A spatially uniform source changes only the zero Fourier mode. Its exact integrated increment over a time step beginning at is
(1)
The unnormalised two-dimensional discrete Fourier transform stores a constant field as in element (1,1).[ref] Keep the existing homogeneous exponential multiplication and add this source increment to that element:
for i = 1:nt
    t = (i-1)*dt;
    u_hat = u_hat .* exp(-alpha*K2*dt);
    dq = exp(-t/10) - exp(-(t+dt)/10);
    u_hat(1,1) = u_hat(1,1) + nx*ny*dq;
end
This replacement uses the original variable names and MATLAB indexing. The uniform forcing of a periodic heat equation is integrated exactly in time here. No source multiplier is applied to the other modes, and the zero-mode homogeneous multiplier equals one. Adding only dq to the unnormalised coefficient would underestimate the physical temperature increment by the factor .

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