Defining a system of equations

In a previous post, a system of three ODE is set as

f = @(t,x) [-x(1)*x(2);x(1)*x(2)-x(2);x(2)]

But I want to define each equation in a separate line and then put all of the three together. Please advise.

[Cesar García Echeverry]

An option could be call a function:

function f=system(x)
f=[-x(1)*x(2);
    x(1)*x(2)-x(2);
    x(2)];
%Other option
% f(1)=-x(1)*x(2);
% f(2)=x(1)*x(2)-x(2);
% f(3)=x(2);
end

Then, in other .m

clc
close all


F=@(X) system (X);

[alpedhuez]

First, can one define function within LiveScript?

Second,

F=@(X) system (X);

But how about time t?

[Cesar García Echeverry]

Do you going to use a set of differential equations? or just a system of non-lienar equations?

Otherwise

function f=system(x,t)
f=[-x(1)*x(2);
    x(1)*x(2)-x(2);
    x(2)];
%Other option
% f(1)=-x(1)*x(2);
% f(2)=x(1)*x(2)-x(2);
% f(3)=x(2);
end

[alpedhuez]

Will run a set of ODEs here. So it will involve t.

[Cesar García Echeverry]

ok call

function f=system(t,x)
f=[-x(1)*x(2);
    x(1)*x(2)-x(2);
    x(2)];
%Other option
% f(1)=-x(1)*x(2);
% f(2)=x(1)*x(2)-x(2);
% f(3)=x(2);
end

in other .m

clear;
clc;
close all

tic

%initial conditions
X0=[5; 5; 6];

%Step size
tf=30;
spam=[0 tf];
% dt=0.01;

%Método ODE (RK4)
options=odeset('RelTol',1e-4,'abstol',1e-4*ones(1,3));
[t,var]=ode45('system',spam,X0,options);

toc

Answer 1

[Ameer Hamza]

You can define it as an anonymous function. Creating a new file is not necessary.

First method

f = @(t,x) [-x(1)*x(2);
            x(1)*x(2)-x(2);
            x(2)];

Second method

f1 = @(t,x) -x(1)*x(2);
f2 = @(t,x) x(1)*x(2)-x(2);
f3 = @(t,x) x(2);

f = @(t,x) [f1(t,x);
            f2(t,x);
            f3(t,x)];

Both are equivalent.

[alpedhuez]

Thank you.

[alpedhuez]

Just to know, what does @(t,x) do? I mean, what does Matlab do when they read @(t,x)?

[Ameer Hamza]

It is a way to specify the input argument of an anonymous function. It is equivalent to t and x in the following line

function y = f(t, x)

If you are familiar with other programming languages, e.g., python, they are similar to lambda functions. See more about anonymous functions here: https://www.mathworks.com/help/matlab/matlab_prog/anonymous-functions.html

[alpedhuez]

I see. This is a data type. Thank you.

[Andrew Newell]

f1, f2, f3 don't need to have t as an argument, since it isn't used.

[alpedhuez]

I would say f1, f2, f3 are defined over x and time t in the dynamical system

[Ameer Hamza]

Yes, it is not necessary to use 't' in f1, f2, and f3. But it makes the code consistent and more readable.

[alpedhuez]

so you mean

f1 = @(x) -x(1)*x(2);
f2 = @(x) x(1)*x(2)-x(2);
f3 = @(x) x(2);
f = @(t,x) [f1(x);
            f2(x);
            f3(x)];        

?

[Ameer Hamza]

alpedhuez, yes. This is also equivalent to the options in my MATLAB.

[alpedhuez]

So this data type @ allows to extend the domain of a function, which is reasonable.