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Generalized Lagrange functions and weighting coefficient formulae for the harmonic differential quadrature method
Author(s) -
Fung T. C.
Publication year - 2003
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.692
Subject(s) - mathematics , gauss–kronrod quadrature formula , tanh sinh quadrature , fourier series , clenshaw–curtis quadrature , mathematical analysis , quadrature (astronomy) , trigonometric interpolation , gauss–jacobi quadrature , lagrange polynomial , trigonometric substitution , collocation method , boundary value problem , trigonometric polynomial , trigonometric integral , differential equation , nyström method , trigonometry , gaussian quadrature , polynomial interpolation , ordinary differential equation , linear interpolation , polynomial , electrical engineering , bicubic interpolation , engineering
In the harmonic differential quadrature method, truncated Fourier series comprising the trigonometric functions are used to approximate the solutions. The generalized Lagrange functions composed of trigonometric functions are constructed as interpolation functions so that the unknowns are the function values, rather than the Fourier coefficients. In the spirit of the differential quadrature method, the derivatives at a sampling grid point are expressed as weighted linear sums of function values at all the sampling grid points. It is shown that the corresponding weighting coefficients of higher order derivatives can be evaluated recursively and the general explicit formulae are given in this paper. The differential quadrature analog of the governing equations can then be established easily at the sampling grid points. For the periodic boundary value problems, the periodic boundary conditions are satisfied automatically and no other boundary conditions are required in general. It is also shown that the harmonic differential quadrature method is related to the trigonometric collocation method and the harmonic balance method. Numerical examples are given to illustrate the validity and efficiency of the present method. Copyright © 2003 John Wiley & Sons, Ltd.