ODE with input a symbolic matrix

Hello All,

I wrote the following function using syms variables and its output (i.e. Sxd) is a system of 12 equations contained inside a symbolic matrix.

function Sxd=S1(t, X)
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x1d x2d x3d x4d x5d x6d x7d x8d x9d x10d x11d x12d x1d x2d k1 k2 k3 k4 V0 u dfdx dfdth Sx Sxd

x1=X(1);
x2=X(2);
x3=X(3);
x4=X(4);
x5=X(5);
x6=X(6);
x7=X(7);
x8=X(8);
x9=X(9);
x10=X(10);
x11=X(11);
x12=X(12);

x1d=k1*x2;
x2d=(u-x2)/(k2+k3*x1);

dfdx=[diff(x1d, x1) diff(x1d, x2); diff(x2d, x1) diff(x2d, x2)];

dfdth=[diff(x1d, k1) diff(x1d, k2) diff(x1d, k3) diff(x1d, k4) diff(x1d, V0) 0; 
    diff(x2d, k1) diff(x2d, k2) diff(x2d, k3) diff(x2d, k4) diff(x2d, V0) 0];

Sx = [x3 x5 x7 x9 x11 x1; x4 x6 x8 x10 x12 x2];

Sxd=dfdx*Sx+dfdth

end

Now I would like to pass this matrix to an ODE solver to solve it over time.

Actually, I am passing to the solver (ODE45):

[t1, sol1] = ode45('S1', tspan1, IC1);

where S1 is the above fn, tspan1 is the time span and IC1 is a vector containing the 12 IC (which are all 0s).

By the way, it doesn't work. I think this is due to the output format of the function (i.e. a symbolic matrix)

How can I solve this issue?

Thank you so much in advance,

Gabri

[Star Strider]

It is not going to work anyway in the form you have it.

To begin with, the first assignment after the syms call must be:

X = sym('X',[1 12]);

Then, it must evaluate that returns a column vector, as a column vector of first-degree differential equations.

When you have that, you can use matlabFunction to create an anonymous function that you can use in the MATLAB numeric differential equation integration functions.

I have no idea what you want to do, so I cannot even begin to put it in the correct form. You must do that.

Answer 1

[Walter Roberson]

If you

syms t
syms X [1 12]

and then apply the body of your function, then Sxd will come out as a 2 x 6 array. However, the output from the ode function must be a column vector.

Symbolic calculations are inefficient. Let's see if we can create an appropriate numeric function:

>> matlabFunction(Sxd(:), 'vars', {t, X})
Error using sym/matlabFunction>checkVarsSubset (line 236)
Free variables 'k1,k2,k3,u' must be included in 'Vars' value.

The answer is NO. Your calculation includes the symbolic variables k1, k2, k3, and u, none of which are defined and none of which are parameters to the function. Your output is inherently symbolic, and that is not permitted for the ode*() series of function.

[Gabriele Galli]

Thank you both Star and Walter for your reply.

Now, I managed to obtain a symbolic column vector of equations. By the way, I cannot pass it to a solver since it looks the solver (ODE45) wants as input a float value and not a symbolic one.

Do you know if there is a way to convert my symbolic column vector into a numeric one?

Thank you!

[madhan ravi]

 k1 = 1; % an example
 k2 = 2;
 k3 = 3;

[Walter Roberson]

If your k1, k2, k3, and u were all numeric, then you could use the commands shown in the first example in the documentation for odeFunction in order to create an anonymous function that could be passed to ode45 or similar.

However, as things stand, those are symbolic variables, not numeric, and none of the ode*() routines can handle that.

You could potentially use dsolve() to get a symbolic solution.