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A new method of fundamental solutions applied to nonhomogeneous elliptic problems

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Abstract

The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples.

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Authors and Affiliations

  1. Centro de Matemática e Aplicações, Departamento de Matemática, Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    C. J. S. Alves

  2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154, USA

    C. S. Chen

Authors
  1. C. J. S. Alves
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  2. C. S. Chen
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Corresponding author

Correspondence to C. J. S. Alves.

Additional information

Communicated by Z. Wu and B.Y.C. Hon

AMS subject classification

35J25, 65N38, 65R20, 74J20

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Alves, C.J.S., Chen, C.S. A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv Comput Math 23, 125–142 (2005). https://doi.org/10.1007/s10444-004-1833-5

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  • DOI: https://doi.org/10.1007/s10444-004-1833-5

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