ON THE CONVERGENCE OF THE MFS–MPS SCHEME FOR 1D POISSON'S EQUATION

2013 ◽  
Vol 10 (02) ◽  
pp. 1341006 ◽  
Author(s):  
C. S. CHEN ◽  
C.-S. HUANG ◽  
K. H. LIN
Keyword(s):  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Basis Functions ◽  
Poisson’S Equation ◽  
Method Of Fundamental Solutions ◽  
Poisson's Equation ◽  
Numerical Process ◽  
Particular Solution ◽  
Inhomogeneous Problems ◽  
Lagrange Interpolating Polynomial

The method of fundamental solutions (MFS) has been an effective meshless method for solving homogeneous partial differential equations. Coupled with radial basis functions (RBFs), the MFS has been extended to solve the inhomogeneous problems through the evaluation of the approximate particular solution and homogeneous solution. In this paper, we prove the the approximate solution of the above numerical process for solving 1D Poisson's equation converges in the sense of Lagrange interpolating polynomial using the result of Driscoll and Fornberg [2002].

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

2015 ◽  
Vol 17 (3) ◽  
pp. 867-886 ◽  
Author(s):  
C. S. Chen ◽  
Xinrong Jiang ◽  
Wen Chen ◽  
Guangming Yao
Keyword(s):  
Helmholtz Equation ◽  
Large Scale ◽  
Solution Process ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Dense Matrix ◽  
Modified Helmholtz Equation ◽  
Large Scale Problems ◽  
Particular Solution

AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

2016 ◽  
Vol 20 (2) ◽  
pp. 512-533 ◽  
Author(s):  
Ji Lin ◽  
C. S. Chen ◽  
Chein-Shan Liu
Keyword(s):  
Three Dimensional ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Singular Problem ◽  
Helmholtz Equations ◽  
Particular Solutions ◽  
Radial Basis ◽  
Particular Solution ◽  
Polyharmonic Spline

AbstractThis paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

scholarly journals THE METHOD OF APPROXIMATE PARTICULAR SOLUTIONS FOR SOLVING ELLIPTIC PROBLEMS WITH VARIABLE COEFFICIENTS

2011 ◽  
Vol 08 (03) ◽  
pp. 545-559 ◽  
Author(s):  
C. S. CHEN ◽  
C. M. FAN ◽  
P. H. WEN
Keyword(s):  
Solution Process ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Left Hand ◽  
Particular Solutions ◽  
Particular Solution ◽  
The Right ◽  
Close Form

A new version of the method of approximate particular solutions (MAPSs) using radial basis functions (RBFs) has been proposed for solving a general class of elliptic partial differential equations. In the solution process, the Laplacian is kept on the left-hand side as a main differential operator. The other terms are moved to the right-hand side and treated as part of the forcing term. In this way, the close-form particular solution is easy to obtain using various RBFs. The numerical scheme of the new MAPSs is simple to implement and yet very accurate. Three numerical examples are given and the results are compared to Kansa's method and the method of fundamental solutions.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

2011 ◽  
Vol 3 (5) ◽  
pp. 572-585 ◽  
Author(s):  
A. Tadeu ◽  
C. S. Chen ◽  
J. António ◽  
Nuno Simões
Keyword(s):  
Fourier Transform ◽  
Helmholtz Equation ◽  
Initial Conditions ◽  
Inverse Fourier Transform ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Static Response ◽  
Particular Solution ◽  
The Fourier Transform ◽  
Transfer Problems

AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

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