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<TITLE>GNU Octave - Differential Equations</TITLE>
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<P><HR><P>
<H1><A NAME="SEC151" HREF="octave_toc.html#TOC151">Differential Equations</A></H1>
<P>
Octave has two built-in functions for solving differential equations.
Both are based on reliable ODE solvers written in Fortran.
</P>
<P>
<A NAME="IDX748"></A>
<A NAME="IDX749"></A>
<A NAME="IDX750"></A>
</P>
<H2><A NAME="SEC152" HREF="octave_toc.html#TOC152">Ordinary Differential Equations</A></H2>
<P>
The function <CODE>lsode</CODE> can be used Solve ODEs of the form
</P>
<PRE>
dx
-- = f (x, t)
dt
</PRE>
<P>
using Hindmarsh's ODE solver LSODE.
</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <B>lsode</B> <I>(<VAR>fcn</VAR>, <VAR>x0</VAR>, <VAR>t</VAR>, <VAR>t_crit</VAR>)</I>
<DD><A NAME="IDX751"></A>
Return a matrix of <VAR>x</VAR> as a function of <VAR>t</VAR>, given the initial
state of the system <VAR>x0</VAR>. Each row in the result matrix corresponds
to one of the elements in the vector <VAR>t</VAR>. The first element of
<VAR>t</VAR> corresponds to the initial state <VAR>x0</VAR>, so that the first row
of the output is <VAR>x0</VAR>.
</P>
<P>
The first argument, <VAR>fcn</VAR>, is a string that names the function to
call to compute the vector of right hand sides for the set of equations.
It must have the form
</P>
<PRE>
<VAR>xdot</VAR> = f (<VAR>x</VAR>, <VAR>t</VAR>)
</PRE>
<P>
where <VAR>xdot</VAR> and <VAR>x</VAR> are vectors and <VAR>t</VAR> is a scalar.
</P>
<P>
The fourth argument is optional, and may be used to specify a set of
times that the ODE solver should not integrate past. It is useful for
avoiding difficulties with singularities and points where there is a
discontinuity in the derivative.
</DL>
</P>
<P>
Here is an example of solving a set of three differential equations using
<CODE>lsode</CODE>. Given the function
</P>
<P>
<A NAME="IDX752"></A>
</P>
<PRE>
function xdot = f (x, t)
xdot = zeros (3,1);
xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \
- 8.375e-06*x(1)^2);
xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
xdot(3) = 0.161*(x(1) - x(3));
endfunction
</PRE>
<P>
and the initial condition <CODE>x0 = [ 4; 1.1; 4 ]</CODE>, the set of
equations can be integrated using the command
</P>
<PRE>
t = linspace (0, 500, 1000);
y = lsode ("f", x0, t);
</PRE>
<P>
If you try this, you will see that the value of the result changes
dramatically between <VAR>t</VAR> = 0 and 5, and again around <VAR>t</VAR> = 305.
A more efficient set of output points might be
</P>
<PRE>
t = [0, logspace (-1, log10(303), 150), \
logspace (log10(304), log10(500), 150)];
</PRE>
<P>
<DL>
<DT><U>Loadable Function:</U> <B>lsode_options</B> <I>(<VAR>opt</VAR>, <VAR>val</VAR>)</I>
<DD><A NAME="IDX753"></A>
When called with two arguments, this function allows you set options
parameters for the function <CODE>lsode</CODE>. Given one argument,
<CODE>lsode_options</CODE> returns the value of the corresponding option. If
no arguments are supplied, the names of all the available options and
their current values are displayed.
</DL>
</P>
<P>
See Alan C. Hindmarsh, <CITE>ODEPACK, A Systematized Collection of ODE
Solvers</CITE>, in Scientific Computing, R. S. Stepleman, editor, (1983) for
more information about the inner workings of <CODE>lsode</CODE>.
</P>
<H2><A NAME="SEC153" HREF="octave_toc.html#TOC153">Differential-Algebraic Equations</A></H2>
<P>
The function <CODE>dassl</CODE> can be used Solve DAEs of the form
</P>
<PRE>
0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0
</PRE>
<P>
using Petzold's DAE solver DASSL.
</P>
<P>
<DL>
<DT><U>Loadable Function:</U> [<VAR>x</VAR>, <VAR>xdot</VAR>] = <B>dassl</B> <I>(<VAR>fcn</VAR>, <VAR>x0</VAR>, <VAR>xdot0</VAR>, <VAR>t</VAR>, <VAR>t_crit</VAR>)</I>
<DD><A NAME="IDX754"></A>
Return a matrix of states and their first derivatives with respect to
<VAR>t</VAR>. Each row in the result matrices correspond to one of the
elements in the vector <VAR>t</VAR>. The first element of <VAR>t</VAR>
corresponds to the initial state <VAR>x0</VAR> and derivative <VAR>xdot0</VAR>, so
that the first row of the output <VAR>x</VAR> is <VAR>x0</VAR> and the first row
of the output <VAR>xdot</VAR> is <VAR>xdot0</VAR>.
</P>
<P>
The first argument, <VAR>fcn</VAR>, is a string that names the function to
call to compute the vector of residuals for the set of equations.
It must have the form
</P>
<PRE>
<VAR>res</VAR> = f (<VAR>x</VAR>, <VAR>xdot</VAR>, <VAR>t</VAR>)
</PRE>
<P>
where <VAR>x</VAR>, <VAR>xdot</VAR>, and <VAR>res</VAR> are vectors, and <VAR>t</VAR> is a
scalar.
</P>
<P>
The second and third arguments to <CODE>dassl</CODE> specify the initial
condition of the states and their derivatives, and the fourth argument
specifies a vector of output times at which the solution is desired,
including the time corresponding to the initial condition.
</P>
<P>
The set of initial states and derivatives are not strictly required to
be consistent. In practice, however, DASSL is not very good at
determining a consistent set for you, so it is best if you ensure that
the initial values result in the function evaluating to zero.
</P>
<P>
The fifth argument is optional, and may be used to specify a set of
times that the DAE solver should not integrate past. It is useful for
avoiding difficulties with singularities and points where there is a
discontinuity in the derivative.
</DL>
</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <B>dassl_options</B> <I>(<VAR>opt</VAR>, <VAR>val</VAR>)</I>
<DD><A NAME="IDX755"></A>
When called with two arguments, this function allows you set options
parameters for the function <CODE>lsode</CODE>. Given one argument,
<CODE>dassl_options</CODE> returns the value of the corresponding option. If
no arguments are supplied, the names of all the available options and
their current values are displayed.
</DL>
</P>
<P>
See K. E. Brenan, et al., <CITE>Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations</CITE>, North-Holland (1989) for
more information about the implementation of DASSL.
</P>
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