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Represent \( I(x) \) as a Fourier series $$ I(x) \approx \sum_{k\in K} b_k e^{ikx} $$
The corresponding sum for \( u \) is $$ u(x,t) \approx \sum_{k\in K} b_k e^{-\dfc k^2t}e^{ikx} $$
Such solutions are also accepted by the numerical schemes, but with an amplification factor \( A \) different from \( \exp{({-\dfc k^2t})} \): $$ u^n_q = A^n e^{ikq\Delta x} = A^ne^{ikx} $$
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