ODE45 Numerical Solution of ODEs
Section: Numerical Methods
Usage
function [t,y] = ode45(f,tspan,y0,options,varargin) function SOL = ode45(f,tspan,y0,options,varargin) ode45 is a solver for ordinary differential equations and initial value problems. To solve the ODE
y'(t) = f(t,y)
y(0) = y0
over the interval tspan=[t0 t1], you can use ode45. For example, to solve the ode y' = y y(0) = 1 whose exact solution is y(t)=exp(t), over the interval t0=0, t1=3, do
--> [t,y]=ode45(@(t,y) y,[0 3],1)
Warning: Newly defined variable error shadows a function of the same name. Use clear error to recover access to the function
k =
2
y =
1.0000 1.0030
t =
1.0e-03 *
0 3.0000
k =
3
y =
1.0000 1.0030 1.0182
t =
1.0e-02 *
0 0.3000 1.8000
k =
4
y =
1.0000 1.0030 1.0182 1.0975
t =
1.0e-02 *
0 0.3000 1.8000 9.3000
k =
5
y =
1.0000 1.0030 1.0182 1.0975 1.4814
t =
0 0.0030 0.0180 0.0930 0.3930
k =
6
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930
k =
7
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930
k =
8
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
k =
9
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 9
4.9185
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 9
1.5930
k =
10
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 10
4.9185 6.6392
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 10
1.5930 1.8930
k =
11
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 11
4.9185 6.6392 8.9620
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 11
1.5930 1.8930 2.1930
k =
12
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 12
4.9185 6.6392 8.9620 12.0975
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 12
1.5930 1.8930 2.1930 2.4930
k =
13
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 13
4.9185 6.6392 8.9620 12.0975 16.3299
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 13
1.5930 1.8930 2.1930 2.4930 2.7930
k =
14
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 14
4.9185 6.6392 8.9620 12.0975 16.3299 20.0854
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 14
1.5930 1.8930 2.1930 2.4930 2.7930 3.0000
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 14
1.5930 1.8930 2.1930 2.4930 2.7930 3.0000
y =
1.0000
1.0030
1.0182
1.0975
1.4814
1.9997
2.6993
3.6437
4.9185
6.6392
8.9620
12.0975
16.3299
20.0854
If you want a dense output (i.e., an output that also contains an interpolating spline), use instead
--> SOL=ode45(@(t,y) y,[0 3],1)
Warning: Newly defined variable error shadows a function of the same name. Use clear error to recover access to the function
k =
2
y =
1.0000 1.0030
t =
1.0e-03 *
0 3.0000
k =
3
y =
1.0000 1.0030 1.0182
t =
1.0e-02 *
0 0.3000 1.8000
k =
4
y =
1.0000 1.0030 1.0182 1.0975
t =
1.0e-02 *
0 0.3000 1.8000 9.3000
k =
5
y =
1.0000 1.0030 1.0182 1.0975 1.4814
t =
0 0.0030 0.0180 0.0930 0.3930
k =
6
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930
k =
7
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930
k =
8
y =
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
t =
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
k =
9
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 9
4.9185
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 9
1.5930
k =
10
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 10
4.9185 6.6392
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 10
1.5930 1.8930
k =
11
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 11
4.9185 6.6392 8.9620
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 11
1.5930 1.8930 2.1930
k =
12
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 12
4.9185 6.6392 8.9620 12.0975
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 12
1.5930 1.8930 2.1930 2.4930
k =
13
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 13
4.9185 6.6392 8.9620 12.0975 16.3299
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 13
1.5930 1.8930 2.1930 2.4930 2.7930
k =
14
y =
Columns 1 to 8
1.0000 1.0030 1.0182 1.0975 1.4814 1.9997 2.6993 3.6437
Columns 9 to 14
4.9185 6.6392 8.9620 12.0975 16.3299 20.0854
t =
Columns 1 to 8
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930 1.2930
Columns 9 to 14
1.5930 1.8930 2.1930 2.4930 2.7930 3.0000
SOL =
x: 1 14 double array
y: 1 14 double array
xe:
ye:
ie:
solver: generic_ode_solver
interpolant: 1 1 functionpointer array
idata: 1 1 struct array
You can view the result using
plot(0:0.01:3,deval(SOL,0:0.01:3))
You will notice that this function is available for "every" value of t, while plot(t,y,'o-') is only available at a few points. The optional argument 'options' is a structure. It may contain any of the following fields: 'AbsTol' - Absolute tolerance, default is 1e-6. 'RelTol' - Relative tolerance, default is 1e-3. 'MaxStep' - Maximum step size, default is (tspan(2)-tspan(1))/10 'InitialStep' - Initial step size, default is maxstep/100 'Stepper' - To override the default Fehlberg integrator 'Events' - To provide an event function 'Projection' - To provide a projection function The varargin is ignored by this function, but is passed to all your callbacks, i.e., f, the event function and the projection function. ==Event Function== The event function can be used to detect situations where the integrator should stop, possibly because the right-hand-side has changed, because of a collision, etc... An event function should look like function [val,isterminal,direction]=event(t,y,...) The return values are: val - the value of the event function. isterminal - whether or not this event should cause termination of the integrator. direction - 1=upcrossings only matter, -1=downcrossings only, 0=both. == Projection function == For geometric integration, you can provide a projection function which will be called after each time step. The projection function has the following signature: function yn=project(t,yn,...); If the output yn is very different from the input yn, the quality of interpolation may decrease.