I have solved a set of three coupled odes in MATLAB. I have to plot another function which is a function of one of these odes. How this can be done?

I was solving the coupled rate equations of a laser by using ODE for obtaining the fluctuations of carrier density, photon number and phase. Now I have to plot a new function for frequency fluctuation which is a function of the ode for phase. Should I include the new function in the function used for defining the set of ODES or just ouside the function?.

The function defined for coupled rate equations of the laser is:

function dy = rate_eq(t,y)

%ode of carrier density

dy(1) = (I/q) - y(1)/te - (g*(y(1)-n0)/(1+(eps*y(2))))*y(2)+((((2*y(1)/te)*(1/delt))^0.5)*1.8339)-((((2*Betasp*y(1)*y(2))/te)*(1/delt)^0.5)*0.5377);

%ode for photon number

dy(2) = (g*(y(1)-n0)/(1+(eps*y(2))))*y(2)- y(2)/tp + Betasp*y(1)/te+((((2*Betasp*y(1)*y(2)/te)*(1/delt))^0.5)*0.5377) ;

%ode for phase

dy(3) = ((a/2)*g)*(y(1)-n_bar)+ (((Betasp*y(1)/2*te*y(2))*(1/delt))^0.5)*(-2.2588);

end

%The frequency fluctuation(deln(t)) is related to the phase ode as

deln=(1/2*pi)*dy(3)

I have to plot deln vs time graph. Kindly help on this

[Askic V]

Please have a lookt at this example:

https://www.mathworks.com/matlabcentral/answers/638120-help-using-ode45-to-solve-3rd-order-ode

Answer 1

[Davide Masiello]

If none of the differential equations you are solving depends on deln, then you can compute it after solving the ODE system.

To retrieve the value of the derivative dy(3), you can use the MatLab function deval.

[Rakhi]

Thank you for the answer Davide. It helped

Answer 2

[Sam Chak]

Hi @Rakhi

Two methods are shown in the example. The 1st method is by logical intuition. If deln depends dy(3), and the equation for dy(3) is known, then it's a direct Plug-and-Plot approach.

The 2nd method employs the deval() function that evaluates differential equation solution structure and extracts the 1st derivative of the numeric solution produced by the ode45 solver.

%% Method 1: manually inserting the equation for dy(3)
[t, y] = ode45(@rate_eq, [0 3], [1 0 3]);

figure(1), subplot(211)
plot(t, y, 'linewidth', 1.5), grid on, legend('show', 'location', 'best'), title('Method 1')
subplot(212)
a     = 1;
g     = 1;
n_bar = 1;
Betasp= 1;
te    = 1;
delt  = 1;
dy3   = ((a/2)*g)*(y(:,1) - n_bar) + (((Betasp*y(:,1)/2*te.*y(:,2))*(1/delt)).^0.5)*(-2.2588);
deln1 = (1/2*pi)*(dy3);
plot(t, deln1, 'color', '#5f86dc', 'linewidth', 1.5), grid on, legend('deln', 'location', 'best'), xlabel('t')

%% Method 2: using deval() function
sol      = ode45(@rate_eq, [0 3], [1 0 3]);
tt       = linspace(0, 3, 31);
[yo, yp] = deval(sol, tt);   % yo is y-out, yp is y-prime (y')

figure(2), subplot(211)
plot(tt, yo, 'linewidth', 1.5), grid on, legend('show', 'location', 'best'), title('Method 2')
subplot(212)
deln2 = (1/2*pi)*yp(3,:);
plot(tt, deln2, 'color', '#5f86dc', 'linewidth', 1.5), grid on, legend('deln', 'location', 'best'), xlabel('t')

function dy = rate_eq(t, y)
    I     = 1;
    q     = 1;
    te    = 1;
    g     = 1;
    n0    = 1;
    delt  = 1;
    Betasp= 1;
    tp    = 1;
    a     = 1;
    n_bar = 1;
    dy    = zeros(3, 1);
    %ode of carrier density
    dy(1) = (I/q) - y(1)/te - (g*(y(1) - n0)/(1 + (eps*y(2))))*y(2) + ((((2*y(1)/te)*(1/delt))^0.5)*1.8339) - ((((2*Betasp*y(1)*y(2))/te)*(1/delt)^0.5)*0.5377);
    %ode for photon number 
    dy(2) = (g*(y(1) - n0)/(1 + (eps*y(2))))*y(2) - y(2)/tp + Betasp*y(1)/te + ((((2*Betasp*y(1)*y(2)/te)*(1/delt))^0.5)*0.5377);
    %ode for phase
    dy(3) = ((a/2)*g)*(y(1) - n_bar) + (((Betasp*y(1)/2*te*y(2))*(1/delt))^0.5)*(-2.2588);
end

[Rakhi]

I am able to plot the graphs now. Thank you for the help Sam.