MATLAB: Solve first-order ordinary differential equations

first-order ordinary differential equationsode45

Hello,
I need to solve the following system of differential equations:
where V(x,y) is a matrix that does not necessarily have a determined value for the integration points (I believe that it is necessary to interpolate to generate such points).
I implemented the following code, but I do not trust the results. I expected a greater variation in the phi angle. Could someone please help validate my code?
function [time, x, y, phi] = my_solution
    load('Data.mat') % attached file
    
    x = linspace(0,50e-3,500);
    y = linspace(0,50e-3,500);
    [Xm, Ym] = meshgrid(x,y); % grid
    
    % Vm is the matrix of velocitys
    grad_xm = gradient(Vm,1); % partial derivative with respect to x
    grad_ym = gradient(Vm,2); % partial derivative with respect to y
    
    % [S] = [s(1) s(2) s(3)]' = [x y phi]'
    % sdot = [S'] = [x' y' phi']' = [V(x,y)cos(phi)   V(x,y)sin(phi)   sin(phi)*grad_xm(x,y)-cos(phi)*grad_ym(x,y)]'
    % functions for interpolation at the integration points.
    V = @(t, s) interp2(Xm,Ym,Vm,s(1),s(2));
    grad_x = @(t, s) interp2(Xm,Ym,grad_xm,s(1),s(2));
    grad_y = @(t, s) interp2(Xm,Ym,grad_ym,s(1),s(2));
    
    sdot = @(t,s)[V(t,s)*cos(s(3)); ...
        V(t,s)*sin(s(3)); ...
        sin(s(3))*grad_x(t,s) - cos(s(3))*grad_y(t,s)];
    
    tspan = [0,100e-6];
    IC = [20e-3 0 pi/3]
    
    [time, state_values] = ode45(sdot,tspan,IC);
    
    x = state_values(:,1);
    y = state_values(:,2);
    phi = state_values(:,3);
    
    figure
    subplot(311)
    plot(time,x)
    ylabel('x')
    subplot(312)
    plot(time,y)
    ylabel('y')
    subplot(313)
    plot(time,phi*180/pi)
    ylabel('\phi')
    xlabel('Time')
end

Best Answer

Your calculation of grad_xm and grad_ym looks peculiar to me. Since you seem to use x and y for the coordinates of Vm, it seems like you'd want to calculate the gradients like this:
[grad_xm,grad_ym] = gradient(Vm,x,y);
If I use that then I get values of phi that varies between 60 and 45 degrees.
Your calculation of grad_xm and grad_ym uses the spacing of 1 and 2 respectively for the spatial coordinates when you intend to have 50e3/499 as spacing.
HTH
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