Reformulation of the Fourier transformed discretized differential equations

The conventional rule for differentiation by Fourier transformation of a discretized function, namely, multiplication of the Fourier transform of the function by $ik$ and a subsequent inverse Fourier transformation, was shown to be a first-order approximation to more complete rules [1]. The $ik$ differentiation formula is an exact relation for a nondiscretized function with a continuum as its independent variable. However, this rule fails, particularly when applied to nonsmooth discretized functions of high-frequency content. A consequence of this result is that Fourier-transformed differential equations must be transformed into the Fourier space not by the $ik$ differentiation rule, but by the appropriate correct differentiation rule if the differential equation is eventually to be solved through discretization. Helmholtz’s equation is discussed, and the solution to the simple diffusion equation is presented using the exact and the approximate differentiation formulas and a high-frequency initial field.