varphi}_1,\ldots,\vf{\varphi}_k$. \end{lemma} \begin{corollary} Let $\vf{A}\in\mathcal{M}_n(\RR)$ and $\sigma(\vf{A})=\{\lambda_1,\ldots,\lambda_n\}$ be the spectrum of $\vf{A}$ such that: @@ -534,7 +534,7 @@ \item $\lambda_1,\ldots,\lambda_{2k}\in\CC\setminus\RR$, $\lambda_{k+i}=\overline{\lambda_i}$ and $\lambda_i=\alpha_i+\ii\beta_i$, $\alpha_i,\beta_i\in\RR$ for $i=1,\ldots,k$. \item $\lambda_{2k+1},\ldots,\lambda_n\in\RR$ \end{itemize} - Then, the general solution of the ode of \mcref{DE:coef-constants} is of the form: + Then, the general solution of theODE of \mcref{DE:coef-constants} is of the form: \begin{multline*} \vf{\varphi}(t)=\sum_{i=1}^k\exp{\alpha_i t}\left(\vf{P}_i(t)\cos(\beta_i t)+\vf{Q}_i(t)\sin(\beta_i t)\right)+\\+\sum_{i=2k+1}^n\exp{\lambda_i t}\vf{R}_i(t) \end{multline*} @@ -549,7 +549,7 @@ \vf{x}(t_0) =\vf{x}_0 \end{cases} \end{equation} - has a unique maximal solution $\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)$ defined on an interval $I_{(t_0,\vf{x}_0,\vf{\lambda})}$. We define the \emph{flow} of the ode $\vf{x}'=\vf{f}(t,\vf{x},\vf{\lambda})$ as: $$\function{\vf{\phi}}{I_{(t_0,\vf{x}_0,\vf{\lambda})}\times\RR\times\RR^n\times\RR^p}{\RR^n}{(t,t_0,\vf{x}_0,\vf{\lambda})}{\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)}$$ + has a unique maximal solution $\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)$ defined on an interval $I_{(t_0,\vf{x}_0,\vf{\lambda})}$. We define the \emph{flow} of theODE $\vf{x}'=\vf{f}(t,\vf{x},\vf{\lambda})$ as: $$\function{\vf{\phi}}{I_{(t_0,\vf{x}_0,\vf{\lambda})}\times\RR\times\RR^n\times\RR^p}{\RR^n}{(t,t_0,\vf{x}_0,\vf{\lambda})}{\vf{\varphi}_{(t_0,\vf{x}_0,\vf{\lambda})}(t)}$$ \end{definition} \subsubsection{Continuous and Lipschitz continuous dependence} \begin{lemma} @@ -577,7 +577,7 @@ If, moreover, $w\in\mathcal{C}^1((a,b))$, then: $$u(t)\leq w(a)\exp{\int_a^tv(r)\dd{r}}+\int_a^tw'(s)\exp{\int_s^tv(r)\dd{r}}\dd{s}\quad\forall t\in[a,b)$$ \end{lemma} \begin{proposition} - Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable with Lipschitz constant $L$. Let $\vf{\phi}$ be the flow of the ode $\vf{x}'=\vf{f}(t,\vf{x})$. Then, $\forall (t_0,\vf{x}_1),(t_0,\vf{x}_2)\in U$ and $\forall t\in I_{(t_0,\vf{x}_1)}\cap I_{(t_0,\vf{x}_2)}$, we have: + Let $U\subseteq\RR\times\RR^n$ be an open set and $\vf{f}:U\rightarrow\RR^n$ be a continuous function and Lipschitz continuous with respect to the second variable with Lipschitz constant $L$. Let $\vf{\phi}$ be the flow of theODE $\vf{x}'=\vf{f}(t,\vf{x})$. Then, $\forall (t_0,\vf{x}_1),(t_0,\vf{x}_2)\in U$ and $\forall t\in I_{(t_0,\vf{x}_1)}\cap I_{(t_0,\vf{x}_2)}$, we have: $$\|\vf{\phi}(t,t_0,\vf{x}_2)-\vf{\phi}(t,t_0,\vf{x}_1)\|\leq\exp{L|t-t_0|}\|\vf{x}_2-\vf{x}_1\|$$ Thus, $\vf{\phi}$ is locally Lipschitz continuous with respect to the third variable. \end{proposition} @@ -651,7 +651,7 @@ which creates a partition of $X$, called \emph{phase portrait}. \end{definition} \begin{definition} - The \emph{phase space} of an ode or system of odes is the space in which all possible states of a system are represented with each possible state corresponding to one unique point in the phase space. + The \emph{phase space} of anODE or system ofODEs is the space in which all possible states of a system are represented with each possible state corresponding to one unique point in the phase space. \end{definition} \begin{center} \begin{minipage}[b]{0.475\linewidth} @@ -676,16 +676,16 @@ Let $(G,X,\Psi)$ be a dynamical system and $x\in X$. We define the following function: $$\function{\Psi_x}{G}{\gamma(x)}{t}{\Psi(t,x)}$$ \end{definition} \begin{lemma} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the flow $\vf{\phi}(t,t_0,\vf{x}_0)$ of the ode $\vf{x}' =\vf{f}(\vf{x})$ is defined for all $t\in\RR$. Then, $(\RR,\RR^n,\vf{\Psi})$ is a dynamical system, where $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$. Furthermore, note that $\vf{\gamma}(\vf{x})=\im(\vf{\phi}(\cdot,0,\vf{x}))$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the flow $\vf{\phi}(t,t_0,\vf{x}_0)$ of theODE $\vf{x}' =\vf{f}(\vf{x})$ is defined for all $t\in\RR$. Then, $(\RR,\RR^n,\vf{\Psi})$ is a dynamical system, where $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$. Furthermore, note that $\vf{\gamma}(\vf{x})=\im(\vf{\phi}(\cdot,0,\vf{x}))$. \end{lemma} \begin{lemma} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that $\exists M,N\in\RR_{\geq 0}$ with $\|\vf{f}(\vf{x})\|\leq M\|\vf{x}\|+N$. Then, the solutions of the ode $\vf{x}' =\vf{f}(\vf{x})$ are defined for all $t\in\RR$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that $\exists M,N\in\RR_{\geq 0}$ with $\|\vf{f}(\vf{x})\|\leq M\|\vf{x}\|+N$. Then, the solutions of theODE $\vf{x}' =\vf{f}(\vf{x})$ are defined for all $t\in\RR$. \end{lemma} \begin{definition} - Let $\vf{f},\vf{g}:\RR^n\rightarrow\RR^n$ be continuous functions and $\vf{x}' =\vf{f}(\vf{x})$, $\vf{x}' =\vf{g}(\vf{x})$ be two odes for which we have existence and uniqueness of solutions. We say that these two odes are \emph{equivalent} if there exists $\vf{h}:\RR^n\rightarrow\RR^n$ such that $\vf{h}(\vf{x})\geq 0$ and $\vf{f}(\vf{x})=\vf{h}(\vf{x})\vf{g}(\vf{x})$ $\forall \vf{x}\in\RR^n$. Therefore, $\vf{f}$ and $\vf{g}$ have the same orbits oriented in the same way. + Let $\vf{f},\vf{g}:\RR^n\rightarrow\RR^n$ be continuous functions and $\vf{x}' =\vf{f}(\vf{x})$, $\vf{x}' =\vf{g}(\vf{x})$ be twoODEs for which we have existence and uniqueness of solutions. We say that these twoODEs are \emph{equivalent} if there exists $\vf{h}:\RR^n\rightarrow\RR^n$ such that $\vf{h}(\vf{x})\geq 0$ and $\vf{f}(\vf{x})=\vf{h}(\vf{x})\vf{g}(\vf{x})$ $\forall \vf{x}\in\RR^n$. Therefore, $\vf{f}$ and $\vf{g}$ have the same orbits oriented in the same way. \end{definition} \begin{corollary} - Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that the ode $\vf{x}' =\vf{f}(\vf{x})$ has existence and uniqueness of solutions for all initial conditions. Then, there exists a continuous function $\vf{g}:\RR^n\rightarrow\RR^n$ such that the autonomous odes induced by $\vf{f}$ and $\vf{g}$ are equivalent and the flow of the ode $\vf{x}' =\vf{g}(\vf{x})$ is defined $\forall t\in\RR$. + Let $\vf{f}:\RR^n\rightarrow\RR^n$ be a continuous function such that theODE $\vf{x}' =\vf{f}(\vf{x})$ has existence and uniqueness of solutions for all initial conditions. Then, there exists a continuous function $\vf{g}:\RR^n\rightarrow\RR^n$ such that the autonomousODEs induced by $\vf{f}$ and $\vf{g}$ are equivalent and the flow of theODE $\vf{x}' =\vf{g}(\vf{x})$ is defined $\forall t\in\RR$. \end{corollary} \begin{lemma} Let $H$ be a proper subgroup of $\RR$ which is closed. Then, $\exists T\in\RR_{\geq 0}$ such that $H=T\ZZ$. @@ -705,7 +705,7 @@ Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system and $\vf{\gamma}(\vf{x})$ be an orbit of $(\RR,\RR^n,\vf{\Psi})$. We say that $\vf{\gamma}(\vf{x})$ is \emph{periodic} of period $T>0$ if $\vf{\gamma}(\vf{x})\cong \S^1$ and $\ker\vf{\Psi_{\vf{x}}}=T\ZZ$. \end{definition} \begin{proposition} - Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system such that $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$, where $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of the ode $\vf{x}' =\vf{f}(\vf{x})$. Let $\vf{p}\in\RR^n$. Then, the following statements are equivalent: + Let $(\RR,\RR^n,\vf{\Psi})$ be a dynamical system such that $\vf{\Psi}(t,\vf{x})=\vf{\phi}(t,0,\vf{x})$, where $\vf{\phi}(t,t_0,\vf{x}_0)$ is the flow of theODE $\vf{x}' =\vf{f}(\vf{x})$. Let $\vf{p}\in\RR^n$. Then, the following statements are equivalent: \begin{enumerate} \item $\{\vf{p}\}$ is a critical point. \item $\vf{\phi}(t,0,\vf{p})=\vf{p}$. @@ -776,7 +776,7 @@ Let $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$ be dynamical systems and $h$ be a conjugacy of class $\mathcal{C}^r$ between them. Then, $h$ is an equivalence of class $\mathcal{C}^r$ between $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$. \end{proposition} \begin{proposition} - Two dynamical systems induced by two equivalent odes are equivalent (as a dynamical systems). + Two dynamical systems induced by two equivalentODEs are equivalent (as a dynamical systems). \end{proposition} \begin{proposition} Let $(G,X,\Psi_1)$ and $(G,X,\Psi_2)$ be dynamical systems and $h:X\rightarrow Y$ be an equivalence of class $\mathcal{C}^r$ between them. Then: @@ -805,10 +805,10 @@ x^{\beta/\alpha} & \text{if }x\geq 0 \\ -{|x|}^{\beta/\alpha} & \text{if }x< 0 \end{cases}$$ - Then, $h$ is a topological conjugation between the systems induced by the odes $x'=\alpha x$ and $y'=\beta y$. + Then, $h$ is a topological conjugation between the systems induced by theODEs $x'=\alpha x$ and $y'=\beta y$. \end{proposition} \begin{proposition} - Let $\vf{A},\vf{B}\in\mathcal{M}_n(\RR)$ be similar matrices, that is, $\exists\vf{P}\in\mathcal{M}_n(\RR)$ such that $\vf{B}=\vf{P}\vf{A}\vf{P}^{-1}$. Then, the function $$\function{\vf{h}}{\RR^n}{\RR^n}{\vf{x}}{\vf{Px}}$$ is a conjugation between the systems induced by the odes $\vf{x}'=\vf{A}\vf{x}$ and $\vf{y}'=\vf{B}\vf{y}$. + Let $\vf{A},\vf{B}\in\mathcal{M}_n(\RR)$ be similar matrices, that is, $\exists\vf{P}\in\mathcal{M}_n(\RR)$ such that $\vf{B}=\vf{P}\vf{A}\vf{P}^{-1}$. Then, the function $$\function{\vf{h}}{\RR^n}{\RR^n}{\vf{x}}{\vf{Px}}$$ is a conjugation between the systems induced by theODEs $\vf{x}'=\vf{A}\vf{x}$ and $\vf{y}'=\vf{B}\vf{y}$. \end{proposition} \subsubsection{Local equivalence and conjugacy of dynamical systems} \begin{definition} @@ -1106,7 +1106,7 @@ \subsection{Qualitative theory of planar differential systems} \subsubsection{Polynomial vectors fields} \begin{definition} - Let $p,q\in\RR[x,y]$. The system of odes + Let $p,q\in\RR[x,y]$. The system ofODEs \begin{equation}\label{DE:poly} \left\{ \begin{aligned} @@ -1449,17 +1449,17 @@ \end{corollary} \subsection{Introduction to partial differential equations} \begin{definition} - Let $U\subseteq \RR^n$ be an open set. A \emph{partial differential equation} (\emph{pde}) of order $k$ is an expression of the form $$F\left(\vf{x},u(\vf{x}),\pdv{u}{\vf{x}},\ldots,\pdv[k]{u}{\vf{x}}\right)=0$$ where $\vf{x}=(x_1,\ldots,x_n)$, $F:U\times\RR\times\RR^{n^1}\times\cdots\times\RR^{n^k}\rightarrow\RR$ is a given function and $u:U\rightarrow \RR$ is an unknown function. The function $u$ is called \emph{solution} of the pde defined by $F$. + Let $U\subseteq \RR^n$ be an open set. A \emph{partial differential equation} (\emph{PDE}) of order $k$ is an expression of the form $$F\left(\vf{x},u(\vf{x}),\pdv{u}{\vf{x}},\ldots,\pdv[k]{u}{\vf{x}}\right)=0$$ where $\vf{x}=(x_1,\ldots,x_n)$, $F:U\times\RR\times\RR^{n^1}\times\cdots\times\RR^{n^k}\rightarrow\RR$ is a given function and $u:U\rightarrow \RR$ is an unknown function. The function $u$ is called \emph{solution} of the PDE defined by $F$. \end{definition} \subsubsection{Quasilinear partial differential equations} \begin{definition} - Let $U\subseteq \RR^n$ be an open set and $u:U\rightarrow\RR$ be a function. A \emph{quasilinear pde} is an expression of the form: + Let $U\subseteq \RR^n$ be an open set and $u:U\rightarrow\RR$ be a function. A \emph{quasilinear PDE} is an expression of the form: \begin{equation}\label{DE:pde1} p_1(\vf{x},u)\pdv{u}{x_1}+\cdots+p_n(\vf{x},u)\pdv{u}{x_n}=q(\vf{x},u) \end{equation} \end{definition} \begin{theorem} - Let $U\subseteq \RR^n$ be an open set and $u:U\rightarrow\RR$ be a function and consider the pde of \mcref{DE:pde1}. Let $H_1,\ldots,H_{n}$ be the $n$ independent first integrals of the system: + Let $U\subseteq \RR^n$ be an open set and $u:U\rightarrow\RR$ be a function and consider the PDE of \mcref{DE:pde1}. Let $H_1,\ldots,H_{n}$ be the $n$ independent first integrals of the system: $$ \left\{ \begin{aligned} @@ -1474,13 +1474,13 @@ \end{theorem} \subsubsection{Heat, wave and Laplace equations} \begin{definition}[Heat equation] - Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{heat equation} is the pde defined by $$\pdv{u}{t}=k\pdv[2]{u}{x}$$ where $k\in\RR$. + Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{heat equation} is the PDE defined by $$\pdv{u}{t}=k\pdv[2]{u}{x}$$ where $k\in\RR$. \end{definition} \begin{proposition} Consider a bar of line $L\in\RR_{>0}$ whose temperature can be modeled by a function $u:\RR\times\RR\rightarrow\RR$, and $f:[0,L]\rightarrow\RR$ be a function. Then, the solution $u(x,t)$ to the heat equation with boundary conditions $u(x,0)=f(x)$ and $u(0,t)=u(L,t)=0$ is: $$u(x,t)=\sum_{n=1}^\infty b_n\sin\left(\frac{\pi n x}{L}\right)\exp{-\frac{n^2\pi^2k}{L^2}t}$$ where $\displaystyle b_n=\frac{1}{L}\int_{-L}^Lf(x)\sin\left(\frac{\pi n x}{L}\right)\dd{x}$. \end{proposition} \begin{definition}[Wave equation] - Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{wave equation} is the pde defined by $$\pdv[2]{u}{t}=c^2\pdv[2]{u}{x}$$ where $c\in\RR$. + Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{wave equation} is the PDE defined by $$\pdv[2]{u}{t}=c^2\pdv[2]{u}{x}$$ where $c\in\RR$. \end{definition} \begin{proposition} Consider a string of line $L\in\RR_{>0}$ whose position can be modeled by a function $u:\RR\times\RR\rightarrow\RR$, and $f,g:[0,L]\rightarrow\RR$ be functions. Then, the solution $u(x,t)$ to the wave equation with boundary conditions $u(x,0)=f(x)$, $u_t(x,0)=g(x)$ and $u(0,t)=u(L,t)=0$ is: $$u(x,t)=\sum_{n=0}^\infty \sin\left(\frac{\pi n x}{L}\right)\left[a_n\cos\left(\frac{\pi n c}{L}t\right)+ b_n\sin\left( \frac{\pi n c}{L}t\right)\right]$$ where: @@ -1496,7 +1496,7 @@ Let $f,g:\RR\rightarrow\RR$ be functions. The solution $u(x,t)$ to the wave equation with boundary conditions $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$ is: $$u(x,t)=\frac{f(x-ct)+f(x+ct)}{2}+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\dd{s}$$ \end{proposition} \begin{definition}[Laplace equation] - Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{Laplace equation} is the pde defined by: $$\pdv[2]{u}{x}+\pdv[2]{u}{y}=\laplacian u=0$$ + Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{Laplace equation} is the PDE defined by: $$\pdv[2]{u}{x}+\pdv[2]{u}{y}=\laplacian u=0$$ \end{definition} \begin{proposition} The Laplacian of a function $u:(0,\infty)\times[0,2\pi]\rightarrow\RR$ in polar coordinates $(r,\theta)$ is: $$\laplacian u=u_{rr}+\frac{u_r}{r}+\frac{u_{\theta\theta}}{r^2}$$ diff --git a/Mathematics/3rd/Differential_geometry/Differential_geometry.tex b/Mathematics/3rd/Differential_geometry/Differential_geometry.tex index 922572d..30a0be6 100644 --- a/Mathematics/3rd/Differential_geometry/Differential_geometry.tex +++ b/Mathematics/3rd/Differential_geometry/Differential_geometry.tex @@ -885,7 +885,7 @@ Let $S\subseteq\RR^3$ be a surface and $\vf{X}$, $\vf{Y}$ be vector fields tangent to $S$ along a curve $\vf\alpha:I\rightarrow S$ of class $\mathcal{C}^\infty$ such that they are parallel. Then, $t\mapsto\langle \vf{X}(t),\vf{Y}(t)\rangle$ is constant. In particular, the norms $\|\vf{X}(t)\|$, $\|\vf{Y}(t)\|$ as well as the angle between $\vf{X}(t)$ and $\vf{Y}(t)$ are constant. \end{proposition} \begin{proposition} - Let $S\subseteq\RR^3$ be a surface, $(V,\vf\varphi(u,v))$ is a parametrization of $S$ and $\vf\alpha: I\rightarrow S$ be a parametrized curve of class $\mathcal{C}^\infty$ such that $\vf\alpha=\vf\varphi(u(t),v(t))$. Then, given $t_0\in I$ and $\vf{w}\in T_{\vf\alpha(t_0)}S$ there exists a unique parallel vector field $\vf{X}=a\vf\varphi_u+b\vf\varphi_v$ along $\vf\alpha$ such that $\vf{X}(t_0)=\vf{w}$. This vector field is called \emph{parallel transport} of the vector $\vf{w}$ along $\vf{\alpha}$, and it is defined on the entire interval $I$. It can be found by solving this system of odes: + Let $S\subseteq\RR^3$ be a surface, $(V,\vf\varphi(u,v))$ is a parametrization of $S$ and $\vf\alpha: I\rightarrow S$ be a parametrized curve of class $\mathcal{C}^\infty$ such that $\vf\alpha=\vf\varphi(u(t),v(t))$. Then, given $t_0\in I$ and $\vf{w}\in T_{\vf\alpha(t_0)}S$ there exists a unique parallel vector field $\vf{X}=a\vf\varphi_u+b\vf\varphi_v$ along $\vf\alpha$ such that $\vf{X}(t_0)=\vf{w}$. This vector field is called \emph{parallel transport} of the vector $\vf{w}$ along $\vf{\alpha}$, and it is defined on the entire interval $I$. It can be found by solving this system ofODEs: $$\left\{ \begin{aligned} a'+\Gamma_{11}^1au'+\Gamma_{12}^1av'+\Gamma_{21}^1bu'+\Gamma_{22}^1bv' & =0 \\ @@ -991,7 +991,7 @@ \end{lemma} \begin{definition} Let $U\subseteq\RR^n$ be an open set and $\vf{X}=\sum X^i\pdv{}{x^i}\in\mathcal{X}(U)$. We say that a parametrized curve $\vf{\gamma}:I\rightarrow\RR^n$ is an \emph{integral curve} of $\vf{X}$ if: $$\vf\gamma'(t)=\vf{X}(\vf\gamma(t))\qquad \forall t\in I$$ - That is, the integral curve $\vf\gamma(t)=(x^1(t),\ldots,x^n(t))$ of $\vf{X}$ satisfies the following system of odes: + That is, the integral curve $\vf\gamma(t)=(x^1(t),\ldots,x^n(t))$ of $\vf{X}$ satisfies the following system ofODEs: $$ \left\{ \begin{aligned} diff --git a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex index 8c4e27a..e32113c 100644 --- a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex +++ b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex @@ -36,7 +36,7 @@ \end{enumerate} \end{theorem} \begin{definition} - Let $f,g:\RR^2\rightarrow\RR$ be two functions and consider the system of odes: + Let $f,g:\RR^2\rightarrow\RR$ be two functions and consider the system ofODEs: \begin{equation}\label{DS:plane} \left\{ \begin{aligned} @@ -645,14 +645,14 @@ A semiestable limit cycle $\Gamma_\mu$ of a family of rotated vector fields splits into two simple limit cycles, one stable and one unstable, as the parameter $\mu$ is varied in one sense and it disappears as $\mu$ is varied in the opposite sense. \end{theorem} \begin{theorem}[Melnikov's method] - Let $\vf{f}\in\mathcal{C}^1(\RR^2)$, $\vf{g}\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following ode: + Let $\vf{f}\in\mathcal{C}^1(\RR^2)$, $\vf{g}\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the followingODE: \begin{equation}\label{DS:melnikov} \vf{x}'=\vf{f}(\vf{x})+\varepsilon\vf{g}(\vf{x},\vf{\mu}) \end{equation} Suppose that for $\varepsilon =0$ the system has a one-parameter family of periodic orbits $\vf\gamma_h(t)$ of period $T_h$. Then for any simple zero $(\vf\mu_0,h_0)$ of the function $$M(\vf\mu, h)=\int_{0}^{T_h}\vf{f}(\vf\gamma_h(t))\times \vf{g}(\vf\gamma_h(t))\dd{t}$$ there exists a unique limit cycle $\vf\Gamma_\varepsilon$ for $\varepsilon\simeq 0$ such that $\displaystyle\lim_{\varepsilon\to 0}\vf\Gamma_\varepsilon=\vf\gamma_{h_0}$. On the other hand, if $M(\vf\mu_0,h_0)\ne 0$, for sufficiently small $\varepsilon$, the system of \mcref{DS:melnikov} with $\vf\mu=\vf\mu_0$ has no limit cycle in any sufficiently small neighborhood of $\vf\gamma_{h_0}$. \end{theorem} \begin{corollary}[Melnikov's method] - Let $H\in\mathcal{C}^2(\RR^2)$, $P,Q\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following system of odes: + Let $H\in\mathcal{C}^2(\RR^2)$, $P,Q\in\mathcal{C}^1(\RR^2\times\RR^m)$ and $\varepsilon\simeq 0$. Consider the following system ofODEs: \begin{equation*} \left\{ \begin{aligned} diff --git a/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex b/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex index 69186d9..834c688 100644 --- a/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex +++ b/Mathematics/4th/Harmonic_analysis/Harmonic_analysis.tex @@ -166,7 +166,7 @@ Let $f(x)=\exp{-a x^2}$. Then, $\F f(\xi)=\sqrt{\frac{\pi}{a}}\exp{-\frac{{(\pi \xi)}^2}{a}}$ and moreover $\F^2f=f$. In particular if $a=\pi$, then $\F f=f$. \end{lemma} \begin{sproof} - $f$ satisfies the ode $y'=-2a x y$. Taking $\ \widehat{}\ $ on this expression and using \mcref{HA:diffFourierXf,HA:diffFourierTransf} we obtain that $\widehat{f}$ must satisfy the following ode: + $f$ satisfies theODE $y'=-2a x y$. Taking $\ \widehat{}\ $ on this expression and using \mcref{HA:diffFourierXf,HA:diffFourierTransf} we obtain that $\widehat{f}$ must satisfy the followingODE: $$y'=-\frac{2\pi^2\xi}{a} y$$ with initial condition $y(0)=\int_{-\infty}^{+\infty}\exp{-a x^2}\dd{x}=\sqrt{\frac{\pi}{a}}$. \end{sproof} @@ -573,7 +573,7 @@ \end{theorem} \subsubsection{Applications of the Fourier transform} \begin{remark} - Probably the most important application of Fourier series is the resolution of pdes and it is a consequence of \mcref{HA:diffFourierTransf}, which reduces any order of a pde in the spatial variable to 1. The procedure is to compute the Fourier transform $\F$ of the pde, solve it, and then get back to the first function using the inverse transform. + Probably the most important application of Fourier series is the resolution of PDEs and it is a consequence of \mcref{HA:diffFourierTransf}, which reduces any order of a PDE in the spatial variable to 1. The procedure is to compute the Fourier transform $\F$ of the PDE, solve it, and then get back to the first function using the inverse transform. \end{remark} \begin{theorem}[Uncertainty principle] Let $f\in L^2(\RR)$ be differentiable such that $x\abs{f}^2\in L^1(\RR)$ and $f'\in L^2(\RR)$. Then: @@ -1266,7 +1266,7 @@ $$ \partial_t \widehat{E}+4\pi^2a^2\norm{\vf\xi}^2\widehat{E}=\delta_t $$ - because $\delta=\delta_{\vf{x}}\delta_t$. It can be seen that a solution of this ode is: + because $\delta=\delta_{\vf{x}}\delta_t$. It can be seen that a solution of thisODE is: $$ \widehat{E}(t,\xi)=\indi{[0,\infty)}(t)\exp{-4\pi^2a^2\norm{\vf\xi}^2t} $$ diff --git a/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex b/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex index 6e39190..1724aca 100644 --- a/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex +++ b/Mathematics/4th/Numerical_calculus/Numerical_calculus.tex @@ -84,7 +84,7 @@ Moreover, we say that the algorithm has \emph{order of convergence} $p$ if $\norm{\vf{e}_n}=\O{h^p}$. \end{definition} \begin{remark} - Note that in a consistent method the difference equation for the method approaches the ode as the step size goes to zero, whereas in a convergent method is the solution to the difference equation that approaches the solution to the ode as the step size goes to zero. + Note that in a consistent method the difference equation for the method approaches theODE as the step size goes to zero, whereas in a convergent method is the solution to the difference equation that approaches the solution to theODE as the step size goes to zero. \end{remark} \begin{theorem}\label{NC:errorLipschitz} Consider a consistent one-step explicit method such that its incremental function $\vf\phi$ is Lipschitz continuous (with constant $L$) with respect to $\vf{x}$. Then: diff --git a/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Numerical_integration_of_partial_differential_equations.tex b/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Numerical_integration_of_partial_differential_equations.tex index 1cdfe68..7dcf227 100644 --- a/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Numerical_integration_of_partial_differential_equations.tex +++ b/Mathematics/4th/Numerical_integration_of_partial_differential_equations/Numerical_integration_of_partial_differential_equations.tex @@ -8,7 +8,7 @@ \subsection{Finite difference schemes} \subsubsection{Introduction} \begin{definition} - A linear system of $n$ first order of pdes for $\vf{u}(t,x)$ is a system of the form: $$\vf{A}(t,x)\vf{u}_t+\vf{B}(t,x)\vf{u}_x=\vf{C}(t,x)\vf{u}+\vf{D}(t,x)$$ + A linear system of $n$ first order of PDEs for $\vf{u}(t,x)$ is a system of the form: $$\vf{A}(t,x)\vf{u}_t+\vf{B}(t,x)\vf{u}_x=\vf{C}(t,x)\vf{u}+\vf{D}(t,x)$$ for certain matrices $\vf{A},\vf{B},\vf{C}, \vf{D}\in \mathcal{M}_q(\RR)$. The system is called \emph{hyperbolic} if $\vf{A}^{-1}\vf{B}$ is diagonalizable. \end{definition} \begin{definition} @@ -70,7 +70,7 @@ \end{definition} \begin{definition} Let $(G_j)$ be a sequence of grids such that the time and space steps $k_j,h_j>0$ of each one satisfy $\displaystyle \lim_{j\to\infty}k_j=\lim_{j\to\infty}h_j=0$. - We say that a finite difference scheme $v$ approximating a pde with initial condition $u_0(x)$ is \emph{unconditionally convergent} if for any solution $u(x,t)$ to the pde we have: + We say that a finite difference scheme $v$ approximating a PDE with initial condition $u_0(x)$ is \emph{unconditionally convergent} if for any solution $u(x,t)$ to the PDE we have: \begin{itemize} \item For all $x\in\domain u_0$ and all increasing sequence $(m_j)\in\NN$ such that $(\cdot,x_{m_j})\in G_j$ and $\displaystyle\lim_{j\to\infty} x_{m_j}=x$, we have $\displaystyle\lim_{j\to\infty} v_{m_j}^0=u_0(x)$. \item For all $(t,x)\in\domain u$ and all increasing sequences $(m_j),(n_j)\in\NN$ such that $(t_{n_j},x_{m_j})\in G_j$ and $\displaystyle\lim_{j\to\infty} x_{m_j}=x$, $\displaystyle\lim_{j\to\infty} t_{n_j}=t$, we have $\displaystyle\lim_{j\to\infty} v_{m_j}^{n_j}=u(t,x)$. @@ -78,7 +78,7 @@ The scheme is \emph{conditionally convergent} if $\forall j\in\NN$, $(k_j,h_j)\in\Lambda$, for some stability region $\Lambda$. \end{definition} \begin{definition} - Let $P$ be a partial differential operator and $\vf{f}$ be a function. Given the pde $P\vf{u}=\vf{f}$ and a finite difference scheme $P_{k,h}\vf{v}=R_{k,h}\vf{f}$ with $R_{k,h}\vf{1}=\vf{1}$, we say that the scheme is \emph{consistent} with the pde if for any smooth function $\vf\phi(t,x)$ we have: $$\lim_{k,h\to 0}R_{k,h}P\vf\phi-P_{k,h}\vf\phi=\vf{0}$$ + Let $P$ be a partial differential operator and $\vf{f}$ be a function. Given the PDE $P\vf{u}=\vf{f}$ and a finite difference scheme $P_{k,h}\vf{v}=R_{k,h}\vf{f}$ with $R_{k,h}\vf{1}=\vf{1}$, we say that the scheme is \emph{consistent} with the PDE if for any smooth function $\vf\phi(t,x)$ we have: $$\lim_{k,h\to 0}R_{k,h}P\vf\phi-P_{k,h}\vf\phi=\vf{0}$$ where the convergence is pointwise at each point $(t,x)$ in the domain of solutions. We say that the consistency is of order $(p,q)$ in time and space if: $$\lim_{k,h\to 0}R_{k,h}P\vf\phi-P_{k,h}\vf\phi=\O{k^p}+\O{h^q}$$ The consistency is a \emph{conditional consistency} if the limit is for $(k,h)\in \Lambda$, for some stability region $\Lambda$. In that case, it makes sense to say that the consistency is of order $r$ in $k=\lambda(h)$ if: $$\lim_{h\to 0}R_{\lambda(h),h}P\vf\phi-P_{\lambda(h),h}\vf\phi=\O{h^r}$$ \end{definition} @@ -86,7 +86,7 @@ The Lax-Friedrichs scheme is consistent if and only if $\displaystyle\lim_{h,k\to 0}\frac{h^2}{k}=0$. \end{lemma} \begin{remark} - The consistency is not enough to guarantee convergence. For example, consider the pde $u_t+au_x=0$, with $a>0$. The forward-time forward-space scheme is consistent with the pde, but it is not convergent if we take the initial condition $u_0(x)=\indi{\{x<0\}}$ on the domain $[-1,1]$. Indeed, looking at \mcref{NIPDE:upwind} we see that from some instant of time, the solution will be $0$ everywhere, which cannot be possible. In that case we should use the forward-time backward-space scheme, which is convergent. The usage of this latter method in these cases is called the \emph{upwind condition}. + The consistency is not enough to guarantee convergence. For example, consider the PDE $u_t+au_x=0$, with $a>0$. The forward-time forward-space scheme is consistent with the PDE, but it is not convergent if we take the initial condition $u_0(x)=\indi{\{x<0\}}$ on the domain $[-1,1]$. Indeed, looking at \mcref{NIPDE:upwind} we see that from some instant of time, the solution will be $0$ everywhere, which cannot be possible. In that case we should use the forward-time backward-space scheme, which is convergent. The usage of this latter method in these cases is called the \emph{upwind condition}. \end{remark} \begin{figure}[H] \centering @@ -204,7 +204,7 @@ \end{align*} \end{proof} \begin{proposition} - Consider the pde of \mcref{NIPDE:traffic} with $\lambda =k/h=\const$ Then: + Consider the PDE of \mcref{NIPDE:traffic} with $\lambda =k/h=\const$ Then: \begin{itemize} \item The FTFS scheme is stable if and only if $a\lambda\in [-1,0]$. \item The FTBS scheme is stable if and only if $a\lambda\in [0,1]$. @@ -397,7 +397,7 @@ It can also be shown that the consistency and convergence imply stability. \end{remark} \begin{theorem} - Consider a scheme of $J$ steps for a 1st-order-in-time linear pde of constant coefficients whose amplification factor is $g$. Let $\Phi(\theta, g)$ be the \emph{amplification polynomial}, that is the polynomial that satisfies $g$ of degree $J-1$. Then, the scheme is stable if and only if: + Consider a scheme of $J$ steps for a 1st-order-in-time linear PDE of constant coefficients whose amplification factor is $g$. Let $\Phi(\theta, g)$ be the \emph{amplification polynomial}, that is the polynomial that satisfies $g$ of degree $J-1$. Then, the scheme is stable if and only if: \begin{itemize} \item for any root $g_j(\theta)$ of $\Phi$ we have $\abs{g_j(\theta)}\leq 1$- \item if $\exists \theta_0$ and $k$ such that $\abs{g_k(\theta_0)}=1$, then this root is simple. @@ -417,9 +417,9 @@ $$ If $\abs{a\lambda}<1$, then $\abs{g_{\pm}}^2=1$ and the two roots are simple $\forall \theta\in\RR$. If $\abs{a\lambda}>1$ and $\theta=\frac{\pi}{2}$, then either $\abs{g_+}>1$ or $\abs{g_-}>1$ and the scheme is unstable. Finally, if $\abs{a\lambda}=1$ and $\theta=\frac{\pi}{2}$, then the scheme is unstable because there is a double root. \end{proof} - \subsubsection{Second order pdes} + \subsubsection{Second order PDEs} \begin{definition} - Consider a second order pde of the form: + Consider a second order PDE of the form: \begin{equation} A u_{tt}+2Bu_{tx} +Cu_{xx}+Du_t+Eu_x+F u=G \end{equation} @@ -431,7 +431,7 @@ u_x(f(s),g(s))=\psi(s) \end{cases} $$ - which are tied to the \emph{compatibility condition} $h'=\phi f'+\psi g'$ that follows from the chain rule. The characteristic curves are the curves from which we cannot find the highest order derivatives of $u$ from the initial conditions and the pde. Differentiating $u_t(s)$ and $u_x(s)$ we get the system of equations for $u_{tt}$, $u_{tx}$ and $u_{xx}$: + which are tied to the \emph{compatibility condition} $h'=\phi f'+\psi g'$ that follows from the chain rule. The characteristic curves are the curves from which we cannot find the highest order derivatives of $u$ from the initial conditions and the PDE. Differentiating $u_t(s)$ and $u_x(s)$ we get the system of equations for $u_{tt}$, $u_{tx}$ and $u_{xx}$: \begin{equation*} \begin{cases} A u_{tt}+2Bu_{tx} +Cu_{xx}=G-D\phi-E\psi -Fh \\ @@ -443,13 +443,13 @@ $$ A{\left(\dv{x}{t}\right)}^2-2B\dv{x}{t}+C=0 $$ - The pde is called \emph{elliptic} if $AC-B^2>0$, \emph{hyperbolic} if $AC-B^2<0$ and \emph{parabolic} if $AC-B^2=0$. + The PDE is called \emph{elliptic} if $AC-B^2>0$, \emph{hyperbolic} if $AC-B^2<0$ and \emph{parabolic} if $AC-B^2=0$. \end{definition} \begin{definition} - Consider a finite difference scheme with $J$ steps for a 2n order homogeneous pde and $\Lambda$ be a stability region. We say that it is \emph{stable} is given $T>0$, there exists $C_T>0$ such that for any grid with $(k,h)\in \Lambda$ and for any initial values $\vf{v}_m^j$, $m\in\ZZ$, $j=0,\ldots,J-1$ we have $$\sum_{m\in\ZZ}\norm{\vf{v}_m^n}^2\leq (1+n^2)C_T\sum_{j=0}^{J-1}\sum_{m\in\ZZ}\norm{\vf{v}_m^j}^2$$ for all $n\in\NN$ such that $0\leq nk\leq T$. + Consider a finite difference scheme with $J$ steps for a 2n order homogeneous PDE and $\Lambda$ be a stability region. We say that it is \emph{stable} is given $T>0$, there exists $C_T>0$ such that for any grid with $(k,h)\in \Lambda$ and for any initial values $\vf{v}_m^j$, $m\in\ZZ$, $j=0,\ldots,J-1$ we have $$\sum_{m\in\ZZ}\norm{\vf{v}_m^n}^2\leq (1+n^2)C_T\sum_{j=0}^{J-1}\sum_{m\in\ZZ}\norm{\vf{v}_m^j}^2$$ for all $n\in\NN$ such that $0\leq nk\leq T$. \end{definition} \begin{theorem} - Consider a finite difference scheme with $J$ steps for a 2n order homogeneous pde whose amplification factor is $g$ and $\Phi(\theta, g)$ is the amplification polynomial. Then, the scheme is stable if and only if: + Consider a finite difference scheme with $J$ steps for a 2n order homogeneous PDE whose amplification factor is $g$ and $\Phi(\theta, g)$ is the amplification polynomial. Then, the scheme is stable if and only if: \begin{itemize} \item for any root $g_j(\theta)$ of $\Phi$ we have $\abs{g_j(\theta)}\leq 1$. \item if $\exists \theta_0$ and $k$ such that $\abs{g_k(\theta_0)}=1$ then this root is at most double. @@ -479,7 +479,7 @@ \end{proposition} \subsubsection{Elliptic equations} \begin{definition} - Let $Pu=f$ be an elliptic pde on $\Omega$. We define the following boundary conditions on $\Fr{\Omega}$: + Let $Pu=f$ be an elliptic PDE on $\Omega$. We define the following boundary conditions on $\Fr{\Omega}$: \begin{enumerate} \item \emph{Dirichlet}: $u=f$ \item \emph{Neumann}: $\pdv{u}{n}=g$ diff --git a/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex b/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex index ac07d72..db14b0b 100644 --- a/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex +++ b/Mathematics/4th/Partial_differential_equations/Partial_differential_equations.tex @@ -135,7 +135,7 @@ \end{definition} \begin{proposition}[Fermat's principle] \emph{Fermat's principle} states that the path taken by a ray between two given points $a$ and $b$ is the path that can be traveled in the least time. Mathematically, we want to minimize the functional: $$\mathcal{T}(\vf{x})=\int_a^b\frac{\abs{\dd{\vf{x}}}}{v(\vf{x})}$$ - So we shall solve the equation $\delta \mathcal{T}=0$, which is equivalent to solve: $$\delta\int_a^bn(\vf{x})\dd{s}=0$$ where $s$ is the arc-length parameter. From the Euler-Lagrange equations, we get the following ode: $$\dv{}{s}\left(n\dv{\vf{x}}{s}\right)=\grad{n}$$ + So we shall solve the equation $\delta \mathcal{T}=0$, which is equivalent to solve: $$\delta\int_a^bn(\vf{x})\dd{s}=0$$ where $s$ is the arc-length parameter. From the Euler-Lagrange equations, we get the followingODE: $$\dv{}{s}\left(n\dv{\vf{x}}{s}\right)=\grad{n}$$ \end{proposition} \begin{proposition}[Eikonal equation] The time $T(x)$ taken by the light to travel from a fixed point $x_0$ to $x$ in a medium of refractive index $n$ is given by: $${\norm{\grad{T}}}^2=n^2$$ @@ -148,7 +148,7 @@ The path taken by a physical system between times $t_1$ and $t_2$ and configurations $\vf{x}_1$ and $\vf{x}_2$ is the one for which the action is stationary (no change) to first order. Mathematically, $\delta \mathcal{S}=0$, where $\delta$ means a \emph{small change}. This value $S(\vf{x},t)$ of the action satisfies the \emph{Hamilton-Jacobi equation}: $$\pdv{S}{t}+\frac{1}{2m}{\norm{\grad S}}^2+V=0$$ \end{proposition} \begin{proposition}[Schrödinger equation] - The \emph{Schrödinger equation} is a pde that governs the \emph{wave function} $\Psi$, which describes the quantum state of an isolated quantum system, of a quantum-mechanical system. This is given by: $$\ii \hbar\pdv{\Psi}{t}=\left(-\frac{\hbar^2}{2m}\laplacian+V\right){\Psi}$$ where $m$ is the mass of the particle and $V$ is the potential in which the particle exists. Furthermore, $\abs{\Psi}^2$ is the probability density function of the position of the particle. + The \emph{Schrödinger equation} is a PDE that governs the \emph{wave function} $\Psi$, which describes the quantum state of an isolated quantum system, of a quantum-mechanical system. This is given by: $$\ii \hbar\pdv{\Psi}{t}=\left(-\frac{\hbar^2}{2m}\laplacian+V\right){\Psi}$$ where $m$ is the mass of the particle and $V$ is the potential in which the particle exists. Furthermore, $\abs{\Psi}^2$ is the probability density function of the position of the particle. \end{proposition} \begin{proposition} Substituting ${\Psi}=\sqrt{\rho}\exp{\ii \frac{S}{\hbar}}$ into the Schrödinger equation and taking the limit $\hbar\to 0$ in the resulting equation yield the Hamilton-Jacobi equation. Moreover, if we define $\vf{v}=\frac{\grad{S}}{m}$, from one real equation (from the original one complex equation) we get the continuous equation (\mcref{PDE:continuous}) and from the imaginary equation taking the limit $\hbar\to 0$ we get the Cauchy momentum equation (\mcref{PDE:cauchy}). @@ -527,7 +527,7 @@ \end{itemize} \end{theorem} \begin{sproof} - Suppose $v$ and $u$ are linearly independent and that $u>0$ and $v>0$ (if $v<0$, $-v>0$ an is also a solution of the same pde) in $(\alpha_1,\alpha_2)$. Then, multiplying the first equation by $-u$ and the second one by $\frac{u^2}{v}$, adding them and integrating we get: + Suppose $v$ and $u$ are linearly independent and that $u>0$ and $v>0$ (if $v<0$, $-v>0$ an is also a solution of the same PDE) in $(\alpha_1,\alpha_2)$. Then, multiplying the first equation by $-u$ and the second one by $\frac{u^2}{v}$, adding them and integrating we get: \begin{align*} 0 & =\int_{\alpha_1}^{\alpha_2}\left[-{(p_1u')}'u +{(p_2(x)v')}'\frac{u^2}{v} +(q_2 -q_1)u^2\right]\dd{x} \\ & =\int_{\alpha_1}^{\alpha_2}\left[p_1{u'}^2+p_2\frac{u^2{v'}^2-2uu'vv'}{v^2}+(q_2 -q_1)u^2\right]\dd{x} @@ -635,7 +635,7 @@ for certain constants $C_1, C_2\in\RR$. \end{proposition} \begin{sproof} - Observe that $u(x,t) = f(\frac{x}{\sqrt{t}})=:f(s)$ and the heat equation is transformed into $-f's=2\alpha f''$. The solution of this ode is straightforward. + Observe that $u(x,t) = f(\frac{x}{\sqrt{t}})=:f(s)$ and the heat equation is transformed into $-f's=2\alpha f''$. The solution of thisODE is straightforward. \end{sproof} \subsubsection{Distributions} \begin{definition} @@ -924,7 +924,7 @@ \subsection{Laplace equation} \subsubsection{General properties and solutions} \begin{definition}[Laplace equation] - Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{Laplace equation} is the pde defined by: $$\laplacian u=0$$ + Let $u:\RR\times\RR\rightarrow\RR$ be an unknown function. The \emph{Laplace equation} is the PDE defined by: $$\laplacian u=0$$ \end{definition} \begin{proposition}[Dirichlet problem in the disc] Let $f:[0,2\pi]\rightarrow\RR$ be a continuous function such that $f(0)=f(2\pi)$. Then, there exists a continuous function $v:\overline{D(0,\rho)}\rightarrow\RR$ that $v\in\mathcal{C}^2(D(0,\rho)\setminus\{0\})$ and such that: diff --git a/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex b/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex index 098c02f..35871f9 100644 --- a/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex +++ b/Mathematics/4th/Stochastic_processes/Stochastic_processes.tex @@ -1165,7 +1165,7 @@ Let ${(X_t)}_{t\geq 0}$ be a CTHMC. Then, ${(X_t)}_{t\geq 0}$ is said to be \emph{stable} if $\forall i\in I$, $q_i<\infty$, and is said to be \emph{conservative} if $\forall i\in I$, $q_i=\sum_{\substack{k\in I\\k\ne i}}q_{ik}$. \end{definition} \begin{theorem} - Let ${(X_t)}_{t\geq 0}$ be a CTHMC and a regular jump process. Then, the two Kolmogorov odes are satisfied. + Let ${(X_t)}_{t\geq 0}$ be a CTHMC and a regular jump process. Then, the two KolmogorovODEs are satisfied. \end{theorem} \subsubsection{Limit and stationary distributions} \begin{definition} diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex index fc73245..ba1b196 100644 --- a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex +++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex @@ -4,24 +4,24 @@ \changecolor{ADS} \begin{multicols}{2}[\section{Advanced dynamical sytems}] \subsection{Discrete maps} - \subsubsection{Maps in \texorpdfstring{$\S^1$}{S1}} + \subsubsection{Maps in \texorpdfstring{$\TT^1$}{S1}} \begin{proposition} - Let $\alpha=\frac{p}{q}\in\QQ$ and let $R_\alpha:\S^1\to \S^1$ be the rotation of angle $\alpha$. Then, all the points of $\S^1$ are periodic for $R_\alpha$ with period $q$. + Let $\alpha=\frac{p}{q}\in\QQ$ and let $R_\alpha:\TT^1\to \TT^1$ be the rotation of angle $\alpha$. Then, all the points of $\TT^1$ are periodic for $R_\alpha$ with period $q$. \end{proposition} \begin{proof} - We identify the elements of $\S^1$ as $\quot{\RR}{\ZZ}$. Let $x\in \S^1$. Then, ${R_\alpha}^q x=x+\alpha q=x+p=x$. And $q$ is the smallest integer such that ${R_\alpha}^q x=x$ because we assume that $p$ and $q$ are coprime. + We identify the elements of $\TT^1$ as $\quot{\RR}{\ZZ}$. Let $x\in \TT^1$. Then, ${R_\alpha}^q x=x+\alpha q=x+p=x$. And $q$ is the smallest integer such that ${R_\alpha}^q x=x$ because we assume that $p$ and $q$ are coprime. \end{proof} \begin{proposition} - Let $\alpha\in\RR\setminus\QQ$ and let $R_\alpha:\S^1\to \S^1$ be the rotation of angle $\alpha$. Then, all the points of $\S^1$ are dense in $\S^1$. + Let $\alpha\in\RR\setminus\QQ$ and let $R_\alpha:\TT^1\to \TT^1$ be the rotation of angle $\alpha$. Then, all the points of $\TT^1$ are dense in $\TT^1$. \end{proposition} \begin{proof} - Let $\varepsilon>0$, $x,y\in \S^1$. Discretize $\S^1$ in intervals of length at most $\frac{1}{\varepsilon}$. Then, $\exists m,n\in \NN$ with $m< n\leq \frac{1}{\varepsilon}+1$ such that ${R_\alpha}^m x$ and ${R_\alpha}^nx$ are in the same interval. Thus, $\abs{{R_\alpha}^{n-m}x-x}<\varepsilon$. Now, concatenating ${R_\alpha}^{n-m}x$ repeatedly, we will eventually have $\abs{{R_\alpha}^{k(n-m)}x - y}<\varepsilon$ for some $k\in \NN$. + Let $\varepsilon>0$, $x,y\in \TT^1$. Discretize $\TT^1$ in intervals of length at most $\frac{1}{\varepsilon}$. Then, $\exists m,n\in \NN$ with $m< n\leq \frac{1}{\varepsilon}+1$ such that ${R_\alpha}^m x$ and ${R_\alpha}^nx$ are in the same interval. Thus, $\abs{{R_\alpha}^{n-m}x-x}<\varepsilon$. Now, concatenating ${R_\alpha}^{n-m}x$ repeatedly, we will eventually have $\abs{{R_\alpha}^{k(n-m)}x - y}<\varepsilon$ for some $k\in \NN$. \end{proof} \begin{corollary} - Let $\alpha\in\RR\setminus\QQ$ and $A\subset \S^1$ be a non-empty closed invariant set for $R_\alpha$. Then, $A=\S^1$. + Let $\alpha\in\RR\setminus\QQ$ and $A\subset \TT^1$ be a non-empty closed invariant set for $R_\alpha$. Then, $A=\TT^1$. \end{corollary} \begin{proof} - Let $x\in \S^1$ and $y\in A$. Then, $\forall k\in\NN$ $\exists n_k\in\NN$ such that $R_\alpha^{n_k}y\in(x-\frac{1}{k},x+\frac{1}{k})$. Thus, $R_\alpha^{n_k}y\to x$ and $x\in A$ because $A$ is closed and $R_\alpha^{n_k}y\in A$ $\forall k\in\NN$. + Let $x\in \TT^1$ and $y\in A$. Then, $\forall k\in\NN$ $\exists n_k\in\NN$ such that $R_\alpha^{n_k}y\in(x-\frac{1}{k},x+\frac{1}{k})$. Thus, $R_\alpha^{n_k}y\to x$ and $x\in A$ because $A$ is closed and $R_\alpha^{n_k}y\in A$ $\forall k\in\NN$. \end{proof} \begin{definition} Consider the set $$\Sigma_m @@ -37,9 +37,9 @@ \begin{proposition} Let $m\in\NN$. Consider the \emph{expansion map} $$ - \function{E_m}{\S^1}{\S^1}{x}{mx} + \function{E_m}{\TT^1}{\TT^1}{x}{mx} $$ - Then, if $\phi:\Sigma_m\to \S^1$ is the map $\phi(x_1,x_2,\ldots)=\sum_{i=1}^\infty \frac{x_i}{m^i}$, we have that $E_m\circ \phi=\phi\circ \sigma_m$. In particular, $\phi$ is a bijection, and thus it is a conjugacy between $E_m$ and $\sigma_m$. + Then, if $\phi:\Sigma_m\to \TT^1$ is the map $\phi(x_1,x_2,\ldots)=\sum_{i=1}^\infty \frac{x_i}{m^i}$, we have that $E_m\circ \phi=\phi\circ \sigma_m$. In particular, $\phi$ is a bijection, and thus it is a conjugacy between $E_m$ and $\sigma_m$. \end{proposition} \begin{proof} Let $x=(x_1,x_2,\ldots)\in \Sigma_m$. Then, $\phi\circ \sigma_m(x)=\sum_{i=1}^\infty \frac{x_{i+1}}{m^i}$. Moreover: @@ -48,7 +48,7 @@ \end{multline*} \end{proof} \begin{remark} - Note that $E$ preserves the Lebesgue measure \textit{backwards}: $\abs{{E_m}^{-1}(A)}=\abs{A}$ for all $A\subseteq \S^1$, but $\abs{E_m(A)}\ne \abs{A}$ in general. + Note that $E$ preserves the Lebesgue measure \textit{backwards}: $\abs{{E_m}^{-1}(A)}=\abs{A}$ for all $A\subseteq \TT^1$, but $\abs{E_m(A)}\ne \abs{A}$ in general. \end{remark} \begin{definition} We define the following distance in $\Sigma_m$. For all $x,x'\in\Sigma_m$: @@ -57,7 +57,7 @@ $$ \end{definition} \begin{proposition} - Periodic points of $E_m$ are dense in $\S^1$. + Periodic points of $E_m$ are dense in $\TT^1$. \end{proposition} \begin{proof} By conjugacy it suffices to show that periodic points of $\sigma_m$ are dense in $\Sigma_m$. Let $x\in \Sigma_m$ and $\varepsilon>0$. Then, $\varepsilon>\frac{1}{2^\ell}$ for some $\ell$. And so the orbit of @@ -67,7 +67,7 @@ is periodic and $d(x,y)<\varepsilon$. So periodic points of $\sigma_m$ are dense in $\Sigma_m$. \end{proof} \begin{proposition} - Le $x\in \S^1$. Then, the positive orbit of $x$ for $E_m$ is dense in $\S^1$. + Le $x\in \TT^1$. Then, the positive orbit of $x$ for $E_m$ is dense in $\TT^1$. \end{proposition} \begin{proof} By conjugacy, we only prove it for $\sigma_m$. But this is clear by taking: @@ -196,8 +196,7 @@ \begin{remark} The Lyapunov exponent measures the exponential growth rate of tangent vectors along orbits. It can rarely be computed explicitly, but if we can show that $\chi(x,\vf{v})>0$ for some $\vf{v}$, then we know that the system is \emph{chaotic}. \end{remark} - \subsection{Hamiltonian systems} - \subsubsection{Introduction} + \subsubsection{Hamiltonian systems} \begin{definition} Let $U\subseteq \RR^n\times \RR^n$ be open and $H:U\rightarrow \RR$ be a $\mathcal{C}^1$ function. We define the \emph{Hamiltonian vector field} associated to $H$ as: \begin{equation}\label{ADS:ham_system} @@ -221,5 +220,122 @@ \end{multline*} where $\vf{X}_H$ is the vector field of \mcref{ADS:ham_system}, and we used that the derivative of the determinant map is the trace. But an easy computation shows that $\div \vf{X}_H=0$. \end{proof} + \subsection{Circle dynamics} + \subsubsection{Generalities} + \begin{definition} + Let $x,x'\in\RR$. We say that $x\sim x'$ if and only if $x-x'\in\ZZ$. We define the \emph{circle} as $\TT^1:=\quot{\RR}{\sim}$. We define the following distance in $\TT^1$: + $$ + d(\overline{x},\overline{y})=\min_{x'\in\overline{x},y'\in\overline{y}}\abs{x'-y'} + $$ + \end{definition} + \begin{proposition}[Existence of a lift]\hfill + \begin{enumerate} + \item For any continuous map $F:\TT^1\to \TT^1$ there exists a \emph{lift} $f$, i.e.\ a continuous map $f:\RR\to \RR$ such that $F\circ \pi=\pi\circ f$, where $\pi:\RR\to \TT^1$ is the canonical projection. + \item If $g$ is another lift of $F$, then $g-f=k\in\ZZ$. + \end{enumerate} + \end{proposition} + \begin{proof} + We only prove the second property. Since, $f$, $g$ are both lifts of $F$, they belong to the same equivalence class. Thus, $f-g\in\ZZ$. And now use the continuity of $f-g$. + \end{proof} + \begin{remark} + Recall that $f:\RR\to\RR$ is a homeomorphism if and only if $f$ is monotone. + \end{remark} + \begin{definition} + We say that a homeomorphism $F$ \emph{preserves orientation} if and only if $f$ is strictly increasing. We define the set of $\Homeo(\TT^1)$ as the set of homeomorphisms of $\TT^1$ that preserve orientation. + \end{definition} + \begin{proposition} + Let $F\in\Homeo(\TT^1)$. Then, $F$ admits a lift $f$ such that $f(x)=x+\varphi(x)$, where $\varphi:\RR\to\RR$ is a 1-periodic function. + \end{proposition} + \begin{proof} + We already now that $F$ admits a lift $f$. A straightforward calculation shows that $f_1:\RR\to\RR$ defined by $f_1(x)=f(x+1)$ is also a lift of $F$. Thus, $f_1-f=k\in \ZZ$. Now, since $f$ must be strictly increasing, we need $k\in \NN$. Moreover, since $F$ is injective, $f|_{[0,1)}$ is injective and its image cannot contain 2 points whose difference is an integer. Thus, $k=1$. Now, define $\varphi(x)=f(x)-x$, which is 1-periodic: + $$ + \varphi(x+1)=f(x+1)-(x+1)=f(x)-x=\varphi(x) + $$ + \end{proof} + \begin{definition} + We define the set: + \begin{multline*} + D^0(\TT^1):\{f:\RR\to\RR:f\text{ increasing and}\\ + \text{ homeomorphism}, f(x+1)=f(x)+1\} + \end{multline*} + Note that we have the projection: + $$ + \function{}{D^0(\TT^1)}{\Homeo(\TT^1)}{f}{F} + $$ + We can define a distance in $D^0(\TT^1)$ as: + $$ + d(f,g)=\max\{ \sup_{x\in\RR}\abs{f(x)-g(x)},\sup_{x\in\RR}\abs{f^{-1}(x)-g^{-1}(x)}\} + $$ + \end{definition} + \begin{lemma} + $D^0(\TT^1)$ is a complete metric space. Moreover: + \begin{enumerate} + \item $f\to f^{-1}$ is continuous, $f\in D^0(\TT^1)$. + \item $(f,g)\to f\circ g$ is continuous, $(f,g)\in D^0(\TT^1)\times D^0(\TT^1)$. + \end{enumerate} + Thus, $D^0(\TT^1)$ is a topological group with the composition. + \end{lemma} + \begin{definition} + Let $\varepsilon\geq 0$ and $\alpha\in\RR$. We define the \emph{Arnold family} as: + $$ + \function{f_{\alpha,\varepsilon}}{\RR}{\RR}{x}{x+\alpha+\varepsilon\sin(2\pi x)} + $$ + \end{definition} + \begin{lemma} + If $0\leq \varepsilon<\frac{1}{2\pi}$, then $f_{\alpha,\varepsilon}\in D^0(\TT^1)$. + \end{lemma} + \begin{proof} + Note that ${f_{\alpha,\varepsilon}}'>0\iff \varepsilon<\frac{1}{2\pi}$. Thus, $f_{\alpha,\varepsilon}$ is strictly increasing, and therefore it is a homeomorphism. + \end{proof} + \subsubsection{Rotation number} + \begin{remark} + Recall that $f=\id+\varphi$ with $\varphi$ 1-periodic. And thus: + $$ + f^n=\id + \sum_{i=0}^{n-1} \varphi\circ f^i=: \id + \varphi_n + $$ + with $\varphi_n$ 1-periodic. + \end{remark} + \begin{lemma}\label{ADS:lema1} + Let $f\in D^0(\TT^1)$ be such that $f=\id +\varphi$, with $\varphi$ 1-periodic. Let $m:=\min_{x\in\RR}\varphi$ and $M:=\max_{x\in\RR}\varphi$. Then, we have $m\leq M< m+1$. + \end{lemma} + \begin{proof} + Let $0\leq x\leq y<1\leq x+1$. Then, $f(y) \liminf_{n\to\infty}\norm{x_n-x_*} + $$ + \end{lemma} + \begin{theorem}[Krasnoselskii-Mann's convergence theorem] + Let $H$ be Hilbert, $T:H\to H$ be nonexpansive. Let $x_0\in H$, $0<\theta<1$ and $x_{k}:=T_\theta^kx_0$, where $T_\theta=(1-\theta)\id+\theta T$. Define + $$ + F:=\{x\in H:Tx=x\}\ne \varnothing + $$ + the set of fixed points of $T$. Then, $\exists x\in F$ such that $x_k\rightharpoonup x_0$. + \end{theorem} + \subsubsection{Extensions} + \begin{theorem} + Let $H$ Hilbert and $(x_k)\in H$ be \textit{inexact} iteration of $T_\theta$: + $$ + \norm{x_{k+1}-T_\theta x_k}\leq \varepsilon_k + $$ + Assume $\sum_{k=0}^\infty \varepsilon_k<\infty$. Then, $\exists x\in F$ such that $x_k\rightharpoonup x$. + \end{theorem} \end{multicols} \end{document} \ No newline at end of file diff --git a/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex b/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex index 3659fd6..60069cd 100644 --- a/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex +++ b/Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex @@ -306,7 +306,7 @@ \end{proposition} \subsubsection{Euler scheme} \begin{definition} - Consider the SDE of \mcref{MM:SDE} and let $h$ be a discretization step. The Euler method consists in: + Consider the SDE of \mcref{MM:SDE} and let $h$ be a discretization step. The \emph{Euler method} consists in: \begin{align*} \vf{X}_{t+h} & =\vf{X}_t+\int_t^{t+h}\vf{b}(\vf{X}_s)\dd{s}+\int_t^{t+h}\vf{\sigma}(\vf{X}_s)\dd{\vf{B}_s} \\ & \approx \vf{X}_t+h\vf{b}(\vf{X}_t)+\vf{\sigma}(\vf{X}_t)(\vf{B}_{t+h}-\vf{B}_t) @@ -319,5 +319,14 @@ \begin{remark} Note that Euler scheme reduces to generating independent increments $\vf{B}_{t_{i+1}}-\vf{B}_{t_i}\sim \sqrt{t_{i+1}-t_i}N_d(0,\vf{I}_d)$. \end{remark} + \begin{remark} + Trying to build an implicit Euler scheme for SDEs is much more complicated than for ODEs, as we need to ensure that the process is still adapted. + \end{remark} + \begin{theorem} + Let $\vf{X}$ be the solution to \mcref{MM:SDE} and $\vf{\tilde{X}}$ be the solution to the Euler scheme. Then, for $p\geq 1$: + $$ + \Exp\left(\sup_{0\leq t\leq T}\norm{\vf{X}_t-\vf{\tilde{X}}_t}_p\right)\leq C_p h + $$ + \end{theorem} \end{multicols} \end{document} \ No newline at end of file diff --git a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex index 5c8a5a8..d072bba 100644 --- a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex +++ b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex @@ -41,7 +41,7 @@ Recall that for these problems to have a unique solution, we need to impose the coercivity and continuity in \mnameref{RFA:laxmilgram}. \end{remark} \begin{proposition} - Consider the homogeneous Dirichlet problem from \mcref{NMPDE:elliptic_pde} and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the pde. Then, the homogeneous Dirichlet problem has a unique solution $u$ in $H_0^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq 0$. In this case, $\exists C>0$ such that: + Consider the homogeneous Dirichlet problem from \mcref{NMPDE:elliptic_pde} and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the PDE. Then, the homogeneous Dirichlet problem has a unique solution $u$ in $H_0^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq 0$. In this case, $\exists C>0$ such that: $$ \norm{u}_{H^1(\Omega)}\leq C\norm{f}_{L^2(\Omega)} $$ @@ -51,13 +51,13 @@ $$ \end{proposition} \begin{proposition} - Consider the Neumann problem from \mcref{NMPDE:elliptic_pde} for $g\in L^2(\Fr\Omega)$ and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the pde. Then, the Neumann problem has a unique solution $u$ in $H^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq\delta> 0$. In this case, $\exists C>0$ such that: + Consider the Neumann problem from \mcref{NMPDE:elliptic_pde} for $g\in L^2(\Fr\Omega)$ and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the PDE. Then, the Neumann problem has a unique solution $u$ in $H^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq\delta> 0$. In this case, $\exists C>0$ such that: $$ \norm{u}_{H^1(\Omega)}\leq C(\norm{f}_{L^2(\Omega)}+\norm{g}_{L^2(\Fr\Omega)}) $$ \end{proposition} \begin{proposition} - Consider the Robin problem from \mcref{NMPDE:elliptic_pde} for $g\in L^2(\Fr\Omega)$ and $\gamma\in L^\infty(\Fr\Omega)$ and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the pde. Then, the Robin problem has a unique solution $u$ in $H^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq\delta\geq 0$ and $\gamma\geq \eta\geq 0$ with either $\delta>0$ or $\eta>0$. In this case, $\exists C>0$ such that: + Consider the Robin problem from \mcref{NMPDE:elliptic_pde} for $g\in L^2(\Fr\Omega)$ and $\gamma\in L^\infty(\Fr\Omega)$ and set $\beta=\alpha^{-1}\sum_{i=1}^n{\norm{b_i}_{L^\infty(\Omega)}}^2$, where $\alpha$ is the ellipticity constant of the PDE. Then, the Robin problem has a unique solution $u$ in $H^1(\Omega)$ if $\forall x\in\Omega$ we have $c-\frac{\beta}{2}\geq\delta\geq 0$ and $\gamma\geq \eta\geq 0$ with either $\delta>0$ or $\eta>0$. In this case, $\exists C>0$ such that: $$ \norm{u}_{H^1(\Omega)}\leq C(\norm{f}_{L^2(\Omega)}+\norm{g}_{L^2(\Fr\Omega)}) $$ diff --git a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex index 7fe5d50..154d02d 100644 --- a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex +++ b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex @@ -959,7 +959,7 @@ X_0^x=x \end{cases} \end{equation} - where $b,\sigma:\RR\to\RR$ are Lipschitz functions. Now, fix $f\in L^\infty(\RR)$ and consider the pde: + where $b,\sigma:\RR\to\RR$ are Lipschitz functions. Now, fix $f\in L^\infty(\RR)$ and consider the PDE: \begin{equation}\label{SC:pde_sde} \begin{cases} \pdv{v}{t}(t,x)=b(x)\pdv{v}{x}(t,x)+\frac{1}{2}\sigma^2(x)\pdv[2]{v}{x}(t,x) \\ @@ -969,7 +969,7 @@ where $v\in \mathcal{C}^{1,2}([0,\infty)\times\RR)$. \begin{theorem}\hfill \begin{enumerate} - \item If $v$ is a bounded solution to the pde of \mcref{SC:pde_sde}, then we must have $\forall (t,x)\in [0,\infty)\times\RR$: + \item If $v$ is a bounded solution to the PDE of \mcref{SC:pde_sde}, then we must have $\forall (t,x)\in [0,\infty)\times\RR$: \begin{equation}\label{SC:sol_v} v(t,x)=\Exp(f(X_t^x)) \end{equation} @@ -980,7 +980,7 @@ The interest of this connection between SDEs and PDEs is two-fold: on the one hand, one can use tools from PDE theory to understand the distribution of $X_t^x$. Conversely, the probabilistic representation of \mcref{SC:sol_v} offers a practical way to numerically solve the PDE of \mcref{SC:pde_sde}, by simulation. \end{remark} \begin{theorem}[Feynman-Kac's formula] - Let $v\in \mathcal{C}^{1,2}([0,\infty)\times\RR)$ be a bounded solution to the pde + Let $v\in \mathcal{C}^{1,2}([0,\infty)\times\RR)$ be a bounded solution to the PDE $$ \begin{cases} \pdv{v}{t}(t,x)=-h(x)v(t,x)+b(x) \pdv{v}{x}(t,x)+\frac{1}{2}\sigma^2(x)\pdv[2]{v}{x}(t,x) \\ diff --git a/Physics/Basic/Classical_mechanics/Classical_mechanics.tex b/Physics/Basic/Classical_mechanics/Classical_mechanics.tex index 851daee..1c0b3d9 100644 --- a/Physics/Basic/Classical_mechanics/Classical_mechanics.tex +++ b/Physics/Basic/Classical_mechanics/Classical_mechanics.tex @@ -300,7 +300,7 @@ \label{CM_RLC-genS} \end{minipage} \end{center} - The ode of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{R}{L\omega-\frac{1}{C\omega}}\right)$. + TheODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{R}{L\omega-\frac{1}{C\omega}}\right)$. And finally: \begin{equation}\label{CM_Vout} V_\text{out}=RI=-\frac{R}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\epsilon_0\sin(\omega t-\delta) @@ -321,7 +321,7 @@ % \label{CM_RLC-genP} % \end{minipage} % \end{center} - % The ode of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+\frac{1}{R^2}}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{1/R}{L\omega-\frac{1}{C\omega}}\right)$. + % TheODE of the system for the charge $q(t)$ is: $$\ddot{q}+\frac{R}{L}\dot{q}+\frac{q}{LC}=\frac{V_\text{in}}{L}$$ Thus, in steady-state part we have: $$q(t)=\frac{\epsilon_0/\omega}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+\frac{1}{R^2}}}\cos(\omega t-\delta)$$ where $\delta=-\arctan\left(\frac{1/R}{L\omega-\frac{1}{C\omega}}\right)$. % And finally: % \begin{equation}\label{CM_Vout} % V_\text{out}=RI=-\frac{R}{\sqrt{{\left(L\omega-\frac{1}{C\omega}\right)}^2+R^2}}\epsilon_0\sin(\omega t-\delta) @@ -341,11 +341,11 @@ f_0 & \text{if }0\leq t\leq \Delta t \\ 0 & \text{if }t>\Delta t \end{cases}$$ - That is, $f$ is a piecewise function\footnote{We shall suppose that the system was at equilibrium for $t<0$.}. Moreover, assuming that $x(0)=\dot{x}(0)=0$, the general solution to this ode when $0\Delta t$, then the general solution to the ode is: + If $t>\Delta t$, then the general solution to theODE is: $$x(t)=\exp{-\beta t}\left(k_1\cos(\tilde{\omega}t)+k_2\sin(\tilde{\omega}t)\right)$$ where $k_1=\text{O}({\Delta t}^2)$, $k_2=\frac{f_0\Delta t}{\tilde{\omega}}+\text{O}({\Delta t}^2)$ Therefore, $\forall t>\Delta t$: $$x(t)=f_0\Delta t\frac{\exp{-\beta t}}{\tilde{\omega}}\sin(\tilde{\omega}t)+\text{O}({\Delta t}^2)$$ \end{proposition} @@ -391,7 +391,7 @@ \end{theorem} \subsubsection{Non linear oscillations} \begin{definition} - Consider a pendulum whose rod (of length $L$) is in a non-small angle $\theta_0$ at initial time. The ode that satisfies $\theta(t)$ is: + Consider a pendulum whose rod (of length $L$) is in a non-small angle $\theta_0$ at initial time. TheODE that satisfies $\theta(t)$ is: $$\ddot{\theta}+\frac{g}{L}\sin\theta=0$$ Then, the period of the pendulum does depend on $\theta_0$. Indeed: \begin{multline*} diff --git a/preamble_formulas.sty b/preamble_formulas.sty index 76afd53..6bb2e41 100644 --- a/preamble_formulas.sty +++ b/preamble_formulas.sty @@ -300,7 +300,7 @@ \newcommand{\topo}{\tau} % symbol for the topology. Feasible options are: \tau, \mathcal{T}... \newcommand{\conn}{\mathrel{\#}} % connected sum. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). \renewcommand{\S}{S} % S of the S ^ n (n-th dimensional sphere) - +\newcommand{\Homeo}{\mathrm{Homeo}^+} %%% GALOIS THEORY \newcommand{\FF}{\ensuremath{\mathbb{F}}} % finite fields