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Comparisons of method of fundamental solutions, method of particular solutions and the MFS-QR;stability analysis

ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2021年 123卷 182-199页

作者： Zhang, Li-Ping Li, Zi-Cai Huang, Hung-Tsai Lee, Ming-Gong Zhejiang Univ Technol Dept Appl Math Hangzhou 310023 Peoples R China Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan Shou Univ Dept Financial & Computat Math Kaohsiung 84001 Taiwan Chung Hua Univ Dept Leisure & Recreat Management PhD Program Engn Sci Hsinchu 30012 Taiwan

The goals of this paper are twofold: selection of pseudo-boundaries for sources nodes in the method of fundamental solutions (MFS), and comparisons of the MFS, the method of particular solutions (MPS) and the MFS-QR of Antunes. To pursue better pseudo-boundaries, we provide new estimates of the condition number (Cond) by the MFS for arbitrary pseudo-boundaries, and propose a new sensitivity index of stability via accuracy. Numerical experiments and comparisons are carried out to verify the analysis made. For five-pedal-flower-like domains, numerical comparisons are made by the sensitivity index. Circular pseudo-boundaries are optimal for highly smooth solutions, but the pseudo-boundaries near the domain boundary may be better for singular solutions. In this paper the gap has been shortened between theoretical analysis and numerical computation of the MFS, to provide some guidance for users. This is the first goal of this paper. The second goal is to compare the MFS, the MPS and the MFS-QR. Characteristics of the MFS-QR are explored. The new basis functions of the MFS-QR are the very particular solutions (PS), and the MFS-QR may be regarded as a special MPS. The MFS-QR is not a variant of the MFS but a variant of the MPS. The MFS-QR also plays a role in bridging from the MFS to the MPS. Both the MFS and the MPS can also be recognized as twins via the MFS-QR in the Trefftz family. The comparisons in this paper are more comprehensive.

关键词： method of fundamental solutions Selection of pseudo-boundaries The sensitivity index method of particular solutions The MFS-QR Stability analysis Comparisons of different methods Laplace's equations Elliptic coordinates

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The Laplace equation in three dimensions by the method of fundamental solutions and the method of particular solutions

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APPLIED NUMERICAL MATHEMATICS 2020年 154卷 47-69页

作者： Zhang, Li-Ping Li, Zi-Cai Chen, Zhen Huang, Hung-Tsai Zhejiang Univ Technol Dept Appl Math Hangzhou 310023 Peoples R China Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan Guizhou Normal Univ Sch Math Sci Guiyang 550001 Peoples R China I Shou Univ Dept Financial & Computat Math Kaohsiung 84001 Taiwan

For Laplace's equation in a bounded simply-connected domain Omega in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al.[10], the theoretical analysis is much behind (Li [23] only for unit sphere Omega). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial convergence rates can be achieved. Numerical experiments are carried out to support the analysis made, and several useful locations of source nodes are investigated numerically. The analysis in this paper may lay a theoretical basis of the MFS for 3D problems, as Bogomolny [8] and [24] for 2D problems. Besides, the method of particular solutions (MPS) in [26] is also studied by using the spherical harmonic functions (SHF). The optimal polynomial convergence rates and the exponential growth of condition number (Cond) are obtained. The source nodes are located based on the abscissas of quadrature rules on surfaces;they are "grid-like". Since most of 3D problems, in reality, can not be simplified to 2D problems, and since the MFS has more advantages for 3D problems in algorithm simplicity and wide application, the study in this paper is essential and important to the MFS. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： method of fundamental solutions method of particular solutions Spherical harmonic functions 3D problems Laplace's equation Error analysis Choices of source nodes "Grid-like" source nodes

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The method of particular solutions for solving nonlinear Poisson problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2019年第2期77卷 501-513页

作者： Dou, Fangfang Liu, Yanshan Chen, C. S. Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China Univ Southern Mississippi Sch Math & Nat Sci Hattiesburg MS 39406 USA

Based on the recent development in the method of particular solutions, we re-exam three approaches using different basis functions for solving nonlinear Poisson problems. We further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions. We also specify the deficiency of some of these methods and provide necessary remedy. The traditional Picard method is introduced to compare with the recent proposed methods using MATLAB optimization toolbox solver for solving nonlinear Poisson equations. Ranking on these three approaches are given based on the results of numerical experiment. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Polynomial basis functions method of particular solutions Nonlinear equation Picard iteration

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The method of particular solutions using trigonometric basis functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2018年 335卷 20-32页

作者： Tian, Zhaolu Li, Xinxiang Fan, C. M. Chen, C. S. Taiyuan Univ Technol Coll Data Sci Taiyuan Shanxi Peoples R China Shanghai Univ Dept Math Shanghai Peoples R China Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA Univ Elect Sci & Technol China Sch Math Sci Chengdu Sichuan Peoples R China

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 provides 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. (C) 2017 Elsevier B.V. All rights reserved.

关键词： method of particular solutions Trigonometric functions particular solution Meshless methods Collocation method

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The method of particular solutions with polynomial basis functions for solving axisymmetric problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2018年 90卷 39-46页

作者： Wang, Meizhen Watson, Daniel Li, Ming Taiyuan Univ Technol Coll Math Taiyuan Shanxi Peoples R China Mississippi Coll Dept Math Clinton MS USA

In this paper, we extend the previous work of Chen et al. (Numer methods Partial Differential Eq 21: 349-367, 2005) on the two-step method of particular solutions (MPS) for solving the Poisson equation with an axisyrnmetric forcing term and boundary conditions in an axisymmetric geometry to general differential equations and time-dependent problems using the one-step MPS. Polynomial basis functions are sufficient for the proposed approach instead of using Chebyshev polynomials. Furthermore, no boundary method is required for solving the homogeneous equation which is required in the two-step approach. In the solution process of the two-step MPS, we only require the closed form particular solution of the Laplacian or Helmholtz equation with respect to the monomial basis functions. The proposed approach is more simplified compared to the previous work and also allows us to solve a large class of partial differential equations including those with variable coefficients. We further extend the proposed approach to time-dependent problems using the Houbolt method, which is a third order time marching finite difference scheme. In the numerical implementation, we compare the results using reduced axisymmetric equations and the original 3D equations. Numerical results show the high simplicity, accuracy, and efficiency of the proposed numerical method. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： method of particular solutions Axisymmetric equation Helmholtz equation Multiple scale technique Polynomial basis Houbolt method

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The localized method of approximate particular solutions for solving an optimal control problem

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Journal of Computational Mathematics and Data Science 2022年 3卷

作者： Acheampong, Kwesi Guan, Hongbo Zhu, Huiqing School of Mathematics and Natural Sciences The University of Southern Mississippi United States College of Mathematics and Information Science Zhengzhou University of Light Industry China

In this paper, we consider the localized method of approximate particular solutions (LMAPS) for solving a two-dimensional distributive optimal control problem governed by elliptic partial differential equations. Both radial basis functions and polynomial basis functions (RBFs) are used in the LMAPS discretization, while the leave-one-out cross-validation is adopted for the selection of the shape parameter appeared in RBFs. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method. © 2022 The Authors

关键词： Meshless method method of particular solutions Optimal control problem Polynomial basis functions Radial basis functions

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method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems

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APPLIED MATHEMATICAL MODELLING 2017年 49卷 452-469页

作者： Lin, Ji Chen, C. S. Wang, Fajie Dangal, Thir Hohai Univ Coll Mech & Mat Int Ctr Simulat Software Engn & Sci State Key Lab Hydrol Water Resources & Hydraul En Nanjing 211100 Jiangsu Peoples R China Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA

The traditional polynomial expansion method is deemed to be not suitable for solving two- and three-dimensional problems. The system matrix is usually singular and highly ill-conditioned due to large powers of polynomial basis functions. And the inverse of the coefficient matrix is not guaranteed for the evaluation of derivatives of polynomial basis functions with respect to the differential operator of governing equations. To avoid these troublesome issues, this paper presents an improved polynomial expansion method for the simulation of plate bending vibration problems. At first, the particular solutions using polynomial basis functions are derived analytically. Then these polynomial particular solutions are employed as basis functions instead of the original polynomial basis functions in the method of particular solutions for the approximated solutions. To alleviate the conditioning of the resultant matrix, we employ the multiple-scale method. Numerical experiments compared with analytical solutions, solutions by the Kansa's method, and reference solutions in references confirm the efficiency and accuracy of the proposed method in the solution of Winkler and thin plate bending problems including irregular shapes. (C) 2017 Elsevier Inc. All rights reserved.

关键词： method of particular solutions Plate bending vibration particular solution Polynomial basis function Multiple-scale method

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Multiple Revolution solutions for the Perturbed Lambert Problem using the method of particular solutions and Picard Iteration

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JOURNAL OF THE ASTRONAUTICAL SCIENCES 2017年第4期64卷 361-378页

作者： Woollands, Robyn M. Read, Julie L. Probe, Austin B. Junkins, John L. Texas A&M Univ Dept Aerosp Engn TAMU 3141 College Stn TX 77843 USA

We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert's problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenth-twelfth order Runge-Kutta integrator. It is well known that the solution to Lambert's problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

关键词： Lambert's problem Perturbed Lambert's problem method of particular solutions Picard-Chebyshev

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Localized method of approximate particular solutions with polynomial basis functions

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2018年 97卷 16-22页

作者： Li, Xinxiang Dangal, Thir Lei, Bin Shanghai Univ Dept Math Shanghai Peoples R China Alcorn State Univ Dept Math & Comp Sci Lorman MS USA Nanchang Univ Sch Civil Engn & Architecture Nanchang 330031 Jiangxi Peoples R China

The method of particular solutions (MPS) using polynomials as the basis functions has been successfully developed for solving a large class of partial differential equations. However, when a large number of collocation points are required, the above mentioned approach is not feasible since the resultant matrix is dense and ill-conditioned. This restriction is common for global methods. One of the alternative approaches is to employ the localized scheme in which only a small number of neighboring points are being used in the solution process. As a result, the resultant matrix of the localized method is sparse and thus can be solved efficiently. In this paper, the localized method has been employed to extend the recently developed MPS using polynomial basis functions for solving large-scale science and engineering problems. Overall, the proposed approach is stable and highly accurate. Four numerical examples are presented to validate the proposed numerical method.

关键词： method of particular solutions Polynomial basis function particular solution Radial basis functions kd-tree

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The closed-form particular solutions of the Poisson?s equation in 3D with the oscillatory radial basis functions in the forcing term

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2023年第1期152卷 148-153页

作者： Lamichhane, A. R. Manns, S. Aiken, Q. Murray, A. Ohio Northern Univ Sch Sci Technol & Math Ada OH 45810 USA Ohio State Univ Dept Math Columbus OH 43210 USA

Several meshless methods that are used to solve the partial differential equations are particular solutions based numerical methods. These numerical methods can only be applied to solve the partial differential equations if researchers have derived a particular solution of some equations beforehand. The main contribution of this article is the derivation of the family of particular solutions of the Poisson's equation in 3D with the oscillatory radial basis functions in the forcing term. Numerical results obtained by solving three elliptic partial differential equations presented here validates the derived particular solutions in the method of particular solutions.

关键词： particular solutions Oscillatory radial basis functions Meshless method Partial differential equations method of particular solutions

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