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Finite‐difference approximation for the u ( k ) ‐derivative with O ( h M − k +1 ) accuracy: An analytical expression
Author(s) -
Dubovsky Vadim,
Yakhot Alexander
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20139
Subject(s) - mathematics , taylor series , inverse , partial derivative , mathematical analysis , combinatorics , function (biology) , series (stratigraphy) , derivative (finance) , matrix (chemical analysis) , mathematical physics , geometry , chemistry , paleontology , chromatography , evolutionary biology , financial economics , economics , biology
An approximation of function u ( x ) as a Taylor series expansion about a point x 0 at M points x i , ∼ i = 1,2,…, M is used where x i are arbitrary‐spaced. This approximation is a linear system for the derivatives u ( k ) with an arbitrary accuracy. An analytical expression for the inverse matrix A −1 where A = [ A ik ] = ${1\over k!}$ ( x i − x 0 ) k is found. A finite‐difference approximation of derivatives u ( k ) of a given function u ( x ) at point x 0 is derived in terms of the values u ( x i ). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006