MATLAB: ODE45 system of equations inaccurate

Question

So I have a system of equations that I have to solve and plot for 3 separate concentrations C1,C2,C3. The initial conditions are c1(0) = .063, c2(0) = 700, and c3(0) = 5.6

There is no way that the graphs I'm getting with this are right. I know the equations are all right but did I miss something or do something incorrectly? I attached an image of the equations.

a = 1;
b = (2*(10^-3));
d = (4*(10^-4));
f = (4*(10^-8));
c1(1) = .063;
c2(1) = 700;
c3(1) = 5.6;
c1prime = @(x1,y1) (-a*(c1^3)) + (b*(c3)) + (d*(c3^4)) - f*(c2^4)*(c1^4);
c2prime = @(x2,y2) (d*(c3^4)) - f*(c2^4)*(c1^4);
c3prime = @(x3,y3) (a*(c1^3)) - (b*c3) - (d*(c3^4)) + f*(c2^4)*(c1^4);
options = odeset('RelTol',10^-7,'AbsTol',10^-9);
[C1 Y1] = ode45(c1prime,[0 5],.063,options);
[C2 Y2] = ode45(c2prime,[0 5],700,options);
[C3 Y3] = ode45(c3prime,[0 5],5.6,options);
plot(C1,Y1,'r',C2,Y2,'b',C3,Y3,'g')

Answer 1

The error you made was in not putting them all in the same system. You did code them correctly otherwise, so all I needed to do was to copy your code to the ‘cprime’ matrix I created, subscript the concentrations, and provide the correct initial conditions vector. You have to plot them using the subplot function or you won’t be able to see the changes.

This works:

a = 1;
b = (2E-3);
d = (4E-4);
f = (4E-8);
C10 = .063;
C20 = 700;
C30 = 5.6;
cprime = @(t,C) [(-a*(C(1).^3)) + (b*(C(3))) + (d*(C(3).^4)) - f*(C(2).^4).*(C(1).^4);  (d*(C(3).^4)) - f*(C(2).^4).*(C(1).^4);  (a*(C(1).^3)) - (b*C(3)) - (d*(C(3).^4)) + f*(C(2).^4).*(C(1).^4)]
options = odeset('RelTol',10^-7,'AbsTol',10^-9);
[X,C] = ode45(cprime, [0 1], [C10; C20; C30], options);
figure(1)
subplot(3,1,1)
plot(X, C(:,1))
ylabel('\itC_1\rm')
grid
subplot(3,1,2)
plot(X, C(:,2))
ylabel('\itC_2\rm')
grid
subplot(3,1,3)
plot(X, C(:,3))
grid
xlabel('\itx\rm')
ylabel('\itC_3\rm')

I experimented with ode15s as well. It gives the same results as ode45.