How to run an ODE Function with a variable outside of the Differential Equation

Hi,

I am trying to solve and model an ODE function. The working code is provided and works exactly as I need, but now I want to vary the I term. I want to vary I so that it changes accordingly to the x. So for each x value to be a different I that is used. Something like the second code I provided, but it does not work.

The working code is provided:

function plotGammaComparison()
clc
clear all %#ok<CLALL> 

figure
hold on;

R0 = .3;
L = 2*pi*R0; 
Q = 0.01;
I = 0.0006;
x = 0:0.001:L; 

gammas = [100, 10, .1, .01, .001];

for i = 1:length(gammas)
    gamma = gammas(i);
    fun = @(y, s) [gamma*Q/(I*s(1)) + 1; sin(s(1)); cos(s(1))];
    tspan = x;
    Y0 = [pi/2 0 0];
    [x, y] = ode45(fun, tspan, Y0);
    X = y(:, 2);
    Y = y(:, 3);
    plot(X, Y, 'DisplayName', sprintf('\\gamma = %.4f', gamma));
end

xlabel(' ');
ylabel(' ');
legend('Location', 'Best');
hold off;

end

The attempted code:

clc
clear all %#ok<CLALL> 

figure
hold on;

R0 = .3;
L = 2*pi*R0; 
Q = 0.01;
x = 0:0.001:L; 
I = -4e8 * x + 4e10;
gammas = [100, 10, .1, .01, .001];

for i = 1:length(gammas)
    gamma = gammas(i);
    fun = @(y, s) [gamma*Q/(I*s(1)) + 1; sin(s(1)); cos(s(1))];
    tspan = x;
    Y0 = [pi/2 0 0];
    [x, y] = ode45(fun, tspan, Y0);
    X = y(:, 2);
    Y = y(:, 3);
    plot(X, Y, 'DisplayName', sprintf('\\gamma = %.4f', gamma));
end

xlabel(' ');
ylabel(' ');
legend('Location', 'Best');
hold off;

Answer 1

[Sam Chak]

@Adam, Do you expect the results as shown?

figure
hold on;

R0 = .3;
L = 2*pi*R0; 
Q = 0.01;
x = 0:0.001:L; 
% I = - 4e8*x + 4e10;
gammas = [100, 10, .1, .01, .001];

for i = 1:length(gammas)
    gamma   = gammas(i);
    fun     = @(y, s) [gamma*Q/((- 4e8*y + 4e10)*s(1)) + 1; % I = - 4e8*x + 4e10;
                       sin(s(1)); 
                       cos(s(1))];
    tspan   = x;    % if x equals to y...
    Y0      = [pi/2 0 0];
    [x, y]  = ode45(fun, tspan, Y0);    % Confusing! Are x = y and y = s?
    X       = y(:, 2);
    Y       = y(:, 3);
    plot(X, Y, 'DisplayName', sprintf('\\gamma = %.4f', gamma));
end

xlabel(' ');
ylabel(' ');
legend('Location', 'Best');
hold off;

[Adam]

Kind of. Yes but different gamma values should change the contour to be smaller or larger. It seems off that they are all the same value.

With the old code, that is about what I expect with various gammas

[Star Strider]

hold

[Sam Chak]

In Case 1, the value of I is 0.0006. However, in Case 2, the value of I is extremely large. Additionally, I kindly request you to verify if the parameter I has been correctly entered in the differential equation. There seem to be conflicting symbols as indicated in the commented sections of the code. Please review and ensure accuracy.

%% Case 2: Varying I
R0  = .3;
L   = 2*pi*R0; 
Q   = 0.01;
x   = 0:0.001:L; 
I   = -4e8*x + 4e10;
plot(x, I), grid on

[Adam]

So yes, x=y and y=s in that case. Apologies on my end.

And yes that is exactly the trend of the I as I expect. That should be entered correctly. But i do see the issue lays with the I magnitude now and overshadows the other variables. Your edits were absolutely perfect and I understand that I just needed to integrate the I function into the ODE.

thank you so much for the help!

[Sam Chak]

Good to hear that, @Adam. Try figuring out the reasonable range of I to obtain the desired results. If you find the solution helpful, please consider clicking 'Accept' ✔ on the answer and voting 👍 for it.