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<div class="title">ODE45 Numerical Solution of ODEs </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_num.html">Numerical Methods</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>function [t,y] = ode45(f,tspan,y0,options,varargin) function SOL = ode45(f,tspan,y0,options,varargin)</p>
<p>ode45 is a solver for ordinary differential equations and initial value problems. To solve the ODE </p>
<pre class="fragment"> y'(t) = f(t,y)
y(0) = y0
</pre><p> over the interval tspan=[t0 t1], you can use ode45. For example, to solve the ode </p>
<pre class="fragment"> y' = y
y(0) = 1
</pre><p>whose exact solution is y(t)=exp(t), over the interval t0=0, t1=3, do</p>
<pre class="fragment">--> [t,y]=ode45(@(t,y) y,[0 3],1)
t =
Columns 1 to 7
0 0.0030 0.0180 0.0930 0.3930 0.6930 0.9930
Columns 8 to 14
1.2930 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
</pre><p>If you want a dense output (i.e., an output that also contains an interpolating spline), use instead</p>
<pre class="fragment">--> SOL=ode45(@(t,y) y,[0 3],1)
SOL =
x: 1x14 double array
y: 1x14 double array
xe:
ye:
ie:
solver: generic_ode_solver
interpolant: 1x1 functionpointer array
idata: 1x1 struct array
</pre><p>You can view the result using </p>
<pre class="fragment"> plot(0:0.01:3,deval(SOL,0:0.01:3))
</pre><p> You will notice that this function is available for "every" value of t, while </p>
<pre class="fragment"> plot(t,y,'o-')
</pre><p>is only available at a few points.</p>
<p>The optional argument 'options' is a structure. It may contain any of the following fields:</p>
<p>'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</p>
<p>The varargin is ignored by this function, but is passed to all your callbacks, i.e., f, the event function and the projection function.</p>
<p>==Event Function==</p>
<p>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...</p>
<p>An event function should look like</p>
<p>function [val,isterminal,direction]=event(t,y,...)</p>
<p>The return values are:</p>
<p>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.</p>
<p>== Projection function ==</p>
<p>For geometric integration, you can provide a projection function which will be called after each time step. The projection function has the following signature: </p>
<pre class="fragment">function yn=project(t,yn,...);
</pre><p>If the output yn is very different from the input yn, the quality of interpolation may decrease. </p>
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