The method of particular solutions using trigonometric basis functions

Abstract

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provide another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method.

Method of Approximate Particular Solutions of Electro-Kinetic Fluids in Micro and Nano channels

“…Durante los últimos años, el MAPS ha sido empleado ampliamente en diversas investigaciones para resolver problemas elípticos como en Zhu [8], problemas de convección-difusión como en Jiang et al [9] (ver también [10] y [11]), así mismo Tian et al [12] utilizan funciones de bases radial trigonométricas para la solución de EDPs y problemas a nivel local Yao et al [13] y [14] y Nath et al en [15]. Es importante tener en cuenta que el esquema desarrollado por Chen et al en [1] se basa en soluciones particulares de Poisson para aproximar las EPDs.…”

Influence of magnetic fields on simultaneous stationary solutions of two-dimensional sudden expansion channel flow at low Re m

“…The authors show that as the polynomial order increases, the error decreases to certain value limited by the ill-conditioning of the solution matrix. Tian et al (2018) employ trigonometric basis function for solving similar problems, showing that it is possible to reduce ill-conditioning issues related to the use of RBF, Chebyshev polynomial and high-order polynomial based particular solution. According to the above, particular solution method have being improved in terms of basis function, ill-conditioning and adjust parameters in order to widen their applicability to three-dimensional and multi-physical problems.…”