Reformulation of the Fourier transformed discretized differential equations
Author:
Behrooz Compani-Tabrizi
Journal:
Quart. Appl. Math. 44 (1986), 529-537
MSC:
Primary 65D25; Secondary 65T05
DOI:
https://doi.org/10.1090/qam/860903
MathSciNet review:
860903
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: 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.
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
E. Butkov, Mathematical Physics, Addison-Wesley (1968)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
65D25,
65T05
Retrieve articles in all journals
with MSC:
65D25,
65T05
Additional Information
Article copyright:
© Copyright 1986
American Mathematical Society