dsolve events

Joe Riel — Thu, 31 Mar 2011 08:55:05 Z

You need to reformulate the equation for a(t) so that it is not in terms of a(t). The dsolve command would normally do that, but the piecewise hinders it. You also need to replace time with t. The following should do what you want

m := 5; F0 := 10; a0 := 5;
deqs := { m*a(t) = piecewise(F0/m*t < a0, F0*t, 0)
          , v(t) = diff(x(t), t)
          , a(t) = diff(v(t), t)
        }:
ics := {x(0) = 0, v(0) = 0};
integ := dsolve( deqs union ics, numeric):

plots:-odeplot(integ, [t,x(t)], 0..3);

Note that for the units to be consistent, F0 has dimensions of force/time.

Followup

I see that does not work past t=2.5. One way around that is to use dsolve event handling. While this can be a bit complicated, in this case we can handle it simply by using a discrete variable (delta) whose value changes from 1 to 0 when F0/m*t-a crosses zero. A better model might pay attention to the direction---not an issue here with the given equations and initial conditions.

deqs := { m*a(t) = F0*t*delta(t)
          , v(t) = diff(x(t), t)
          , a(t) = diff(v(t), t)
        }:

ics := {x(0) = 0, v(0) = 0, delta(0)=1};

# Use a single event that is triggered when the acceleration
# reaches the threshold.  At that time, change delta to 0
# which causes the effective acceleration to go to 0.
evts := [[F0/m*t-a0=0, delta(t)=0]]:

integ := dsolve( deqs union ics, 'numeric'
                 , 'events' = evts
                 , 'discrete_variables' = [delta(t)::float]
               ):

plots:-odeplot(integ, [t,v(t)], 0..3);

dsolve events

Macrohard — Thu, 31 Mar 2011 18:39:27 Z

Thank you. I'll try to resolve this problem.

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dsolve events

dsolve events