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@@ -10,13 +10,17 @@
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\chapter{Theoretical background}
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\label{chap:background}
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+This chapter is set to introduce the theoretical knowledge on which the work in this thesis is founded on. First we talk about domain mathematics and differential equations.
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+In these sections both the explanations and the approach are strongly based on the book on analysis by W. Rudin\cite{Rudin2007} and the book about ordinary differential equations
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+by M. Tenenbaum and H. Pollard. %\cite{Tenenbaum1985}. %TODO introduce other sections
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+
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% -------------------------------------------------------------------
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-\section{Domain Mathematics}
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+\section{Domain Mathematics and Functions}
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\label{sec:domain}
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-The mathematics of domains work with spaces of numbers on which several mathematical operations are performed to retrieve a number in a different space.
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-For the definition of a function is essential. Which is described by Rudin\cite{Rudin2007}. A function $f$ assigns each element $x$ of the set $A$ to an
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+% describe the meaning of a domain and codomain space to its problem and then shivvy to the function that translates from one to another.
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+When looking at problems from a mathematical it is possible to describe it by putting it into a system. Then the base A function $f$ assigns each element $x$ of the set $A$ to an
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element of the set $B$.
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\begin{equation}
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f: A \rightarrow B
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@@ -32,8 +36,26 @@ Functions are able to describe the condition of a system given certain parameter
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\section{Basics of Differential Equations}
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\label{sec:differentialEq}
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-- different information about the domain (domain -> codomain) -> (domain -> change of the funcion)
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-- example
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+In the real world states of system are under constant change. While functions are able to show the state of a system for a certain set of parameters that
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+are living in the domain space, they can only indirectly give information about the change of the system under different sets of input. This shows the
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+need of a way to retrieve the information of change from a function. One way would be taking the rate of change across a certain interval $[a, b]\subseteq\mathbb{R}$ of
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+the domain of a function $f$, by calculating
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+\begin{equation}
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+ m = \frac{f(b) - f(a)}{a-b}
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+\end{equation}
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+$m$ is the average rate of change across the interval $[a, b]$. Since in most cases we want to find the rate of change in a specific spot and the average will
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+not be sufficient. For this reason instead of looking across the whole interval we single out every $x\in[a, b]$. We narrow the interval down to be infinitesimal small
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+and then calculate the average rate of change
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+\begin{equation} \label{eqn:differential}
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+ \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x}
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+\end{equation}
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+where this value exists. $\frac{df}{dx}$ is the rate of the change or derivative of the function $f$ undergoes in respect of its parameter $x$. This now is able to give information about
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+the rate of change for a specific set of parameters. By re-iterating this process it is possible for us to calculate the rate of the rate of change as well, which is called the derivative of the second
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+order. Equation \ref{eqn:differential} shows how to theoretically come from the function to its assigned derivative, but in many cases (applications) differential equations are built from
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+semantics and logics. For this thesis we would like to concentrate on ordinary differential equations, which have only one input parameter.
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+In our case this is $t$ the point of time. \\
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+
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+For illustrating the functionality of a derivative we will look upon the specific problem. For this we
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% -------------------------------------------------------------------
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Phillip Rothenbeck