commit 0e11e431821801f58da57acb07d47a7241c78978
parent dcabfda0a4c4fb548221e8061b759a7d7f685a6c
Author: miksa234 <milutin@popovic.xyz>
Date: Sun, 27 Feb 2022 09:41:35 +0100
appl_ana finished
Diffstat:
44 files changed, 5499 insertions(+), 139 deletions(-)
diff --git a/appl_ana/assingments/Exercises5.pdf b/appl_ana/assingments/Exercises5.pdf
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diff --git a/appl_ana/sesh1/main.tex b/appl_ana/sesh1/main.tex
@@ -46,7 +46,9 @@
\pagestyle{myheadings}
-\markright{Popovic\hfill Applied Analysis\hfill}
+\newcommand{\eps}{\varepsilon}
+
+\markright{Popović\hfill Applied Analysis\hfill}
\title{University of Vienna\\ Faculty of Mathematics\\
@@ -62,14 +64,13 @@
\subsection{Fall from high}
We consider a free fall ($\dot{x}(t=0)=0$) of an object with mass $20\
-\text{kg}$ from a height
-$x(0) = h = 20\; \text{km}$, such that the gravitational force depends on the hight $x(t)$ in
-the following way
+\text{kg}$ from a height $x(0) = h = 20\; \text{km}$, such that the
+gravitational force depends on the height $x(t)$ in the following way
\begin{align}\label{eq: free fall}
- \ddot{x} = -g\frac{R^2}{(x(t) + R)^2},
+ \ddot{x}(t) = -g\frac{R^2}{(x(t) + R)^2},
\end{align}
where $R$ is the radius of the earth $R \approx 6000\; \text{km}$ and $g
-\approx 9.91\ \frac{m}{s^2}$ is the gravitational acceleration on the surface
+\approx 9.81\ \frac{m}{s^2}$ is the gravitational acceleration on the surface
of the earth. For this problem there are two possible non-dimensionalisations,
but first let us rewrite the variables in terms of non-dimensional variables
and some dimensional constants, a priori let
@@ -105,13 +106,13 @@ The first scaling is done by reducing $\Pi_1$ and $\Pi_3$ to 1, by setting
\begin{align}
x_c = R, \;\;\;\; t_c = \sqrt{\frac{R}{g}},
\end{align}
-refolmulating the initial problem in equation \ref{eq: free fall} to
+reformulating the initial problem in equation \ref{eq: free fall} to
\begin{align}
&\ddot{\xi} = -\frac{1}{(\xi + 1)^2},\;\;\;\;
\text{with} \nonumber\\
&\xi(0) = \frac{h}{R}, \;\;\;\; \dot{\xi}(0) = 0.
\end{align}
-Reducing the problem, meaining letting $R \rightarrow \infty$ makes the first
+Reducing the problem, meaning if $\frac{h}{R} \rightarrow 0$ makes the first
initial condition $\xi(0) \rightarrow 0$. We can conclude that this scaling
is bad since it changes the initial condition in the reduced problem.
@@ -119,7 +120,7 @@ The second scaling option reduces $\Pi_2$ and $\Pi_3$ to 1, by setting
\begin{align}
x_c = h, \;\;\;\; t_c = \sqrt{\frac{h}{g}},
\end{align}
-refolmulating the initial problem in equation \ref{eq: free fall} to
+reformulating the initial problem in equation \ref{eq: free fall} to
\begin{align}
&\ddot{\xi} = -\frac{1}{(\frac{h}{R}\xi + 1)^2},\;\;\;\;
\text{with} \nonumber\\
@@ -129,14 +130,21 @@ By letting $R \rightarrow \infty$ we get the following reduced problem
\begin{align}\label{eq: free fall reduced}
\ddot{\xi} = -1.
\end{align}
-Integrating and solving for $\xi(\tau = \frac{T}{t_c}) = 0$ for when the
-object hits the ground we get a familliar solution
+Integrating and solving for $\xi(\tau = T\sqrt{\frac{g}{h}}) = 0$ for when the
+object hits the ground we get a familiar solution
\begin{align}
T = \sqrt{\frac{2h}{g}}
\end{align}
-Now in the reduced problem the time untill the object hits the ground is
-shorter since the acceleration is constant, but in the original one the
-acceleration increases as the object comes closer to earth.
+Note that in the reduced problem the time until the object hits the ground is
+\textbf{(much) shorter} since the acceleration is at its maximum $\ddot{x}(t) =
+g$ for all $t$. Yet in the original problem the acceleration (gravitation
+force) \textbf{increases} as the object comes \textbf{closer} to earth . For
+instance, if we let an object fall down from the height $h = R$ then its
+gravitational force (acceleration) at that height would be $\ddot{x}(0) =
+g/2$ and upon landing on earth the gravitational force $\ddot{x}(T) = g$,
+while in the reduced solution its gravitational force would be $\ddot{x}(t) =
+g$ for all $t$.
+
Additionally we can calculate the velocity at impact we need to integrate the
reduced problem \ref{eq: free fall reduced} once and put in the initial
condition
@@ -144,11 +152,60 @@ condition
\dot{\xi}(\tau = \frac{T}{t_c}) &= -\tau = -\sqrt{2} \\
\text{and} \;\;\; \dot{x} &= \frac{x_c}{t_c}\dot{\xi} =
\sqrt{gh}\; \dot{\xi}\\
- \Rightarrow \dot{x}(T) &= -\sqrt{2gh}
+ \Rightarrow \dot{x}(T) &= -\sqrt{2gh},
+\end{align}
+The result is exactly the same as we would get from energy conservation
+\begin{align}
+ \frac{m}{2}\dot{x}^2 = mgh \quad \Rightarrow \quad \dot{x} = \sqrt{2gh}.
+\end{align}
+The vertical throw allows for an additional scaling because the
+initial conditions are different, $x(0) = 0$ and $\dot{x}(0) = v$. Thus
+the solution too.
+
+To summarize, the assumptions that used for modeling and simplifying the
+equation are
+\begin{itemize}
+ \item no relativistic influence,
+ \item closed system, no outside influence (gravitation of the sun, air
+ resistance),
+ \item spherical symmetry of the earth (thereby center of mass can be
+ set in the middle of earth).
+\end{itemize}
+By looking at our assumptions a question arises:\textbf{Is it a good
+approximation to replace the attractive force of the earth by the attraction
+of the whole mass concentrated at the center?}.
+
+To answer this question more or less simply we look at the Poisson's equation
+for gravity,
+\begin{align}
+ \ddot{\vec{x}}(\vec{r}) = -\nabla \phi(\vec{r}) \\
+ \Delta \phi = 4\pi G\varrho(\vec{r}).
+\end{align}
+for a gravitational potential $\phi$ and the mass density of earth
+$\varrho$. We assume that \textbf{the earth can be approximated by a sphere}
+and then we integrate both sides along the sphere (and use the Gaussian law
+for integration)
+\begin{align}
+ \int_{S} \nabla \ddot{\vec{x}}\ dS =
+ \int_{\partial S}\ddot{\vec{x}}\ d\vec{s} = -4\pi
+ G \int_S\varrho(\vec{r})\ ds = -4\pi GM.
+\end{align}
+Obviously $\ddot{\vec{x}}$ and $d\vec{s}$ point in the same direction. We
+choose (rotate) the coordinate system such hat $\ddot{\vec{x}} =
+\ddot{x}\ \mathbf{\hat{n}}$ and $d\vec{s} = \mathbf{\hat{n}}\ ds$, thereby
+we get
+\begin{align}
+ &\ddot{x}\int_{\partial S} ds = 4\pi r^2 \ddot{x},\\
+ \Rightarrow &\ddot{x} = -\frac{GM}{r^2}.
+\end{align}
+The further derivation to get the exact equation of motion as in \ref{eq:
+free fall}, we have to keep in mind that $r = x + R$, because by our
+assumptions we are not in the sphere only outside or on the border $R$.
+Lastly by reformulating the constants $gR^2 = GM$ gets us to our equation of
+motion
+\begin{align}
+ \ddot{x}(t) = -g\frac{R^2}{(x(t) + R)^2}.
\end{align}
-The vectical throw allows for different scaling because the initial
-conditions are different, and thus the solution too $x(0) = 0$ and
-$\dot{x}(0) = v$.
\subsection{Scaling The Van der Pol equation}
The Van der Pol equation is a perturbation of the oscillation equation
@@ -190,74 +247,61 @@ non-dimensionalisations. Let us begin by setting
\end{align}
where $I_c$ and $t_c$ have the dimension of $I(t)$ and $t$ accordingly and
$\psi(\tau)$ and $\tau$ are dimensionless
-The first and second derivative in time is
+The \textbf{first} and second derivative in time is
\begin{align}
\frac{d}{dt} &= \frac{1}{t_c}\frac{d}{d\tau}\\
\frac{d^2}{dt^2} &= \frac{1}{t_c^2}\frac{d^2}{d\tau^2}.
\end{align}
We can rewrite the Van-Der-Pol equation in terms of $\psi$ and $\tau$
\begin{align}
- &\frac{LC}{t_c^2}\ddot{\psi} - \frac{g_1C}{t_c}\dot{\psi}
- \frac{3g_3CI_c}{t_c}\dot{\psi}\psi = -\psi\\
+ &\frac{LC}{t_c^2}\ddot{\psi} - \left(\frac{3g_3I_c^2}{g_1}\psi^2 -
+ 1\right)\frac{g_1C}{t_c}\dot{\psi}= -\psi\\
&\psi(0) = \frac{I_0}{I_c} \;\;\;\; \dot{\psi}(0) = 0
\end{align}
There are a total of four constants that we can eliminate
\begin{align}
- \Pi_1 &= \frac{I_0}{I_c},\;\;\;\;
+ \Pi_1 &= \frac{I_0}{I_c}, \qquad
\Pi_2 = \frac{LC}{t_c^2},\nonumber\\
- \Pi_3 &= \frac{-g_1C}{t_c},\;\;\;\;\;
- \Pi_4 = \frac{3g_3CI_C}{t_c}.
-\end{align}
-The first scaling is
-\begin{align}
- I_c = I_0,\;\;\; t_c=\frac{1}{\sqrt{LC}}.
-\end{align}
-Thereby we get the following problem
-\begin{align}
- &\ddot{\psi}
- -\sqrt{\frac{C}{L}}g_1\dot{\psi}+\sqrt{\frac{C}{L}}3g_3I_0\dot{\psi}\psi
- = -\psi\\
- &\psi(0) = 1 \;\;\;\; \dot{\psi}(0) = 0
+ \Pi_3 &= \frac{3g_3I_c^2}{g_1}, \qquad
+ \Pi_4 = \frac{g_1C}{t_c}.
\end{align}
-
-
-The second scaling is
+The \textbf{first} scaling is
\begin{align}
- I_c = I_0,\;\;\; t_c=g_1C.
+ I_c = \sqrt{\frac{g_1}{3g_3}},\;\;\; t_c=\sqrt{LC}.
\end{align}
Thereby we get the following problem
\begin{align}
- &\frac{L}{g_1^2C}\ddot{\psi}
- -\dot{\psi}+\frac{3g_3}{g_1}\dot{\psi}\psi
- = -\psi\\
- &\psi(0) = 1 \;\;\;\; \dot{\psi}(0) = 0
+ \ddot{\psi} + (\psi^2 - 1)\eps \dot{\psi} = -\psi, \qquad \psi(0) =
+ \sqrt{\frac{3g_3}{g_1}}I_0,
\end{align}
+where $\eps = g_1\sqrt{\frac{C}{L}}$.
-The third scaling is
+The \textbf{second} scaling is
\begin{align}
- I_c = I_0,\;\;\; t_c=g_3CI_0.
+ I_c = \sqrt{\frac{g_1}{3g_3}},\;\;\; t_c=g_1C.
\end{align}
-Thereby we get the following problem
+Thereby we get the following
\begin{align}
- &\frac{L}{g_3^2CI_0^2}\ddot{\psi}
- -\frac{g_1}{g_3I_0}\dot{\psi}+3\dot{\psi}\psi
- = -\psi\\
- &\psi(0) = 1 \;\;\;\; \dot{\psi}(0) = 0
+ \eps \psi'' +(\psi^2 +1)\psi' = -\psi, \qquad \psi(0) =
+ \sqrt{\frac{3g_3}{g_1}}I_0,
\end{align}
+where $\eps = \frac{L}{g_1^2C}$. We could also consider scaling $I_c = I_0$
+with $t_c = \sqrt{LC}$ or $t_c = g_1C$ but they wouldn't develop significant
+model hierarchies like the above two scaling.
\subsection{Scale the Schrödinger Equation}
-The well known Schrödinger equation that describes quantum physics of one
-particle can be written as
+The well known Schrödinger equation that describes quantum physics of the one
+particle system is
\begin{align}
- &i\hbar \partial_t\psi = -\frac{\hbar}{2m}\Delta \psi + V\psi \nonumber\\
+ &i\hbar \partial_t\psi = -\frac{\hbar^2}{2m}\Delta \psi + V\psi \nonumber\\
&\psi(t=0) =\psi_0
\end{align}
-where $\hbar$ is the Planks constant, $\psi=\psi(x, t)$ the wave function,
+where $\hbar$ is the reduced Plank constant, $\psi=\psi(x, t)$ the wave function,
$m$ the mass and $V = V(x)$ the potential in which the wave function is. The
dimensions are
\begin{align}
- [\hbar] = Js, \;\;\;\; V = J, \;\;\;\; [\psi]= m^{-d/2}
+ [\hbar] = js, \;\;\;\; [V] = j, \;\;\;\; [\psi]= m^{-d/2}
\end{align}
-for the special dimension $d$. The standard scaling ansatz is
+where $d$ is the spacial dimension. The standard scaling ansatz is
\begin{align}
&\psi = \psi_c \phi \\
&t = t_c \tau \;\;\;\; x = x_c \xi,
@@ -296,6 +340,7 @@ simplifies the Schrodinger equation without the potential to
i\Delta_{\vec{\xi}}\ \phi = \partial_\tau \phi,
\end{align}
with the initial condition $\phi(\tau=0) = 1$.
+.
Now consider $V = 0$, $x\in[0, L]$ and $t \in [0, T]$, the Schrodinger
equation is the same only with one spacial dimension as above, we can set
@@ -306,19 +351,21 @@ Thus we get
\begin{align}
i\partial_{\xi}^2 \phi = \partial_\tau \phi,
\end{align}
-with the initial condition $\phi(\tau=0) = 1$.
+with the initial condition $\phi(\tau=0) = 1$, where $\xi \in [0, 1]$ and
+$\tau \in [0, \frac{\hbar T}{2mL^2}]$.
+.
-As a last example let us consider the quantum harmonic oscillator, that is
-$V(x) = m\omega^2 x^2$ for $x\in \mathbb{R}$, where $\omega$ is the
-frequency, the equation is the following
+In the last example let us consider the quantum harmonic oscillator
+represented by the potential $V(x) = m\omega^2 x^2$ for $x\in \mathbb{R}$,
+where $\omega$ is the frequency. The equation is the following
\begin{align}
i \hbar \partial_t \psi = -\frac{\hbar^2}{2m}\partial^2_x \psi
- m\omega^2x^2 \psi.
+ +m\omega^2x^2 \psi.
\end{align}
-By inserting the scaling
+By inserting the standard scaling ansatz we get
\begin{align}
i\partial_\tau \phi = -\frac{t_c\hbar}{2mx_c^2}\partial_\xi^2 \phi
- \frac{t_cm\omega^2x_c^2}{\hbar} \xi^2 \phi
+ +\frac{t_cm\omega^2x_c^2}{\hbar} \xi^2 \phi,
\end{align}
The dimensional constants are
\begin{align}
@@ -336,7 +383,5 @@ Thereby getting the following problem
\end{align}
with $\phi(\tau = 0) = 1$.
-
-
%\printbibliography
\end{document}
diff --git a/appl_ana/sesh2/main.aux b/appl_ana/sesh2/main.aux
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diff --git a/appl_ana/sesh2/main.pdf b/appl_ana/sesh2/main.pdf
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diff --git a/appl_ana/sesh2/main.tex b/appl_ana/sesh2/main.tex
@@ -59,18 +59,18 @@
\section{Sheet 2}
\subsection{Problem 4}
-We consider a quadratic equation with two ways to perturb it by $\eps =
-0.001$ \begin{align}
+We consider a quadratic equation with two ways to perturb it by $\eps$:
+\begin{align}
x^2 + 2\eps x -1 = 0, \label{eq: (1)}\\
\nonumber \\
\eps x^2 + 2x - 1 = 0.\label{eq: (2)}
\end{align}
Equation \ref{eq: (2)} is singular, because the reduced problem ($\eps
-\rightarrow 0$) has only one solution while the at $x = \frac{1}{2}$. While
-the reduced problem in \ref{eq: (1)} has two solutions for $x = \pm 1$, which
-is the regular case. Let us thereby calculate the asymptotic expansion of the
-regular case up to $O(\eps^2)$, we take the ansatz for the asymptotic
-expansion
+\rightarrow 0$) has only one solution at $x = \frac{1}{2}$. While the reduced
+problem in \ref{eq: (1)} has two solutions for $x = \pm 1$, which is the case
+for this non reduced equation. Let us thereby calculate the asymptotic
+expansion of the regular case up to $O(\eps^2)$, we take the ansatz for the
+asymptotic expansion
\begin{align}\label{eq: p4 ansatz}
x_\eps = x_0 + \eps x_1 + \eps^2 x_2 + O(\eps^3).
\end{align}
@@ -78,7 +78,7 @@ By substituting $x_\eps$ into \ref{eq: (1)} and factoring out the orders of
$\eps$ we get
\begin{align}
\eps^0 (x_0^2 - 1) + \eps^1(2x_0 + 2x_0x_1) + \eps^2(x_1^2+2x_2x_0
- 2x_1) = O(\eps^3)
+ +2x_1) + O(\eps^3) = 0
\end{align}
By solving the equations in order of $\eps$, for the coefficients
$x_0$, $x_1$ and $x_2$ we get
@@ -91,8 +91,8 @@ By substituting into the equation \ref{eq: p4 ansatz} we get
\end{align}
For $\eps = 0.001$ we get
\begin{align}
- x_\eps = -1.00150 + O(\eps^3)\\
- x_\eps = -1.00100 + O(\eps^2).
+ &x_\eps = -1.0010005 + O(\eps^3), &x_\eps = 0.9990005 + O(\eps^3),\\
+ &x_\eps = -1.001 + O(\eps^2), &x_\eps = 0.999 + O(\eps^2).
\end{align}
\subsection{Problem 5}
Consider the following equations
@@ -104,7 +104,7 @@ Consider the following equations
\label{eq: p5 4}\eps^2 y' + y = x \;\;\;\;\;\; y(0) = \eps\\
\nonumber\\
\label{eq: p5 5}y' + \eps y = x \;\;\;\;\;\; y(0) = 1\\
- \label{eq: p5 6}y' + y = \eps x \;\;\;\;\;\; y(0) = 1\\
+ \label{eq: p5 6}y' + y = \eps x \;\;\;\;\;\; y(0) = 1
\end{align}
We will go through the equations and elaborate on if the perturbation is
regular or singular, if regular we will compute the asymptotic expansion up
@@ -129,7 +129,7 @@ But by doing the ansatz for the asymptotic expansion
plugging in into \ref{eq: p5 2} and separating coefficients in terms of
$\eps$, we get
\begin{align}
- \eps^0 (y_0 -x) + \eps^1(y_0' + y_1) + \eps^2(y_1' + y_2) = O(\eps^3)
+ \eps^0 (y_0 -x) + \eps^1(y_0' + y_1) + \eps^2(y_1' + y_2) + O(\eps^3) = 0
\end{align}
The solutions to these equations are
\begin{align}
@@ -144,7 +144,8 @@ first order coefficient of $\eps$, $y_2$, has the boundary condition $y_2(0)
= 0$. But by applying the ansatz of the asymptotic expansion and plugging
into the equation we get
\begin{align}
- \eps^2(y_0 - x) + \eps^1 (y_0' + y_1) + \eps^2(y_1' + y_2) = O(\eps^3).
+ \eps^2(y_0 - x) + \eps^1 (y_0' + y_1) + \eps^2(y_1' + y_2) + O(\eps^3) =
+ 0.
\end{align}
Solving these equations we get
\begin{align}
@@ -169,7 +170,7 @@ Equation \ref{eq: p5 5} on the first sight does not indicate for any
contradictions, we may plug the ansatz of the asymptotic expansion into the
equation and see what happens
\begin{align}
- \eps^0(y_0 -x) + \eps^1(y_1' + y_0) + \eps^2(y_2' +y_1) = O(\eps^2),
+ \eps^0(y_0 -x) + \eps^1(y_1' + y_0) + \eps^2(y_2' +y_1) + O(\eps^2) = 0,
\end{align}
with the initial conditions $y_0(0) = 1$, $y_1(0) = y_2(0) = 0$.
\begin{align}
@@ -187,8 +188,8 @@ Thereby we can conclude that \ref{eq: p5 5} is \textbf{regularly perturbed}.
The last equation \ref{eq: p5 6} is also regular, let us do the asymptotic
expansion of the equation and order the equation in orders of $\eps$.
\begin{align}
- \eps^0(y_0' + y_0) + \eps^1(y_1' + y_1 -x) + \eps^2(y_2 + y_2) =
- O(\eps^3).
+ \eps^0(y_0' + y_0) + \eps^1(y_1' + y_1 -x) + \eps^2(y_2' + y_2)+
+ O(\eps^3) = 0 .
\end{align}
by solving these differential equations with the boundary conditions $y_0(0)
= 1$, $y_1(0) = y_2(0) = 0$ we get
@@ -216,14 +217,14 @@ y_\eps(x) = y_0 + \eps y_1 + \eps^2 y_2 + O(\eps^3).
The ODE then expands to
\begin{align}
\eps^0(y_0' + y_0) + \eps(y_1' + y_1 - y_0^2) + \eps^2(y_2' + y_2 -
- 2y_0y_1) = O(\eps^3).
+ 2y_0y_1) + O(\eps^3) = 0.
\end{align}
Equations in order of $\eps$ and $\eps^2$ are non homogeneous ODE's, a little
bit of a hassle to solve but, still easy by first solving the non homogeneous
part, getting a constant dependent on $x$ we can plug the solution into the
non homogeneous equation and calculate for the constant (...). The solution to
these three coefficients with the boundary conditions $y_0(0) = 1$, $y_1(0) =
-y_2(0) = 1$ we get
+y_2(0) = 0$ we get
\begin{align}
y_0 = e^{-x}, \;\;\;\; y_1 = -e^{-2x} + e^{-x}, \;\;\;\; y_2 = e^{-3x} -
2e^{-2x} + e^{-x}.
@@ -273,20 +274,20 @@ in the order of $\eps$ we get
\begin{align}
&\eps^0 (\partial_t u_0 + \partial_x^2 u_0) + \\
&\eps^1 (\partial_t u_1 + \partial_x^2 u_1 - u_0^2) +\\
- &\eps^2 (\partial_t u_2 + \partial_x^2 u_2 - 2u_1u_0) = O(\eps^3)
+ &\eps^2 (\partial_t u_2 + \partial_x^2 u_2 - 2u_1u_0) + O(\eps^3) = 0.
\end{align}
We can solve the reduced problem with the initial condition $\tilde{u}_0 =
-\sin(\pi x)$ by separation of variables. Setting $u(x, t) = \xi(x) \tau(t)$
+\sin(\pi x)$ by separation of variables. Setting $u(x, t) = \psi(x) \phi(t)$
and substituting into the equation we get two ordinary differential equation
\begin{align}
- \underbrace{\frac{\xi_{xx}}{\xi}}_{=k}
- \underbrace{\frac{\tau_t}{\tau}}_{=-k} = 0
+ \underbrace{\frac{\psi_{xx}}{\psi}}_{=k}
+ +\underbrace{\frac{\phi_t}{\phi}}_{=-k} = 0,
\end{align}
for some $k$. Solving these two by the exponential ansatz.
\begin{align}
- \xi(x) &= A_1 e^{\sqrt{k} x} +A_2 e^{-\sqrt{k} x},\\
- \tau(t) &= A_3 e^{-kt}.
+ \psi(x) &= A_1 e^{\sqrt{k} x} +A_2 e^{-\sqrt{k} x},\\
+ \phi(t) &= A_3 e^{-kt}.
\end{align}
With the initial condition we get the conditions that
\begin{align}
@@ -296,7 +297,7 @@ With the initial condition we get the conditions that
we choose $A_1 = A_3 = 1$ , $A_2 = -1$. We get the following solution to the
PDE
\begin{align}
- u(x, t) = \xi(x)\tau(t) = \sin(\pi x) e^{-\pi^2 t}.
+ u(x, t) = \psi(x)\phi(t) = \sin(\pi x) e^{-\pi^2 t}.
\end{align}
%\printbibliography
diff --git a/appl_ana/sesh2/main.toc b/appl_ana/sesh2/main.toc
@@ -0,0 +1,5 @@
+\contentsline {section}{\numberline {1}Sheet 2}{1}{section.1}%
+\contentsline {subsection}{\numberline {1.1}Problem 4}{1}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}Problem 5}{2}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Problem 6}{3}{subsection.1.3}%
+\contentsline {subsection}{\numberline {1.4}Problem 7}{4}{subsection.1.4}%
diff --git a/appl_ana/sesh3/main.aux b/appl_ana/sesh3/main.aux
@@ -0,0 +1,27 @@
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diff --git a/appl_ana/sesh3/main.tex b/appl_ana/sesh3/main.tex
@@ -0,0 +1,250 @@
+\documentclass[a4paper]{article}
+
+
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{mlmodern}
+
+%\usepackage{ngerman} % Sprachanpassung Deutsch
+
+\usepackage{graphicx}
+\usepackage{geometry}
+\geometry{a4paper, top=15mm}
+
+\usepackage{subcaption}
+\usepackage[shortlabels]{enumitem}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{mathtools}
+\usepackage{braket}
+\usepackage{bbm}
+\usepackage{graphicx}
+\usepackage{float}
+\usepackage{yhmath}
+\usepackage{tikz}
+\usetikzlibrary{patterns,decorations.pathmorphing,positioning}
+\usetikzlibrary{calc,decorations.markings}
+
+%\usepackage[backend=biber, sorting=none]{biblatex}
+%\addbibresource{uni.bib}
+
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+
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+ text=black, rectangle,
+ right, minimum height=.7cm]
+
+
+\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref}
+\usepackage[parfill]{parskip}
+\usepackage{lipsum}
+
+
+\usepackage{tcolorbox}
+\tcbuselibrary{skins,breakable}
+\newcommand{\eps}{\varepsilon}
+\pagestyle{myheadings}
+
+\markright{Popović\hfill Applied Analysis\hfill}
+
+
+
+\title{Applied Analysis Problems}
+\author{Milutin Popović}
+
+\begin{document}
+\maketitle
+\tableofcontents
+
+\section{Sheet 3}
+\subsection{Problem 8}
+Let us look at functions $f: \mathcal{D} \mapsto \mathbb{R}$ that show
+boundary layer behavior at the following manifolds.
+
+The \textbf{first} for $\mathcal{D} = \mathbb{R}^2$ and $S = \{0\}$ we have a
+function e.g.
+\begin{align}
+ f_{\eps}(x, y) = e^{-\frac{x}{\eps}} + y,
+\end{align}
+with the reduced equation
+\begin{align}
+ \lim_{\eps \rightarrow 0} f_{\eps}(x, y) =
+ \begin{cases}
+ y \;\;\;\;\;\;\;\;\;\; x > 0\\
+ 1+y \;\;\;\; x = 0\\
+ \end{cases}
+\end{align}
+
+The \textbf{second} example is $\mathcal{D} = \mathbb{R}^n$ and $S = \{|x| = 1\}$.
+\begin{align}
+ f_\eps(x_1,\dots,x_n) = \tanh\left(\frac{|x| - 1}{\eps} \right),
+\end{align}
+with the reduced equation
+\begin{align}
+ \lim_{\eps \rightarrow 0} f_{\eps}(x_1,\dots, x_n) =
+ \begin{cases}
+ -1 \;\;\;\; |x| < 0\\
+ 1 \;\;\;\;\;\;\; |x| > 0\\
+ \end{cases}
+\end{align}
+
+The \textbf{third} example is $\mathcal{D} = \mathbb{R}^3$ and $S = \{x_1 =
+1\}$
+\begin{align}
+ f_\eps(x_1, x_2, x_3) = \tanh\left(\frac{x_1 - 1}{\eps}\right)+x_2x_3
+\end{align}
+with the reduced equation
+\begin{align}
+ \lim_{\eps \rightarrow 0} f_{\eps}(x_1,x_2,x_3) =
+ \begin{cases}
+ -1 + x_2x_3 \;\;\;\; x_1 < 0\\
+ 1 + x_2x_3 \;\;\;\;\;\;\; x_1 > 0\\
+ \end{cases}
+\end{align}
+\subsection{Problem 9}
+Consider a linear BVP
+\begin{align}
+ Lu := -\eps u'' + b(x)u' + c(x)u = f(x),\\
+ u(0) = u(1) = 0,
+\end{align}
+for $0 < \eps \ll \eps_0$ and $b, c, f \in C([0,1])$ with the conditions
+\begin{align}
+ c(x) \geq 0, \qquad b(x) \geq \beta > 0 \qquad x\in[0, 1]
+\end{align}
+We are to show that for all $x\in[0, 1)$ the reduced solution $u_0$ of the
+above BVP satisfies
+\begin{align}
+ \lim_{\eps \rightarrow 0} u_\eps(x) = u_0(x)))),
+\end{align}
+where the reduced solution $u_0$ is the solution to the following
+differential equation
+\begin{align}
+ b(x)u' + c(x)u = f(x), \quad u(0) = 0.
+\end{align}
+The hint was given: Set
+\begin{align}
+ w_1(x) = e^{\beta x} \quad w_2(x) = e^{-\beta\frac{1-x}{\eps}},
+\end{align}
+such that $Lw_1 \geq \gamma > 0$ for some suitable $\gamma$ and $Lw_2 \geq
+0$. Then for
+\begin{align}
+ v = \pm (u_\eps - u_0), \qquad w = A\eps w_1 + B\eps w_2,
+\end{align}
+for some suitable $A, B$. The following comparison principal is applicable:
+IF
+\begin{align}
+ &Lv(x) \leq Lw(x) \quad \forall x \in (0, 1) \label{eq:cond1}\\
+ &v(0) \leq w(0) \label{eq:cond2}\\
+ &v(1) \leq w(1) \label{eq:cond3}\\
+\end{align}
+then
+\begin{align}
+ \Longrightarrow v(x) \leq w(x) \quad \forall x\in(0, 1)
+\end{align}
+which holds for $u, v \in C^2((0, 1)) \cap C([0, 1])$. Thus a boundary layer
+is possible only at $x=1$. On the other hand, for $b(x) \leq \beta < 0$ it
+follows that the boundary layer is possible only at $x=0$.
+
+We shall go through the chronological order of the conditions\ref{eq:cond1},
+\ref{eq:cond2}, \ref{eq:cond3} and check them. So for \ref{eq:cond1}
+we have that
+\begin{align}
+ Lw(x) &= A\eps Lw_1(x) + B Lw_2(x) \\
+ &\geq A\eps Lw_1(x) = A\eps e^{\beta x} \left(-\eps \beta^2 -
+ b(x)\beta+c(x)\right)\\
+ &\geq A\eps \beta e^{\beta x} \left(1-\eps\right)\\
+ &\geq \eps A\beta^2 e^{\beta}(1-\eps) = \gamma > 0
+\end{align}
+And obviously
+\begin{align}
+ Lv(x) \leq 0 ,
+\end{align}
+by that we have that
+\begin{align}
+ Lv(x) \leq \gamma \leq Lw(x).
+\end{align}
+For the condition \ref{eq:cond2} we have
+\begin{align}
+ w(0) &= A\eps w_1(0) Bw_2(0) = Be^{-\frac{\beta}{\eps}},\\
+ v(0) &= \pm\left(u_\eps(0) - u_0(0)\right) = 0.
+\end{align}
+By the simple choice $B \geq 0$ we satisfy the condition
+\begin{align}
+ v(0) \leq w(0).
+\end{align}
+Now for the last condition \ref{eq:cond3} we have
+\begin{align}
+ w(1) &= A\eps e^\beta + B \geq A\eps e^\beta,\\
+ v(1) &= \mp u_0(1) = 0.
+\end{align}
+And choose $A = \frac{\pm u(1)}{\eps} e^{-\beta}$, which satisfies the last
+condition
+\begin{align}
+ v(1) \leq w(1).
+\end{align}
+Thereby we have
+\begin{align}
+ &v(x) &\leq w(x)\\
+ &\Rightarrow \lim_{\eps \rightarrow 0} v(x) &\leq \lim_{\eps \rightarrow
+ 0} w(x) = 0\\
+ &\Rightarrow \lim_{\eps \rightarrow 0} v(x) = 0
+\end{align}
+uniformly on $(0, 1)$.
+\subsection{Problem 10}
+Consider the following BVP
+\begin{align}
+ -\eps u'' + (1 + x)u' + u = 2, \qquad u(0) = u(1) - 0,
+\end{align}
+for $0 < \eps \ll 1$. \textbf{Where can this problem have a boundary layer?}
+To answer this question we need to look at the reduced problem
+\begin{align}
+ -(1+x)u' + u = 2.
+\end{align}
+The solution to the equation is
+\begin{align}
+ \bar{u}(x) = 2 + A(x+1).
+\end{align}
+According to the boundary conditions it is unclear what the value of the
+constant is, according to $\bar{u}(0)=0$ we get $A = -2$ or according to
+$\bar{u}(1)=0$ we get $A = -1$. Ultimately this means that there exists a
+boundary layer near $x=1$ or $x=0$. We choose $x=0$ and according to this the
+local variable $\xi = x\eps^{-\alpha}$ ($x = \xi \eps^{-\alpha}$). The
+derivatives of $u$ are calculated using the chain rule
+\begin{align}
+ \frac{du}{dx}&= \frac{du}{d\xi}\frac{d\xi}{dx} = \eps^{-\alpha} \dot{u}\\
+ \frac{d^2u}{dx^2}&= \eps^{-\alpha} \frac{d^2u}{d\xi^2}\frac{d\xi}{dx} =
+ \eps^{-2\alpha} \ddot{u}.
+\end{align}
+The BVP transforms as follows
+\begin{align}
+ -\eps^{1-\alpha}\ddot{u} - \dot{u} + \eps(u - \xi\dot{u} - 2) =
+ \begin{cases}
+ -\ddot{u} - \dot{u} = 0 \;\;\;\;\; \alpha=1\\
+ -\dot{u} = 0 \;\;\;\;\;\;\;\; 0<\alpha<1
+ \end{cases}
+\end{align}
+Choosing $\alpha = 1$ for a reasonable solution
+\begin{align}
+ \hat{u}(\xi) = Be^{-\xi},
+\end{align}
+which converges in the local limit (!). Thereby we have a asymptotic
+representation up to the degree of $\eps$
+\begin{align}
+ u_\eps(x) &= \bar{u}(x) + \hat{u}(\psi) + O(\eps)\\
+ &= 2 + A(1+x) + B e^{-\frac{x}{\eps}} + O(\eps)
+\end{align}
+And by the boundary conditions
+\begin{align}
+ u_\eps(0) = 2+A+B=0, \qquad u_\eps(1) = 2+2A+B=0,
+\end{align}
+we get that the constants are
+\begin{align}
+ A = -4, \qquad B = 2.
+\end{align}
+The asymptotic representation is thereby
+\begin{align}
+ u_\eps(x) = 2 - 4(1+x) + 2 e^{-\frac{x}{\eps}} + O(\eps)
+\end{align}
+%\printbibliography
+\end{document}
diff --git a/appl_ana/sesh3/main.toc b/appl_ana/sesh3/main.toc
@@ -0,0 +1,4 @@
+\contentsline {section}{\numberline {1}Sheet 3}{1}{section.1}%
+\contentsline {subsection}{\numberline {1.1}Problem 8}{1}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}Problem 9}{2}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Problem 10}{3}{subsection.1.3}%
diff --git a/appl_ana/sesh4/main.pdf b/appl_ana/sesh4/main.pdf
Binary files differ.
diff --git a/appl_ana/sesh4/main.tex b/appl_ana/sesh4/main.tex
@@ -78,13 +78,13 @@ and are given by
Let us compute the Fourier series of $f(t) = t$ for $t \in [-\frac{1}{2},
\frac{1}{2}]$. The Fourier coefficients are
\begin{align}
- a_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} f(t) \cos(2\pi n t)\ dt = 0
+ a_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} t \cos(2\pi n t)\ dt = 0
\;\;\;\;\; \text{(odd: g(-t) = -g(t))},\\
\nonumber\\
- b_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} f(t) \sin(2\pi n t)\ dt = \\
+ b_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} t \sin(2\pi n t)\ dt = \\
&= 2 \left(-\frac{1}{2\pi n} \cos(2\pi n
t)\bigg|_{-\frac{1}{2}}^{\frac{1}{2}}
- \int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{2 \pi n}\cos(2\pi n t)\ dt
+ +\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{2 \pi n}\cos(2\pi n t)\ dt
\right) =\\
&= -\frac{1}{\pi n}\left( -\cos(\pi n) + \frac{1}{\pi n }\sin(\pi
n)\right) =
@@ -101,7 +101,7 @@ The truncation error of the trigonometric polynomial $(Sf_N)$ of degree $N$ is
\sum_{|k| > N} |\hat{f}(k)|^2 = \lVert f - S_N\rVert_2^2 =
\int_{-\frac{1}{2}}^{\frac{1}{2}} |E_N(t)|^2 dt.
\end{align}
-Computations for $N = 3, 9$ were done in python with a integration error of
+Computations for $N = 3$ and $N = 9$ were done in python with a integration error of
around $10^{-15}$, resulting in the overall truncation errors of
\begin{align}
\sum_{|k| > 3} |\hat{f}(k)|^2 = 0.0053,\\
@@ -136,12 +136,12 @@ Furthermore there exists a linear projection $\Phi\ : \mathcal{H}
An example of an orthonormal basis, which spans $L^2([-\frac{p}{2},
\frac{p}{2}])$ is $\mathcal{T}_p = \{e_n := \frac{e^{2\pi i
-\frac{n}{p}x}}{\sqrt{p}}\}_{n\in\mathbb{Z}}$. The $e_n$ are orthonormal in $L^2$ which
-can be easily seen by using the scalar product of $L^2$, so for $n, m \in
-\mathbb{Z}$
+\frac{n}{p}x}}{\sqrt{p}}\}_{n\in\mathbb{Z}}$. The $e_n$'s are orthonormal in
+$L^2$ which can be easily seen by using the scalar product of $L^2$, so for
+$n, m \in \mathbb{Z}$
\begin{align}
\langle e_n, e_m\rangle_{L^2([-\frac{p}{2}, \frac{p}{2})} &=
- \frac{1}{p}\int_{[-\frac{p}{2}, \frac{p}{2}]}e_n \cdot \bar{e}_m \ dx=\\
+ \frac{1}{p}\int_{[-\frac{p}{2}, \frac{p}{2}]}e_n \cdot e_m^* \ dx=\\
&=\frac{1}{p}\int_{[-\frac{p}{2}, \frac{p}{2}]} e^{2\pi i \frac{(n-m)}{p} x} \ dx=\\
&=\frac{\sin(\pi (n-m))}{\pi(n-m)} =
\begin{cases}
@@ -170,8 +170,9 @@ And furthermore the Dirichlet Kernel satisfies
\begin{align}
D_t(x) = \prod_{i=1}^d \frac{e_{t+1}(x_i) - e_{-t}(x_i)}{e_1(x_i) - 1}
\end{align}
-We start off by applying the orthogonal projection into the trigonometric
-polynomials $S_t$ onto a function $f \in L(\mathbb{T}^d)$
+To show the convolution property, we start off by applying the orthogonal
+projection into the trigonometric polynomials $S_t$ onto a function $f \in
+L(\mathbb{T}^d)$
\begin{align}
(S_tf) &= \sum_{\lvert k\rVert_\infty \leq t} \int_{I^d} f(y) e^{-2\pi i
\langle k, y\rangle}\ dy\ e^{2\pi i\langle k, x\rangle} =\\
@@ -179,15 +180,16 @@ polynomials $S_t$ onto a function $f \in L(\mathbb{T}^d)$
k, (x- y)\rangle}\ dy =\\
&= (f * D_t) (x) = \int_{I^d} f(y) D_t(x - y)\ dy.
\end{align}
-The Dirichlet kernel can be simply rewritten
+To show the reformulation of the Dirichlet kernel, we need to simply
+calculate it directly
\begin{align}
\sum_{\lVert k \rVert_\infty \leq t} e^{2\pi i \langle k , x\rangle} &=
- \prod_{i=1}^d \sum_{k_i = -t}^t e^{2\pi i k_i x_i} =\\
- &= \prod_{i=1}^d e^{-2\pi i t x_i} \sum_{k_i = 0}^{2t} e^{2\pi i k_i
- x_i}=;\;\;\;\; \text{(trigonometric series)}\\
- &= \prod_{i=1}^d e^{-2\pi i t x_i} \frac{e^{2\pi i (2t +1}x_i -
- 1}{e^{2\pi i x_i} - 1} =\\
- &= \prod_{i = 1} \frac{e_{t+1}(x_i) - e_{-t}(x_i)}{e_1(x_i) - 1}.
+ \prod_{j=1}^d \sum_{k_j = -t}^t e^{2\pi i k_j x_j} =\\
+ &= \prod_{j=1}^d e^{-2\pi i t x_j} \sum_{k_j = 0}^{2t} e^{2\pi i k_j
+ x_j}=;\;\;\;\; \text{(trigonometric series)}\\
+ &= \prod_{j=1}^d e^{-2\pi i t x_j} \frac{e^{2\pi i (2t + 1)x_j} -
+ 1}{e^{2\pi i x_j} - 1} =\\
+ &= \prod_{j = 1} \frac{e_{t+1}(x_j) - e_{-t}(x_j)}{e_1(x_j) - 1}.
\end{align}
%\printbibliography
\end{document}
diff --git a/appl_ana/sesh5/main.pdf b/appl_ana/sesh5/main.pdf
Binary files differ.
diff --git a/appl_ana/sesh5/main.tex b/appl_ana/sesh5/main.tex
@@ -93,7 +93,7 @@ The following are three identities of the Fourier transform
(2) & $e^{2\pi i \xi_0 x} f(x)$ & $f(\xi - \xi_0)$ \\ \hline
(3) & $f(ax)$ & $\frac{1}{a} \hat{f}(\frac{\xi}{a})$\\ \hline
\end{tabular}
- \caption{Identities of the Fourier transform for $0 < a\in\mathbb{R},
+ \caption{Identities of the Fourier transform for $a > 0,
\xi_0, x \in \mathbb{R}$}
\end{table}
We start with (1)
@@ -117,9 +117,10 @@ For (2) we have
For (3) we have
\begin{align}
\widehat{f(ax)}
- &= \int_\mathbb{R} f(ax) e^{-2\pi i \xi x}\ dx = \;\;\;\;\; (y=ax)\\
+ &= \int_\mathbb{R} f(ax) e^{-2\pi i \xi x}\ dx = \qquad \text{sub:
+ $(y=ax)$}\\
&= \int_\mathbb{R} \frac{1}{a}f(y) e^{-2\pi i \frac{\xi}{a} y}\ dy=\\
- &= \frac{1}{a} \hat{f}(\frac{\xi}{a}).
+ &= \frac{1}{a} \hat{f}\left(\frac{\xi}{a}\right).
\end{align}
\subsection{The Box-Function}
Consider the following Box-Function
diff --git a/appl_ana/sesh6/main.aux b/appl_ana/sesh6/main.aux
@@ -0,0 +1,25 @@
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+\global\let\hyper@last\relax
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+\@writefile{toc}{\contentsline {section}{\numberline {1}Sheet 6}{1}{section.1}\protected@file@percent }
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+\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Discrete Matrix Notation}{3}{subsection.1.4}\protected@file@percent }
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diff --git a/appl_ana/sesh6/main.log b/appl_ana/sesh6/main.log
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diff --git a/appl_ana/sesh6/main.pdf b/appl_ana/sesh6/main.pdf
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diff --git a/appl_ana/sesh6/main.tex b/appl_ana/sesh6/main.tex
@@ -63,16 +63,18 @@
Consider the function $f(x)$, which has a Fourier Transform $\hat{f}(\xi)$,
now let us compute the Fourier transform of
\begin{align}
- g(x) = f(3x-1) \sin(x).
+ h(x) = f(3x-1) \sin(x) .
\end{align}
-We know that the Fourier transform of the convolution is
+We know that the Fourier transform of the convolution is (we use somewhat of
+the inverse convolution theorem).
\begin{align}
- (\widehat{f(3x-1)*g(x)}) = \widehat{f(3x-1)} \cdot \hat{g}(\xi).
+ \widehat{(f(3x-1)*g(x))} = \widehat{f(3x-1)} \cdot \hat{g}(\xi).
\end{align}
The Fourier transform of $f(3x-1)$ is simply done by substituting a new
variable
\begin{align}
- \widehat{f(3x-1)} = \frac{1}{3}e^{2\pi i\frac{\xi}{3}} f(\frac{\xi}{3}).
+ \widehat{f(3x-1)} = \frac{1}{3}e^{2\pi i\frac{\xi}{3}}\
+ f\left(\frac{\xi}{3}\right).
\end{align}
The Fourier transform of $\sin(x)$ can be calculated when looking at the
Fourier transform of the Dirac-delta function
@@ -105,12 +107,13 @@ substitutions and thereby get the following result
\end{align}
The whole result is thereby
\begin{align}
- \widehat{f(3x-1)} \cdot \widehat{sin(x)}
- =& \frac{1}{6i} \left(
+ \widehat{f(3x-1)} * \widehat{sin(x)}
+ =& \frac{1}{6i} \bigg(
e^{2\pi
- i(\frac{\xi}{3}-\frac{1}{2\pi})}\hat{f}(\frac{\xi}{3}-\frac{1}{2\pi})-
- e^{2\pi i(\frac{\xi}{3}+\frac{1}{2\pi})}\hat{f}(\frac{\xi}{3}+\frac{1}{2\pi})
- \right)
+ i(\frac{\xi}{3}-\frac{1}{6\pi})}\hat{f}\big(\frac{\xi}{3}-\frac{1}{6\pi}\big)-
+ e^{2\pi
+ i(\frac{\xi}{3}+\frac{1}{6\pi})}\hat{f}\big(\frac{\xi}{3}+\frac{1}{6\pi}\big)
+ \bigg)
\end{align}
\subsection{More Fourier Transforms}
Consider the function
@@ -130,8 +133,13 @@ The Fourier transform of this function is
\end{align}
Let us use this result to solve the following integral
\begin{align}
- \int_\mathbb{R} \frac{\cos(a\xi)}{(2\pi \xi)^2 + 1}\ d\xi &=
- \frac{1}{2}\int_\mathbb{R} \hat{f}(\xi) \text{Re}(e^{ia\xi})\ dx=\\
+ \int_\mathbb{R} \frac{\cos(a\xi)}{(2\pi \xi)^2 + 1}\ d\xi =
+ \frac{1}{2}\int_\mathbb{R} \hat{f}(\xi) \text{Re}(e^{ia\xi})\ dx,\\
+\end{align}
+where we used the fact that $\text{Re}(e^{ia\xi}) = \cos(a\xi)$ and
+$\hat{f}(\xi) = \frac{2}{1+(2\pi \xi)^2}$, thereby
+\begin{align}
+ \frac{1}{2}\int_\mathbb{R} \hat{f}(\xi) \text{Re}(e^{ia\xi})\ dx
&= \frac{1}{2}\text{Re}\left(
\int_\mathbb{R}\hat{f}(\xi)e^{ia\xi}\ d\xi
\right)=\\
@@ -139,7 +147,7 @@ Let us use this result to solve the following integral
\int_\mathbb{R} \hat{f}(\xi) e^{2\pi i \frac{a}{2\pi}\xi}\ d\xi
\right)=\\
&= \frac{1}{2}\text{Re}\left(f(\frac{a}{2\pi})\right)=\\
- &= \frac{1}{2} e^{-\frac{|a|}{2\pi}}
+ &= \frac{1}{2} e^{-\frac{|a|}{2\pi}}.
\end{align}
\subsection{Finite discrete Fourier transform}
Consider $s\in \mathbb{C}^N$ with entries
@@ -151,7 +159,7 @@ for same $0 < \xi_0 < N$. The finite discrete Fourier transform of $s$ is
\hat{s}[k] &= \frac{1}{N} \sum_{n=0}^{N-1} \sin\left(2\pi\xi_0\frac{n}{N}\right)
e^{-2\pi i \frac{k}{N} n} =\\
&=\frac{1}{2iN}\left(
- \sum_{n=0}^{N-1}e^{2\pi i \frac{n}{N}(\xi_0 -k)} + e^{-2\pi i
+ \sum_{n=0}^{N-1}e^{2\pi i \frac{n}{N}(\xi_0 -k)} - e^{-2\pi i
\frac{n}{N}(\xi_0 +k)}
\right).
\end{align}
diff --git a/appl_ana/sesh6/main.toc b/appl_ana/sesh6/main.toc
@@ -0,0 +1,5 @@
+\contentsline {section}{\numberline {1}Sheet 6}{1}{section.1}%
+\contentsline {subsection}{\numberline {1.1}Fourier Transform of the convolution}{1}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}More Fourier Transforms}{2}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Finite discrete Fourier transform}{3}{subsection.1.3}%
+\contentsline {subsection}{\numberline {1.4}Discrete Matrix Notation}{3}{subsection.1.4}%
diff --git a/appl_ana/sesh7/main.pdf b/appl_ana/sesh7/main.pdf
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diff --git a/appl_ana/sesh7/main.tex b/appl_ana/sesh7/main.tex
@@ -71,18 +71,26 @@ The Dirac train or Dirac comb on defined in the following way
0\;\;\;\;\;\; \text{else}
\end{cases}
\end{align}
+The dirac comb can be represented in a series of discrete dirac delta's
+\begin{align}
+ \Sha_m[n] = \sum_{l=-N}^N \delta[n - lm],
+\end{align}
+where $\delta[s] = 1$ if $s = 0$ else $0$, for $s \in \mathbb{Z}$.
The discrete Fourier transform of the Dirac comb in $\mathbb{C}^N$ is
\begin{align}
\widehat{\Sha_m[n]}
&=\frac{1}{N}\sum_{n=0}^{N-1} \Sha_m[n] e^{-2\pi i \frac{k}{N}n}=\\
&=\frac{1}{N}\sum_{n=0}^{N-1}
\left(
- m\sum_{l\in\mathbb{Z}} \delta(n-lm)
+ \sum_{l=-N}^N \delta(n-lm)
\right)
- e^{-2\pi i \frac{k}{N}n}=\\
- &=\frac{m}{N}\sum_{l\in\mathbb{Z}} e^{-2\pi i \frac{k}{N}lm}=
- \;\;\;\;\;\;\;\;\;\;\;\;\; (m = \frac{N}{m'})\\
- &= \frac{1}{m'} \sum_{l\in\mathbb{Z}} e^{-2\pi i \frac{k}{m'}l} =\\
+ e^{-2\pi i \frac{k}{N}n},
+\end{align}
+where the summation happens exactly $\frac{N}{m}$ times, then
+\begin{align}
+ &\frac{1}{m}\sum_{l=-N}^N e^{-2\pi i \frac{k}{N}lm}=\\
+ &= \frac{1}{m} \sum_{l=-N}^N \delta[k - l\cdot \frac{N}{m}]\qquad
+ \text{(Poisson's summation formula)} \\
&= \frac{1}{m}\Sha_{\frac{N}{m}}[k]
\end{align}
\subsection{Schwartz Space}
@@ -146,17 +154,13 @@ The Fourier transform of $\xi$ is
\begin{align}
\hat{\xi}(\varphi)
&=\xi(\hat{\varphi})
- = \int_\mathbb{R} \xi \hat{\varphi}(\xi)\ d\xi=\\
- &= \int_\mathbb{R}\xi \int_\mathbb{R} \varphi(x) e^{-2\pi i \xi x}\ dx\
- d\xi=\\
- &=\int_\mathbb{R}\int_\mathbb{R} \xi e^{-2\pi i \xi x}\ d\xi \varphi(x)\
- dx=\\
- &=\int_\mathbb{R} \frac{i}{2\pi}\frac{d}{dx}\int_\mathbb{R}e^{-2\pi i \xi
-x} \ d\xi \varphi(x)\ dx=\\
- &= \int_\mathbb{R}i\frac{1}{2\pi} \frac{d}{dx}\left(2\pi \delta(x)\right)
- \varphi(x)\ dx=\\
- &= \int_\mathbb{R} i\delta'(x)\varphi(x)\ dx\\
- &= i \delta'(\varphi).
+ = \int_\mathbb{R} \xi \hat{\varphi}(\xi)\ d\xi\\
+ &= \int_{\mathbb{R}^2}\xi \varphi(x) e^{2\pi i\xi x}\ dxd\xi\\
+ &= \int_{\mathbb{R}^2}\varphi(x) \xi e^{2\pi i \xi x}\ dxd\xi\\
+ &=\int_{\mathbb{R}^2}\varphi(x)\frac{i}{2\pi} \frac{\partial}{\partial x}
+ e^{2\pi i \xi x}\ dxd\xi =\\
+ &=\frac{i}{2\pi}\int_{\mathbb{R}^2}\varphi(x)\delta'(x)\ dx=\\
+ &=\frac{i}{2\pi} \delta'(\varphi).
\end{align}
\subsection{Fourier transform of the Dirac Comb}
The general case of the Dirac Comb as a distribution is
diff --git a/appl_ana/sesh8/fdft.png b/appl_ana/sesh8/fdft.png
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diff --git a/appl_ana/sesh8/main.aux b/appl_ana/sesh8/main.aux
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diff --git a/appl_ana/sesh8/main.log b/appl_ana/sesh8/main.log
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diff --git a/appl_ana/sesh8/main.tex b/appl_ana/sesh8/main.tex
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+\documentclass[a4paper]{article}
+
+
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{mlmodern}
+
+%\usepackage{ngerman} % Sprachanpassung Deutsch
+
+\usepackage{graphicx}
+\usepackage{geometry}
+\geometry{a4paper, top=15mm}
+
+\usepackage{subcaption}
+\usepackage[shortlabels]{enumitem}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{mathtools}
+\usepackage{braket}
+\usepackage{bbm}
+\usepackage{graphicx}
+\usepackage{float}
+\usepackage{yhmath}
+\usepackage{tikz}
+\usetikzlibrary{patterns,decorations.pathmorphing,positioning}
+\usetikzlibrary{calc,decorations.markings}
+
+%\usepackage[backend=biber, sorting=none]{biblatex}
+%\addbibresource{uni.bib}
+
+\usepackage[framemethod=TikZ]{mdframed}
+
+\tikzstyle{titlered} =
+ [draw=black, thick, fill=white,%
+ text=black, rectangle,
+ right, minimum height=.7cm]
+
+
+\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref}
+\usepackage[parfill]{parskip}
+\usepackage{lipsum}
+
+\usepackage[OT2,T1]{fontenc}
+\DeclareSymbolFont{cyrletters}{OT2}{wncyr}{m}{n}
+\DeclareMathSymbol{\Sha}{\mathalpha}{cyrletters}{"58}
+
+\usepackage{tcolorbox}
+\tcbuselibrary{skins,breakable}
+
+\pagestyle{myheadings}
+
+\markright{Popović\hfill Applied Analysis\hfill}
+
+
+\title{University of Vienna\\ Faculty of Mathematics\\
+\vspace{1cm}Applied Analysis Problems
+}
+\author{Milutin Popovic}
+
+\begin{document}
+\maketitle
+\tableofcontents
+
+\section{Sheet 8}
+\subsection{Finite Discrete Fourier Transform (FDFT)}
+Consider the vector $\begin{pmatrix}a & b & c & d\end{pmatrix}^T \in
+\mathbb{C}^4$ with a FDFT $\begin{pmatrix}A & B & C & D\end{pmatrix}^T$. We
+can show that the vector
+\begin{align}
+ \begin{pmatrix}a & 0 & b & 0 & c & 0 & d & 0\end{pmatrix}^T,
+\end{align}
+has the FDFT of
+\begin{align}
+ \frac{1}{2}\begin{pmatrix}A & 0 & B & 0 & C & 0 & D & 0\end{pmatrix}^T.
+\end{align}
+For the $N=4$, $n\in\{0,\dots,3\}$ the coefficients $a, b, c, d$ are denoted in
+$f[n]$. The FDFT is
+\begin{align}
+ \hat{f}[k] &= \frac{1}{4} * \sum_{n=0}^3 f[n] e^{-2\pi i \frac{n}{4}k} \\
+ &=\frac{1}{4}\left(
+ a + be^{-\pi i \frac{k}{2}}
+ + ce^{-\pi i k}+ de^{-\frac{3\pi i k}{2}}
+ \right) = \\
+ (&=\begin{pmatrix}A & B & C & D\end{pmatrix}^T)
+\end{align}
+for $k \in \{0,\dots, 3\}$ accordingly. For the $N=8$, $\mathbb{C}^8$ case
+ we have $f_2[n]$ for $n \in \{0,\dots 7\}$,
+ \begin{align}
+ \hat{f}_2[k] &= \frac{1}{8} * \sum_{n=0}^7 f_2[n] e^{-2\pi i \frac{n}{8}k} \\
+ &=\frac{1}{2}\frac{1}{4}\left(
+ a + be^{-\pi i \frac{k}{2}}
+ + ce^{-\pi i k}+ de^{-\frac{3\pi i k}{2}}
+ \right) = \\
+ (&=\frac{1}{2}\begin{pmatrix}A & B & C & D & A & B & C & D\end{pmatrix}^T)
+ \end{align}
+for $k \in \{0,\dots, 7\}$ accordingly. We may generalize now for
+$\mathbb{C}^{4N}$, and the sequence for $a, b, c, d, 0$ represented by the
+function $g[n]$ for $n \in \{0,\dots, 4N-1\}$,
+\begin{align}
+ g[n] =\begin{cases}
+ f[n] \qquad n\in \{0, N, 2N, 3N\}\\
+ 0 \qquad \text{else}
+ \end{cases}.
+\end{align}
+Now we can compute the FDFT for $k \in \{0,\dots, 4N-1\}$
+\begin{align}
+ \hat{g}[k] &= \frac{1}{4N}\sum_{n=0}^{4N-1} g[n]e^{-2\pi i
+ \frac{n}{4N}k}\\
+ &=\frac{4}{N}\sum_{n=0}{3}f[n]e^{-2\pi i \frac{n}{4}k}\\
+ &=\frac{1}{N}\left(\frac{1}{4}\sum_{n=0}^3 f[n] e^{-2\pi i
+ \frac{n}{4}k} \right) \\
+ &= \frac{1}{N} \underbrace{\begin{pmatrix}A & B & C & D & \dots &
+ \dots & A & B & C & D\end{pmatrix}^T)}_{\text{$4N$ entries, $N$
+ sequences}}.
+\end{align}
+\subsection{More FDFT}
+Consider the discrete complex exponential with frequency of $1Hz$ in
+$\mathbb{C}^8$, for $n \in \{0, \dots , 7\}$,
+\begin{align}
+ \exp[n] = e^{2\pi i n/8}.
+\end{align}
+The FDFT for $k \in \{0, \dots, 7\}$ is
+\begin{align}
+ \hat{\exp}[k] &= \frac{1}{8}\sum_{n=0}^7 e^{2\pi i \frac{n}{8}}e^{-2\pi i
+ n \frac{k}{8}} \\
+ &= \frac{1}{8} \sum_{n=0}^7e^{-2\pi i (k-1)\frac{n}{8}}\\
+ &=
+ \begin{cases}
+ 1\quad k=1\\
+ 0 \qquad k\neq 1
+ \end{cases}.
+\end{align}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.49\textwidth]{./fdft.png}
+ \includegraphics[width=0.49\textwidth]{./normal.png}
+ \caption{Test in Julia}
+\end{figure}
+\subsection{Sampling Sinusoids}
+Consider the following continuous signal
+\begin{align}
+ f(t) = sin(20\pi t) + sin(40\pi t)
+\end{align}
+with frequencies $\omega = 2\pi \nu$, $\nu_1 = 10\ \text{Hz}$ and $\nu_2 = 20\
+\text{Hz}$. Sketching its Fourier transform would be something like this
+\begin{figure}[H]
+ \centering
+\begin{tikzpicture}[
+ axisline/.style={very thick, -stealth},
+ xscale = 1.5,
+ yscale = 1.5
+ ]
+ \draw[axisline] (-3,0)--(3,0) node[right]{$\nu$};
+ \draw[axisline] (0,-1.5)--(0,1.5) node[above]{$\hat{f}$};
+ \draw[->] (-1,0) -- (-1, -1) node[below] {$-\delta(\nu - 10)$};
+ \draw[->] (-2,0) -- (-2, 1) node[above] {$\delta(\nu - 20)$};
+ \draw[->] (1,0) -- (1, 1) node[above] {$\delta(\nu - 10)$};
+ \draw[->] (2,0) -- (2, 1) node[above] {$\delta(\nu - 20)$};
+\end{tikzpicture}
+\end{figure}
+The Nyquist frequency for sampling would be
+\begin{align}
+ \nu_{\text{Nyquist}} = 2\nu_\text{max} = 2\nu_2 = 40\ \text{Hz},
+\end{align}
+If we choose $50\ \text{Hz}$ for sampling we would get aliasing with the
+following frequencies
+\begin{align}
+ n \cdot 50\ \text{Hz} - 20\ \text{Hz} = 30\ \text{Hz},80\ \text{Hz}, 130\
+ \text{Hz}, \dots
+\end{align}
+\subsection{Short-Time Fourier Transform (STFT)}
+The Definition of the STFT is
+\begin{align}
+ \text{STFT}\{f\} &= S_\varphi f(\tau, \omega) = \int_\mathbb{R} f(t)
+ \overline{\text{M}_\omega \text{T}_\tau \varphi}dt \\
+ &=\int_\mathbb{R} f(t)
+ \bar{\varphi}(t - \tau)e^{-2\pi i \omega t}\ dt \\
+\end{align}
+Then we have the following identity
+\begin{align}
+ S_\varphi(\text{T}_u\text{M}_\eta f)(x,\omega)
+ &= \int_\mathbb{R}
+ \left(\text{T}_u \text{M}_\eta f(t)\right) \bar{\varphi}(t-x) e^{-2\pi i
+ \omega t}\ dt\\
+ &= \int_\mathbb{R} e^{2\pi i \eta(t-u)}f(t-u) e^{-2\pi i \omega
+ t}\bar{\varphi}(t-x)\ dt \qquad \text{(sub: $s = t-u$)}\\
+ &= \int_\mathbb{R} f(s)\bar{\varphi}(s-(x-u))e^{2\pi i \eta s}e^{-2\pi i
+ \omega s} e^{-2\pi i \omega u}\ ds \\
+ &=e^{-2\pi i \omega u}\int_\mathbb{R} f(s) \bar{\varphi}(s-(x-u))e^{-2\pi i
+ (\omega - \eta)s}\ ds\\
+ &=e^{-2\pi i \omega u}\int_\mathbb{R}
+ f(s)\overline{ \text{M}_{(\omega-\eta)} \text{T}_{(x-u)}\varphi(s)}\ ds\\
+ &=e^{-2\pi i \omega u} S_\varphi f\left(x-u,\ \omega -\eta\right).
+\end{align}
+The second identity we can show
+\begin{align}
+ S_\varphi f(x, \omega)
+ &= \langle f, \overline{\text{M}_\omega \text{T}_x \varphi}\rangle \\
+ &= \langle\mathcal{F} f, \mathcal{F} \overline{\text{M}_\omega \text{T}_x \varphi}\rangle \\
+ &= \int_\xi \hat{f}(\xi)\int_t \overline{\text{M}_\omega \text{T}_x
+ \varphi}(t) e^{-2\pi i \xi t}\ dt\ d\xi \\
+ &= \int_\xi \hat{f}(\xi)\int_t \hat{\bar{\varphi}}(t-x) e^{2\pi i \omega
+ t} e^{-2\pi i \xi t}\ dt\ d\xi \\
+ &= \int_\xi \hat{f}(\xi)\int_t \hat{\bar{\varphi}}(t-x)e^{-2\pi i (\xi
+ -\omega)t}\ dt\ d\xi \qquad \text{sub $u=t-x$}\\
+ &= \int_\xi \hat{f}(\xi)\int_t \hat{\bar{\varphi}}(u)e^{-2\pi i (\xi
+ -\omega)u}e^{-2\pi i (\xi -\omega)x} \ dt\ d\xi\\
+ &= \int_\xi \hat{f}(\xi)e^{-2\pi i (\xi -\omega)x}\int_t
+ \hat{\varphi}(u)e^{-2\pi i (\xi -\omega)u} \ dt\ d\xi\\ &= e^{2\pi i
+ \omega x}\int_\xi \hat{f}(\xi) \hat{\bar{\varphi}}(\xi - \omega) e^{-2\pi
+ i \xi x}d\xi\\
+ &= e^{2\pi i \omega x} S_{\hat{\varphi}} \hat{f}(\omega, -x).
+\end{align}
+% printbibliography
+\end{document}
diff --git a/appl_ana/sesh8/main.toc b/appl_ana/sesh8/main.toc
@@ -0,0 +1,5 @@
+\contentsline {section}{\numberline {1}Sheet 8}{1}{section.1}%
+\contentsline {subsection}{\numberline {1.1}Finite Discrete Fourier Transform (FDFT)}{1}{subsection.1.1}%
+\contentsline {subsection}{\numberline {1.2}More FDFT}{2}{subsection.1.2}%
+\contentsline {subsection}{\numberline {1.3}Sampling Sinusoids}{2}{subsection.1.3}%
+\contentsline {subsection}{\numberline {1.4}Short-Time Fourier Transform (STFT)}{3}{subsection.1.4}%
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