Generalized Lagrange functions and weighting coefficient formulae for the harmonic differential quadrature method

Abstract

SUMMARYIn the harmonic di erential 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 coe cients. In the spirit of the di erential 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 coe cients of higher order derivatives can be evaluated recursively and the general explicit formulae are given in this paper. The di erential 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 satisÿed automatically and no other boundary conditions are required in general. It is also shown that the harmonic di erential quadrature method is related to the trigonometric collocation method and the harmonic balance method. Numerical examples are given to illustrate the validity and e ciency of the present method.