Matlab system of differential equations

Question

I have here a little program that I do not know how to tackle, I have previously written a code that would solve for me a system of two differential equations which works fine.

However, I now have to solve a system of three differential equations:

d(alphai)/dt = ui = k * alphai*f(tethai)

d(tethai)/dt = alphai*f(tethai) - (tethai - tethaex)

d(tethaex)/dt = sigma*(Sum(1.N)(tethai - tethaex)) - phi*tethaex

Here are my codes so far but I get some errors:

%Script for differential derivatives specifcation

   function dy = NRateDE(T,Y,p)
   %Extract parameters from P
   u0 = p(1);
   g = p(2);
   k = p(3);
   a = p(4);
   b = p(5);
   N = p(6);

   %Format output as a column vector
   dy = zeros(3,1);

   %Set differential equations
   dy(1) = (u0*exp(-g*T)) - (k*Y(1)*exp(Y(2)));
   dy(2) = Y(1)*exp(Y(2)) - (Y(2)) - Y(3);
A = Sum(Y,N);
dy(3) = (a .* A) - (b .* Y(3));
end

function A = Sum(Y,N)
    for i=1:N
        A = A + (Y(2) - Y(3));
    end
end

And finally here is the main function:

%% Initialize the environment
close all;
clear all;
clc;

%%Define parameters and initial conditions
u0 = 0.12; g = 0; k = 0.09; N = 2; a = 0; b = 0.1; z = 0;A=0; %Parameters
MaxTime = 300;
T = 0:0.01:MaxTime; %integration time
Yinitial = 0; Yminitial = 0; %intial conditions

%%Solve and generate equations
%#Numerical approximation
    for z=1:N
        %Create random numbers to change the value of u0
        a1 = - 0.01;
        b1 = 0.01;
        r = (b1-a1).*rand(1) + a1;
        u0 = u0 +r; %value changes on each iteration
        [T,Y] = ode23s(@NRateDE,[T],[Yminitial Yinitial],[],[u0 g k a b N A]);
    end   
%#Analytical approximation
    tau = 0:0.2:MaxTime;              
    tetha = (u0*exp(-g*tau))/k;
    alpha = u0*exp(-g*tau).*exp(tetha);

With N = 2, I should get two set of matrices of [T,Y] but I have no clue of how should I correct the code. I am fairly new to Matlab so please forgive any silly mistake haha'.

Any idea, hint?

Thank you for your help in advance!

EDIT: I Have slightly changed the first function and the "RateDE" was not intentional. Also here is the error message:

Attempted to access Y(3); index out of bounds because numel(Y)=2.

Error in NRateDE (line 18) dy(2) = Y(1)*exp(Y(2)) - (Y(2)) - Y(3);

Error in odearguments (line 88) f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode15s (line 149) [neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn, ...

Error in RunRateNparticles (line 20) [T,Y] = ode15s(@NRateDE,[T],[Yminitial Yinitial],[],[u0 gkab NA]);

---

Right!

I understand the error messages and have corrected them by adding an element on Y:

[T,Y] = ode23s(@NRateDE,[T],[Yminitial Yinitial Ylinitial],[],[u0 g k a b N A]);

However, I am still unable to solve the set of three differential equations given on the first message.

Here are the current error messages:

Undefined function or variable "A".

Error in NRateDE (line 19) A = A + (Y(2) - Y(3));

Error in odearguments (line 88) f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode23s (line 120) [neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn, ...

Error in RunRateNparticles (line 20) [T,Y] = ode23s(@NRateDE,[T],[Yminitial Yinitial Ylinitial],[],[u0 gkab NA]);

Answer 1

The solution to the original problem is outlined in your own answer already.
Y needs to have 3 elements, so you can just add another initial condition.

The new problem is a missing assignment. The Sum function tries to add something to A , but this variable doesn't exist yet. Fixing your function like this should work:

function A = Sum(Y,N)
    A = 0;
    for i=1:N
        A = A + (Y(2) - Y(3));
    end
end