summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
authorPuneeth Chaganti2009-11-20 00:36:58 +0530
committerPuneeth Chaganti2009-11-20 00:36:58 +0530
commit3367a6ddb66ae86c4a52d08ab843bda4cf96e965 (patch)
treec1d90a1ee92021ecb4982ff81878ce048703a777
parente7f852e3347277714f2543d3259923c8a018084c (diff)
downloadworkshops-3367a6ddb66ae86c4a52d08ab843bda4cf96e965.tar.gz
workshops-3367a6ddb66ae86c4a52d08ab843bda4cf96e965.tar.bz2
workshops-3367a6ddb66ae86c4a52d08ab843bda4cf96e965.zip
Added ODE section to cheatsheet6, day1.
-rwxr-xr-xday1/cheatsheet6.tex30
1 files changed, 21 insertions, 9 deletions
diff --git a/day1/cheatsheet6.tex b/day1/cheatsheet6.tex
index 5a9f652..ee988f6 100755
--- a/day1/cheatsheet6.tex
+++ b/day1/cheatsheet6.tex
@@ -15,6 +15,7 @@ showstringspaces=false,
\usepackage{ae,aecompl}
\usepackage{mathpazo,courier,euler}
\usepackage[scaled=.95]{helvet}
+\usepackage{amsmath}
\begin{document}
\date{}
\vspace{-1in}
@@ -27,7 +28,7 @@ Condier following sets of equations:\\
\begin{align*}
3x + 2y - z & = 1 \\
2x - 2y + 4z & = -2 \\
- -x + $\frac{1}{2}$y -z & = 0
+ -x + \frac{1}{2}y -z & = 0
\end{align*}\\
The matrix representation is:\\
\begin{center}
@@ -84,7 +85,7 @@ Out[96]: 0.90689968211710881
\subsection{Roots of non-linear eqations}
For Finding the roots of a non linear equation(defined as $f(x)=0$), around a starting estimate we use \typ{fsolve}:\\
\typ{In []: from scipy.optimize import fsolve}\\
-\typ{fsolve} is not part of \typ{pylab}, instead it is part of \textbf{optimize} package of \textbf{scipy}, and hence we \textbf{import} it.\\
+\typ{fsolve} is not a part of \typ{pylab}, instead is a function in the \textbf{optimize} module of \textbf{scipy}, and hence we \textbf{import} it.\\
%\typ{fsolve} takes first argument as name of function, which evaluates $f(x)$, whose roots one wants to find. And second argument is starting estimate, around which roots are found.
For illustration, we want to find roots of equation:
\begin{center}
@@ -103,16 +104,27 @@ Out[]: -0.66623943249251527
\end{lstlisting}
\section{ODE}
+
+We wish to solve an (a system of) Ordinary Differential Equation. For this purpose, we shall use \typ{odeint}.\\
+\typ{In []: from scipy.integrate import odeint}\\
+\typ{odeint} is a function in the \textbf{integrate} module of \textbf{scipy}.\\
+As an illustration, let us solve the ODE below:\\
+$\frac{dy}{dt} = ky(L-y)$, L = 25000, k = 0.00003, y(0) = 250\\
+We define a function (as below) that takes $y$ and time as arguments and returns the right hand side of the ODE.
\begin{lstlisting}
- In []: def epid(y, t):
+In []: def f(y, t):
.... k, L = 0.00003, 25000
.... return k*y*(L-y)
....
-
- In []: t = arange(0, 12, 0.2)
-
- In []: y = odeint(epid, 250, t)
-
- In []: plot(t, y)
\end{lstlisting}
+Next we define the time over which we wish to solve the ODE. We also note the initial conditions given to us.
+\begin{lstlisting}
+In []: t = linspace(0, 12, 61)
+In []: y0 = 250
+\end{lstlisting}
+To solve the ODE, we call the \typ{odeint} function with three arguments - the function \typ{f}, initial conditions and the time vector.
+\begin{lstlisting}
+In []: y = odeint(f, y0, t)
+\end{lstlisting}
+Note: To solve a system of ODEs, we need to change the function to return the right hand side of all the equations and the system and the pass the required number of initial conditions to the \typ{odeint} function.
\end{document}