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发表于 2004-8-24 21:08:24
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< >在matlab6.5中没有ode15这个函数，</P><>有ode15s，ode23，ode45，ode23s等</P><>ODE15S Solve stiff differential equations and DAEs, variable order method.
[T,Y] = ODE15S(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates the
system of differential equations y' = f(t,y) from time T0 to TFINAL with
initial conditions Y0. Function ODEFUN(T,Y) must return a column vector
corresponding to f(t,y). Each row in the solution array Y corresponds to
a time returned in the column vector T. To obtain solutions at specific
times T0,T1,...,TFINAL (all increasing or all decreasing), use
TSPAN = [T0 T1 ... TFINAL].
[T,Y] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
integration properties replaced by values in OPTIONS, an argument created
with the ODESET function. See ODESET for details. Commonly used options
are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
of absolute error tolerances 'AbsTol' (all components 1e-6 by default).
[T,Y] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS,P1,P2...) passes the additional
parameters P1,P2,... to the ODE function as ODEFUN(T,Y,P1,P2...), and to
all functions specified in OPTIONS. Use OPTIONS = [] as a place holder if
no options are set.
The Jacobian matrix df/dy is critical to reliability and efficiency. Use
ODESET to set 'Jacobian' to a function FJAC if FJAC(T,Y) returns the
Jacobian df/dy or to the matrix df/dy if the Jacobian is constant. If the
'Jacobian' option is not set (the default), df/dy is approximated by
finite differences. Set 'Vectorized' 'on' if the ODE function is coded so
that ODEFUN(T,[Y1 Y2 ...]) returns [ODEFUN(T,Y1) ODEFUN(T,Y2) ...]. If
df/dy is a sparse matrix, set 'JPattern' to the sparsity pattern of
df/dy, i.e., a sparse matrix S with S(i,j) = 1 if component i of f(t,y)
depends on component j of y, and 0 otherwise.
ODE15S can solve problems M(t,y)*y' = f(t,y) with mass matrix M(t,y). Use
ODESET to set the 'Mass' property to a function MASS if MASS(T,Y) returns
the value of the mass matrix. If the mass matrix is constant, the matrix
can be used as the value of the 'Mass' option. Problems with
state-dependent mass matrices are more difficult. If the mass matrix does
not depend on the state variable Y and the function MASS is to be called
with one input argument T, set 'MStateDependence' to 'none'. If the mass
matrix depends weakly on Y, set 'MStateDependence' to 'weak' (the
default) and otherwise, to 'strong'. In either case the function MASS is
to be called with the two arguments (T,Y). If there are many differential
equations, it is important to exploit sparsity: Return a sparse
M(t,y). Either supply the sparsity pattern of df/dy using the 'JPattern'
property or a sparse df/dy using the Jacobian property. For strongly
state-dependent M(t,y), set 'MvPattern' to a sparse matrix S with S(i,j)
= 1 if for any k, the (i,k) component of M(t,y) depends on component j of
y, and 0 otherwise.
If the mass matrix is non-singular, the solution of the problem is
straightforward. See examples FEM1ODE, FEM2ODE, BATONODE, or
BURGERSODE. If M(t0,y0) is singular, the problem is a differential-
algebraic equation (DAE). ODE15S solves DAEs of index 1. DAEs have
solutions only when y0 is consistent, i.e., there is a yp0 such that
M(t0,y0)*yp0 = f(t0,y0). Use ODESET to set 'MassSingular' to 'yes', 'no',
or 'maybe'. The default of 'maybe' causes ODE15S to test whether M(t0,y0)
is singular. You can provide yp0 as the value of the 'InitialSlope'
property. The default is the zero vector. If y0 and yp0 are not
consistent, ODE15S treats them as guesses, tries to compute consistent
values close to the guesses, and then goes on to solve the problem. See
examples HB1DAE or AMP1DAE.
[T,Y,TE,YE,IE] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS...) with the 'Events'
property in OPTIONS set to a function EVENTS, solves as above while also
finding where functions of (T,Y), called event functions, are zero. For
each function you specify whether the integration is to terminate at a
zero and whether the direction of the zero crossing matters. These are
the three vectors returned by EVENTS: [VALUE,ISTERMINAL,DIRECTION] =
EVENTS(T,Y). For the I-th event function: VALUE(I) is the value of the
function, ISTERMINAL(I)=1 if the integration is to terminate at a zero of
this event function and 0 otherwise. DIRECTION(I)=0 if all zeros are to
be computed (the default), +1 if only zeros where the event function is
increasing, and -1 if only zeros where the event function is
decreasing. Output TE is a column vector of times at which events occur.
Rows of YE are the corresponding solutions, and indices in vector IE
specify which event occurred.
SOL = ODE15S(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
used with DEVAL to evaluate the solution at any point between T0 and
TFINAL. The steps chosen by ODE15S are returned in a row vector SOL.x.
For each I, the column SOL.y(:,I) contains the solution at SOL.x(I).
If events were detected, SOL.xe is a row vector of points at which events
occurred. Columns of SOL.ye are the corresponding solutions, and indices
in vector SOL.ie specify which event occurred. If a terminal event has
been detected, SOL.x(end) contains the end of the step at which the event
occurred. The exact point of the event is reported in SOL.xe(end).
Example
[t,y]=ode15s(@vdp1000,[0 3000],[2 0]);
plot(t,y(:,1));
solves the system y' = vdp1000(t,y), using the default relative error
tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
component, and plots the first component of the solution.
See also
other ODE solvers: ODE23S, ODE23T, ODE23TB, ODE45, ODE23, ODE113
options handling: ODESET, ODEGET
output functions: ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT
evaluating solution: DEVAL
ODE examples: VDPODE, FEM1ODE, BRUSSODE, HB1DAE
NOTE:
The interpretation of the first input argument of the ODE solvers and
some properties available through ODESET have changed in this version
of MATLAB. Although we still support the v5 syntax, any new
functionality is available only with the new syntax. To see the v5 help
type in the command line
more on, type ode15s, more off
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发表于 2004-8-24 06:42:46