I have tried Runge Kutta method on this coupled nonlinear ode and get the error respect to the matrix dimension. Please help me with this.

% Define spatial domain
Ly = 2;             % Length of domain 
Lz = 1;
N = 200;           % Number of discretization points
M = 200;
dy = Ly/N;
dz = Lz/M;
y = -1; % Define x domain % for fixed values of y
z = 0:dz:Lz;
[Y,Z] = meshgrid(y,z);
% Define discrete wavenumbers
eta = (2*pi/Ly)*[-N/2:N/2];
eta = fftshift(eta');    % Re-order fft wavenumbers
jeta = (2*pi/Lz)*[-M/2:M/2];
jeta = fftshift(jeta');    % Re-order fft wavenumbers

[Eta,Jeta] = meshgrid(eta,jeta);
%% Time domain %%
t(1) = 0; dt = 0.05; tf = 10; L = ceil(tf/dt)+1;

dt = 0.0050

tsol = 0

%% function handle %%
f = @(t,F,G) (-g1.*J-F.*((F).*(2.*pi.*1i.*jeta))-((nu+nu_t).*(jeta.^2+eta.^2).*4.*pi.^2)-(nu_t.*4.*pi.^2.*jeta.*eta)-(G.*2.*pi.*1i.*eta));
 
g = @(t,F,G) (-g1.*J.*P-G.*((G).*(2.*pi.*1i.*eta))-((nu+nu_t).*(jeta.^2+eta.^2).*4.*pi.^2)-(nu_t.*4.*pi.^2.*jeta.*eta)-(F.*2.*pi.*1i.*jeta));



%% Initial condition %%
F(1) = 0; G(1) = 0;


%% Main calculation %%

 
 for i = 1:L+1
     t(i+1) = t(i)+dt
     
     k1F = f(t(i),      F(i),           G(i));
     k1G = g(t(i),      F(i),           G(i));
     k2F = f(t(i)+dt./2,F(i)+dt./2.*k1F,G(i)+dt./2.*k1G);
     k2G = g(t(i)+dt./2,F(i)+dt./2.*k1F,G(i)+dt./2.*k1G);
     k3F = f(t(i)+dt./2,F(i)+dt./2.*k2F,G(i)+dt./2.*k2G);
     k3G = g(t(i)+dt./2,F(i)+dt./2.*k2F,G(i)+dt./2.*k2G);
     k4F = f(t(i)+dt,   F(i)+dt.*k3F,   G(i)+dt.*k3G);
     k4G = g(t(i)+dt,   F(i)+dt.*k3F,   G(i)+dt.*k3G);
     F(i+1) = F(i) + (dt./6).*(k1F + 2.*k2F + 2.*k3F + k4F);
     G(i+1) = G(i) + (dt./6).*(k1G + 2.*k2G + 2.*k3G + k4G)
 end

t = 1×201

0 0.0050 0.6000 0.9000 1.2000 1.5000 1.8000 2.1000 2.4000 2.7000 3.0000 3.3000 3.6000 3.9000 4.2000 4.5000 4.8000 5.1000 5.4000 5.7000 6.0000 6.3000 6.6000 6.9000 7.2000 7.5000 7.8000 8.1000 8.4000 8.7000

Unable to perform assignment because the left and right sides have a different number of elements.

[Torsten]

We can't check your code because several variables in the definition of f and g are undefined.