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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: In this paper, the mixed-boundary value problem for a typical strip is reduced to a singular integral equation, which is solved numerically by the use of Fourier series techniques, and a check of the accuracy of the computations is provided by the balance of rates of energies.
99 citations
99 citations
TL;DR: In this paper, clean analytical solutions of Richards' equation for three-dimensional unsaturated groundwater flow were derived for finite difference/volume/element computer programs using architectures ranging from PCs to parallel high-performance computers.
Abstract: [1] This technical note derives clean analytical solutions of Richards' equation for three-dimensional unsaturated groundwater flow. Clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional Fourier series terms. Two-dimensional versions of these solutions are also given. The primary purpose for the solutions is to test linear and nonlinear solvers in finite difference/volume/element computer programs for accuracy and scalability using architectures ranging from PCs to parallel high-performance computers. This derivation starts from the quasi-linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content. This allows a transformation to be used to create a linear partial differential equation. Separation of variables and Fourier series are then used to obtain the final solution. Physically reasonable material properties are also used. A total of four solutions are given in this technical note (steady state and transient solutions for two different boundary conditions of the sample problem).
99 citations
TL;DR: In this article, the Fourier expansion-based diAerential quadrature (FDQ) was applied to solve the Helmholtz eigenvalue problems with periodic and non-periodic boundary conditions.
Abstract: SUMMARY Based on the same concept as generalized diAerential quadrature (GDQ), the method of Fourier expansionbased diAerential quadrature (FDQ) was developed and then applied to solve the Helmholtz eigenvalue problems with periodic and non-periodic boundary conditions. In FDQ, the solution of a partial diAerential equation is approximated by a Fourier series expansion. The details of the FDQ method and its implementation to sample problems are shown in this paper. It was found that the FDQ results are very accurate for the Helmholtz eigenvalue problems even though very few grid points are used. #1997 John Wiley & Sons, Ltd.
99 citations
TL;DR: A detailed description of spectral multigrid methods, which includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems, is provided.
99 citations