MATLAB: Optimizing parameters in ODE

differential equations

Hi I have a set of experimental data. I want to fit this experimental data to first order differential equation of the form dy/dt = -a*n-b*n^2-c*n^3 to optimize the value of constants a,b and c. Can anyone help in this regards? I am new to matlab as this question might be too simple for others. Thanks in advance.

Best Answer

This is a simple, separable differential equation that you can likely solve by hand.
Using the Symbolic Math Toolbox: 
syms a b c n y(t) y0 
DEqn = diff(y) == -a*n-b*n^2-c*n^3;
Eqn = dsolve(DEqn, y(0)==y0)
fcn = matlabFunction(Eqn, 'Vars',{[a,b,c],t,n,y0})
produces: 
Eqn =
y0 - t*(c*n^3 + b*n^2 + a*n)
 
fcn =
  function_handle with value:
    @(in1,t,n,y0) y0-t.*(in1(:,1).*n+in1(:,2).*n.^2+in1(:,3).*n.^3)
or more conveniently: 
    fcn = @(in1,t,n,y0) y0-t.*(in1(:,1).*n+in1(:,2).*n.^2+in1(:,3).*n.^3);
with ‘in1’ corresponding to [a,b,c] in that order. Supply values for ‘n’ and ‘y0’, then present it to the nonlinear parameter estimation function of your choice as: 
    objfcn = @(in1,t) fcn(in1,t,n,y0)
Or, since it is ‘linear in the parameters’ you can re-write it as a design matrix and use linear methods such as mldivide,\ to solve it as well.
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