Defining 2N ODEs with 2N variables when using ode45

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William 2023-6-23

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https://ww2.mathworks.cn/matlabcentral/answers/1987478-defining-2n-odes-with-2n-variables-when-using-ode45

编辑： Torsten 2023-6-23

I am new to Matlab and am using it to complete a university project. I am trying to solve a system of 2N ODEs for the position of N vortices in a 2D fluid. They interact with each other and have a circulation strenth Γ, and each vortex has a location given by the

and

coordinate for each index (See below the system of ODEs). I am able to set up the equations explicitly with a couple of 'for' loops but am struggling to figure out how to define each equation when it is in a 'function' environment and without a nested 'for' loop.

Each vortex has an initial position for x,y which I have each stored in 2 Nx1 vectors and I want to define each differential equation in terms of x and y as shown in the screenshot below (note the prime on the sum denotes an ommision of the α= β term). I also am trying to use ode45 to solve these and then plot the output which I am able to do, it's just the defining of the ODEs where I am stuck.

Hope this makes sense and thanks in advance for any help.

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William 2023-6-23

Thanks I have sorted it now. Also what do you mean by painfully? Do you mean by way of the for loops?

Torsten 2023-6-23

编辑：Torsten 2023-6-23

在 MATLAB Online 中打开

I meant that ODE45 spent so much effort computing the solution and you just carelessly overwrite it :-)

function dzdt = vortex_pos(t,z)
  N = numel(z)/2;
  x = z(1:N);
  y = z(N+1:end);
  dzdt = zeros(2*N,1);
  %% ODEs
  for i = 1:N
    for j = setdiff(1:N, i)
        dzdt(i) = dzdt(i) + (-1/(2*pi))*(gamma(j)*(y(i)-y(j))/((x(i)-x(j))^2 + (y(i)-y(j))^2));
        dzdt(N+i) = dzdt(N+i) + (1/(2*pi))*(gamma(j)*(x(i)-x(j))/((x(i)-x(j))^2 + (y(i)-y(j))^2));
    end
  end
end