Fourier representation

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 $\exp{({-\dfc k^2t})}$: $$u^n_q = A^n e^{ikq\Delta x} = A^ne^{ikx}$$