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Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions
Author(s) -
Reutskiy S. Y.,
Chen C. S.
Publication year - 2006
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1679
Subject(s) - mathematics , partial differential equation , chebyshev polynomials , chebyshev nodes , polynomial , approximation theory , chebyshev equation , trigonometric functions , mathematical analysis , orthogonal polynomials , classical orthogonal polynomials , geometry
A two‐stage numerical procedure using Chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation. The purpose of such numerical schemes is crucial for the evaluation of particular solutions of a large class of partial differential equations. Our proposed scheme provides a highly efficient and accurate approximation of multivariate functions and particular solution of certain partial differential equations simultaneously. Numerical results on the approximation of eight two‐dimensional test functions and their derivatives are given. To demonstrate that the scheme for the approximation of functions can be easily extended to evaluate the particular solution of certain partial differential equations, we solve a modified Helmholtz equation. Near machine precision can be achieved for all these test problems. Copyright © 2006 John Wiley & Sons, Ltd.