@c Copyright (C) 1996 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Differential Equations, Optimization, Nonlinear Equations, Top
@chapter Differential Equations

Octave has two built-in functions for solving differential equations.
Both are based on reliable ODE solvers written in Fortran.

@menu
* Ordinary Differential Equations::  
* Differential-Algebraic Equations::  
@end menu

@cindex Differential Equations
@cindex ODE
@cindex DAE

@node Ordinary Differential Equations, Differential-Algebraic Equations, Differential Equations, Differential Equations
@section Ordinary Differential Equations

@findex lsode
The function @code{lsode} can be used Solve ODEs of the form
@iftex
@tex
$$
 {dx\over dt} = f (x, t)
$$
@end tex
@end iftex
@ifinfo

@example
dx
-- = f (x, t)
dt
@end example
@end ifinfo

@noindent
using Hindmarsh's ODE solver LSODE.

@example
lsode (@var{fcn}, @var{x0}, @var{t_out}, @var{t_crit})
@end example

The first argument is the name of the function to call to
compute the vector of right hand sides.  It must have the form

@example
@var{xdot} = f (@var{x}, @var{t})
@end example

@noindent
where @var{xdot} and @var{x} are vectors and @var{t} is a scalar.

The second argument specifies the initial condition, and the third
specifies a vector of output times at which the solution is desired,
including the time corresponding to the initial condition.

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.

@findex lsode_options
Tolerances and other options for @code{lsode} may be specified using the
function @code{lsode_options}.

Here is an example of solving a set of two differential equations using
@code{lsode}.  The function

@example
function xdot = f (x, t) 

  r = 0.25;
  k = 1.4;
  a = 1.5;
  b = 0.16;
  c = 0.9;
  d = 0.8;

  xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
  xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);

endfunction
@end example

@noindent
is integrated with the command

@example
x = lsode ("f", [1; 2], (t = linspace (0, 50, 200)'));
@end example

@noindent
producing a set of 200 values stored in the variable @var{x}.  Note that
this example takes advantage of the fact that an assignment produces a
value to store the values of the output times in the variable @var{t}
directly in the function call   The results can then be plotted using
the command

@example
plot (t, x)
@end example

See Alan C. Hindmarsh, @cite{ODEPACK, A Systematized Collection of ODE
Solvers}, in Scientific Computing, R. S. Stepleman, editor, (1983) for
more information about this family of ODE solvers.

@node Differential-Algebraic Equations,  , Ordinary Differential Equations, Differential Equations
@section Differential-Algebraic Equations

@findex dassl
The function @code{dassl} can be used Solve DAEs of the form
@iftex
@tex
$$
 0 = f (\dot{x}, x, t), \qquad x(t=0) = x_0, \dot{x}(t=0) = \dot{x}_0
$$
@end tex
@end iftex
@ifinfo

@example
0 = f (x-dot, x, t),    x(t=0) = x_0, x-dot(t=0) = x-dot_0
@end example
@end ifinfo

@example
dassl (@var{fcn}, @var{x_0}, @var{xdot_0}, @var{t_out}, @var{t_crit})
@end example

The first argument is the name of the function to call to
compute the vector of residuals.  It must have the form

@example
@var{res} = f (@var{x}, @var{xdot}, @var{t})
@end example

@noindent
where @var{x}, @var{xdot}, and @var{res} are vectors, and @var{t} is a
scalar.

The second and third arguments to @code{dassl} 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.

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.

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.

@findex dassl_options
Tolerances and other options for @code{dassl} may be specified using the
function @code{dassl_options}.

See K. E. Brenan, et al., @cite{Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations}, North-Holland (1989) for
more information about the implementation of DASSL.