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==History and recent developments==
In recent decades, the [[dual reciprocity method]] (DRM)<ref>Partridge PW, Brebbia CA, Wrobel LC, The dual reciprocity boundary element method. Computational Mechanics Publications, 1992</ref> and [[multiple reciprocity method]] (MRM)<ref>Nowak AJ, Neves AC, The multiple reciprocity boundary element method. Computational Mechanics Publication, 1994</ref> have been emerging as the two most promising techniques to evaluate the particular solution of nonhomogeneous [[partial differential equations]] in conjunction with the boundary discretization techniques, such as [[boundary element method]] (BEM). For instance, the so-called DR-BEM and MR-BEM are popular BEM techniques in the numerical solution of nonhomogeneous problems.
The DRM has become de facto the method of choice in the boundary methods to evaluate the particular solution, since it is easy to use, efficient, and flexible to handle a variety of problems. However, the DRM demands the inner nodes to guarantee the convergence and stability. Therefore, the method is not truly boundary-only.
It has been claimed in literatures that the MRM has the striking advantage over the DRM in that it does not require using inner nodes at all for nonhomogeneous problems. However, the traditional MRM does also have disadvantages compared with the DRM. Firstly, the MRM is computationally much more expensive in the construction of the different interpolation matrices. Secondly, the method has limited applicability to general nonhomogeneous problems due to its conventional use of high-order Laplacian operators in the annihilation process.
An improved multiple reciprocity method, called the recursive composite multiple reciprocity method (RC-MRM),<ref name="Chena">Chen W, Meshfree boundary particle method applied to Helmholtz problems. Engineering Analysis with Boundary Elements 2002,26(7): 577–581</ref><ref name="Chenb">Chen W, Fu ZJ, Jin BT, A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique. Engineering Analysis with Boundary Elements 2010,34(3): 196–205</ref>
The boundary particle method (BPM) is then developed to a boundary-only discretization of inhomogeneous partial differential equation by combining the RC-MRM with a variety of the strong-form meshless boundary collocation discretization schemes, such as the [[method of fundamental solution]] (MFS), [[boundary knot method]] (BKM), [[regularized meshless method]] (RMM), [[singular boundary method]] (SBM), and [[Trefftz method]] (TM). The BPM has since applied to a variety of problems such as nonhomogeneous [[Helmholtz]] and [[convection-diffusion equation]]. Numerical experiments are very encouraging. It is worthy of noting that the BPM interpolation representation is in fact of a [[wavelet]] series.
For the application of the BPM to [[Helmholtz]],<ref name="Chena" />
▲For the application of the BPM to [[Helmholtz]]<ref name="Chena" />, [[Poisson]]<ref name="Chenb" /> and [[plate]] bending problems<ref>Fu ZJ, Chen W, Yang W, Winkler plate bending problems by a truly boundary-only boundary particle method. Computational Mechanics 2009,44(6): 757–563</ref>, the high-order [[fundamental solution]] or general solution, harmonic function<ref>Hon YC, Wu ZM, A numerical computation for inverse boundary determination problem. Engineering Analysis with Boundary Elements 2000,24(7–8): 599–606</ref> or [[Trefftz]] function (T-complete functions)<ref>Chen W, Fu ZJ, Qin QH, Boundary particle method with high-order Trefftz functions. CMC: Computers, Materials & Continua 2010,13(3): 201–217</ref> are often used, for instance, those of [[Berger]], [[Winkler]], and vibrational thin plate equations<ref>Chen W, Shen ZJ, Shen LJ, Yuan GW, General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates. Engineering Analysis with Boundary Elements 2005,29(7): 699–702</ref>. Thanks to its truly boundary-only merit, the BPM is more appealing in the solution of [[optimization]] and [[inverse problems]], where only a part of boundary data is usually accessible. The method has successfully been applied to inverse Cauchy problem associated with [[Poisson]]<ref>Fu ZJ, Chen W, Zhang CZ, Boundary particle method for Cauchy inhomogeneous potential problems. Inverse Problems in Science and Engineering 2012,20(2): 189–207</ref> and nonhomogeneous [[Helmholtz]] equations<ref>Chen W, Fu ZJ, Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations. Journal of Marine Science and Technology–Taiwan 2009,17(3): 157–163</ref>.
==Further comments==
The BPM may encounter difficulty in the solution of problems having complex source functions, such as non-smooth, large-gradient functions, or a set of discrete measured data. The road map for the BPM solution of such problems is briefly outlined below:
(1) The complex functions or a set of discrete measured data can be interpolated by a sum of [[polynomial]] or [[trigonometric]] function series. Then, the RC-MRM can easily reduce the nonhomogeneous equation to a high-order homogeneous equations, and the BPM can be simply implemented to solve these problems with boundary-only discretization.
(2) The [[___domain decomposition]] may be used to in the BPM boundary-only solution of large-gradient source functions problems.
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==References==
{{Reflist}}
==External links==
* [http://www.ccms.ac.cn/fuzj/Boundary%20Particle%20Method.htm Boundary Particle Method]
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