Reformulation of the Fourier transformed discretized differential equations
Abstract: The conventional rule for differentiation by Fourier transformation of a discretized function, namely, multiplication of the Fourier transform of the function by and a subsequent inverse Fourier transformation, was shown to be a first-order approximation to more complete rules . The 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 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.
-  Behrooz Compani-Tabrizi and Richard G. Geyer, Differentiation by Fourier transformation and its connection with differentiation by finite differencing, Quart. Appl. Math. 44 (1986), no. 3, 519–528. MR 860902, https://doi.org/10.1090/S0033-569X-1986-0860902-4
-  E. Butkov, Mathematical Physics, Addison-Wesley (1968)
- B. Compani-Tabrizi and R. G. Geyer, Differentiation by Fourier transformation and its connection with differentiation by finite differencing, Research Bulletin B85-2R or Technical Report TR85-15R. Marathon Oil Company, Research Center, Littleton, Colorado; Quart. Appl. Math. This issue MR 860902
- E. Butkov, Mathematical Physics, Addison-Wesley (1968)