Change input at each time step of the ODE solver 'ode45'

I am not sure how to change an input parameter 'β' at each time step. My code is below - which gives me an error. Can anybody help please!

t = [7 14 21 28 35 42 49 56 63 70 77 84];
for i=1:12;
 beta(i) = 0.43e-08 + (4.28e-08 - 0.43e-08)*exp(-0.20*t(i)); 
end

f = @(t,x) [3494-0.054*x(1)-beta*x(1)*x(3); beta*x(1)*x(3) - 0.41*x(2); ...
            50000*x(2) - 23*x(3)];
[t,xa1] = ode45(f,t,[64700 0 0.0033],beta);

[Jan]

And the error message is:

Error using vertcat
Dimensions of matrices being concatenated are not consistent.
Error in @(t,x)[3494-0.054*x(1)-beta*x(1)*x(3);beta*x(1)*x(3)-0.41*x(2);50000*x(2)-23*x(3)]

Answer 1

[Jan]

Please consider, that Matlab's ODE integrators cannot handle non-smooth functions sufficiently. See http://www.mathworks.com/matlabcentral/answers/59582#answer_72047 .

The only reliable method to run the integration is a loop over the time intervals:

function yourIntegration

tResult = [];
xResult = [];
tStep = [7 14 21 28 35 42 49 56 63 70 77 84];
x0    = [64700 0 0.0033];
for index = 2:numel(tStep)
  % Integrate:
  beta = 0.43e-08 + (4.28e-08 - 0.43e-08) * exp(-0.20*t(index - 1))
  af   = @(t,x) f(t, x, beta);
  t    = tStep(index-1:index);
  [t, x] = ode45(af, t, x0);

    % Collect the results:
    tResult = cat(1, tResult, t);
    xResult = cat(1, xResult, x);

    % Final value of x is initial value for next step:
    x0 = x(end, :);
  end

function dx = f(t,x, beta)
dx = [3494-0.054*x(1)-beta*x(1)*x(3); ...
      beta*x(1)*x(3) - 0.41*x(2); ...
      50000*x(2) - 23*x(3)];

[KC]

Thank you Jan!

[sam]

Trying to run this code, I encountered the error: Undefined function or variable "t".

Error in yourIntegration (line 8) beta = 0.43e-08 + (4.28e-08 - 0.43e-08) * exp(-0.20*t(index - 1))

Could you provide a correct version of the code?

[Walter Roberson]

In that line replace t with tStep

[sam]

Thanks walter.

[sam]

@Jan Simon

Hi Jan,

How to do integration inside the ode45 "h" intervals as well? I know how to do integration in my defined interval, but the ode45 will treat the variable as constant when running its own intervals. Thank you.

[Saiprasad Gore]

Thanks a lot, I had a similar problem. I wanted to switch the eqn depending on condition after every step. I hope this will work in my case too. Can you tell me how to give ode45 just 1 step without intermediate adaptive steps?

[Jan]

@Saiprasad Gore: This is not possible with ode45.

Answer 2

[Walter Roberson]

f = @(T,x) [3494-0.054*x(1)-interp1(t,beta,T,'linear','extrap')*x(1)*x(3); interp1(t,beta,T,'linear','extrap')*x(1)*x(3) - 0.41*x(2); ...
          50000*x(2) - 23*x(3)];

[KC]

Thanks Walter!

[sam]

@Walter Roberson

Hi Walter,

Why do we have to do interpolation if we already know the exact expression of the variables? Couldnt we just input the exact expression of the variables into the Matlab ode45 solver? If we could, could you kindly tell me how to do this? Thanks.