## 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)
    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

    -->
    quit

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)

    SOL =
        x: [[1 14] double]
        y: [[1 14] double]
        xe: []
        ye: []
        ie: []
        solver: ['generic_ode_solver']
        interpolant: {[1 1] function pointer array }
        idata: [[1 1] struct array]
    -->
    quit

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.