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On Fourier series for higher order (partial) derivatives of functions
Author(s) -
Sun Weiming,
Zhang Zimao
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4133
Subject(s) - mathematics , fourier series , fourier sine and cosine series , conjugate fourier series , fourier transform , series (stratigraphy) , mathematical analysis , fourier analysis , partial derivative , order (exchange) , discrete fourier series , partial differential equation , fourier inversion theorem , constant coefficients , fractional fourier transform , short time fourier transform , paleontology , finance , economics , biology
This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2 r th order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.