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== History and recent developments ==
In recent decades, the dual reciprocity method (DRM)[1] and multiple reciprocity method (MRM)[2] 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.
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For the application of the BPM to Helmholtz[3], Poisson[4] and plate bending problems[5], the high-order fundamental or general solutions, harmonic [6] or Trefftz functions (T-complete functions)[7] are often used, for instance, those of Berger, Winkler, and vibrational thin plate equations[8].
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[9] and nonhomogeneous Helmholtz equations[10].
== 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 inhomogeneous 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.
== See also ==
* Meshfree method
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