[?] Set an endpoint to NDSolve?

Questions
1

I’m trying to tell NDSolve to stop at a specific point when a variable reaches a certain value. Here is a minimal example

ode1 = {x'[t] == -1, x[0] == 1, y'[t] + y[t] x[t]^2 == 100, y[0] == 10}; sol = NDSolve[ode1, {x, y}, {t, 0, tf}];

I need tf to be the point when x reaches zero.

2

Hi,

Does WhenEvent work for you?

ode1 = {x'[t] == -1, x[0] == 1, y'[t] + y[t] x[t]^2 == 100,  y[0] == 10, WhenEvent[x[t] == 0, "StopIntegration"]}; 
sol1 = NDSolve[ode1, {x, y}, {t, 0, 3}];

If you look at the resulting InterpolatingFunction, it lives on the domain from 0 to 1.

Cheers,

Marco

3

Thanks for your answer. But how do I know the value of t for x[t] == 0? Also I tried a plot and it looks like x[t] continues beyond x=0 which is not what I’d excpect

ode1 = {x'[t] == -1, x[0] == 1, y'[t] + y[t] x[t]^2 == 100, 
   y[0] == 10, WhenEvent[x[t] == 0, "StopIntegration"]};
sol1 = NDSolve[ode1, {x, y}, {t, 0, 10}]; Plot[{x[t] /. sol1}, {t, 0, 
  10}]
4

Well, you said

“stop at a specific point when a variable reaches a certain value”

so you probably need to know the value? Or a condition that needs to be fulfilled to stop the integration?

And yes, x[t] will continue, but that is because you get an interpolating function that will extrapolate outside the known interval.

Best wishes,

Marco

5

Ammar,

You can print the result. Also, As Marco points out, Since Plot extrapolates beyond the end of your integration (Evaluate sol1 to see this) , you can save the end value and use it in the plot.

I would do:

ode1 = {x'[t] == -1, x[0] == 1, y'[t] + y[t] x[t]^2 == 100, 
   y[0] == 10, 
   WhenEvent[x[t] == 0, "StopIntegration"; 
    Print["EndTime: ", stoptime = t]]};
sol1 = NDSolve[ode1, {x, y}, {t, 0, 10}]; Plot[{x[t] /. sol1}, {t, 0, 
  stoptime}]

Regards,

Neil

6

Another way would be to do this:

ode1 = {x'[t] == -1, x[0] == 1, y'[t] + y[t] x[t]^2 == 100, 
y[0] == 10, WhenEvent[x[t] == 0, "StopIntegration"]}; sol = NDSolve[ode1, {x, y}, {t, 0, 3}];

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
ParametricPlot[{x[t], y[t]} /. sol, Evaluate[Flatten[{t, InterpolatingFunctionDomain[x /. sol[[1]]][[1]]}]], 
AspectRatio -> 1]

Cheers,

Marco

7

Many thanks. Your answers were really helpful, “WhenEvent” does it nicely :slight_smile: