How to address initial conditions for coupled system in Matlab - matlab

Question

We have a coupled system of 10 ode each. The coupling presents in the last equation.
I thought about using a matrix 10 by 2 as initial conditions.I also followed a similar question with the same title here, but I still get the same errors ('Too many input arguments.')
time = [0 5];
x1_0 = [0 0 0 0 0 0 0 0 0 0];
x2_0 = [0 0 0 0 0 0 0 0 0 0];
initial = [x1_0;x2_0];
x = NaN(length(initial),2*length(time));
[t,x] = ode45(#ode,time,initial);
function [dxdt] = ode(x)
N = 2;
dxdt = NaN(10,2);
for i = 1:N
dxdt(1,i) = x(6,i);
dxdt(6,i) = (x(3,i)-x(4,i)+x(5,i)) - x(6,i) - x(1,i);
dxdt(2,i) = x(7,i);
dxdt(7,i) = (x(3,i)-x(4,i)+x(5,i)) - x(7,i) - x(2,i);
dxdt(3,i) = x(8,i);
dxdt(8,i) = (x(1,i)) - x(8,i) - x(3,i);
dxdt(4,i) = x(9,i);
dxdt(9,i) = (x(2,i)) - x(9,i) - x(4,i);
dxdt(5,i) = x(10,i);
if i == 1
j = 2;
elseif i == 2
j = 1 ;
end
dxdt(10,i) = (x(3,j)-x(4,j)+x(5,j)) - x(10,i) - x(5,i);
end
end
If my mistake is the use of a matrix initial conditions instead of a vector, using a 1 by 20 vector, and adjust the ode form accordingly would be not practical, I think
- What would be another more efficient way to address the initial conditions
-What is unnecessary from the inputs I have given and why?
-Is there any other computational way to represent the coupling?
EDIT: If I use the option: "function dxdt = ode(t,x)" the errors are the following:
Error in Test11>ode (line 33)
dxdt(10,i) = (x(3,j)-x(4,j)+x(5,j)) - x(10,i) - x(5,i);
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 115)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options,
varargin);
Error in Test11 (line 9)
[t,x] = ode45(#ode,time,initial);
If I change the way I call the function, so that:
"[t,x] = ode45(#(t,x)ode(x), time, initial);"
the results are the following:
Not enough input arguments.
Error in Test11>ode (line 19)
dxdt(1,i) = x(6,i);
Error in Test11>#(t,x)ode(x)
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 115)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options,
varargin);
Error in Test11 (line 9)
[t,x] = ode45(#(t,x)ode(x), time, initial);

Answer 1

The error 'Too many input arguments.' is caused by the fact that your ode function only accepts one variable, x, while ode45 is passing it two variables, time instance t and state x. Even if your ode equations are independent off the current time, you do need to handle this input argument.
Either let your ode function accept two input arguments:
function dxdt = ode(t,x)
% etc
or, change the way you call your ode function:
[t,x] = ode45(#(t,x)ode(x), time, initial);
Furthermore, I don't think ode45 can handle a 'state matrix', so I would indeed concatenate your two state vectors vertical to one large state vector. If your don't want to rewrite your code, add some reshapes before and after the loop:
function [dxdt] = ode(xv)
N = 2;
dxdt = NaN(10,2);
% reshape to two column vectors
x = reshape(xv,[],2); % xv is one large column vector
for ki = 1:N
dxdt(1,ki) = x(6,ki);
dxdt(6,ki) = (x(3,ki)-x(4,ki)+x(5,ki)) - x(6,ki) - x(1,ki);
dxdt(2,ki) = x(7,ki);
dxdt(7,ki) = (x(3,ki)-x(4,ki)+x(5,ki)) - x(7,ki) - x(2,ki);
dxdt(3,ki) = x(8,ki);
dxdt(8,ki) = (x(1,ki)) - x(8,ki) - x(3,ki);
dxdt(4,ki) = x(9,ki);
dxdt(9,ki) = (x(2,ki)) - x(9,ki) - x(4,ki);
dxdt(5,ki) = x(10,ki);
if ki == 1
kj = 2;
elseif ki == 2
kj = 1;
end
dxdt(10,ki) = (x(3,kj)-x(4,kj)+x(5,kj)) - x(10,ki) - x(5,ki);
end
% reshape dxdt to one column vector
dxdt = reshape(dxdt,[],1);
end