IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v382y2020ics0096300320302988.html
   My bibliography  Save this article

A fictitious points one–step MPS–MFS technique

Author

Listed:
  • Zhu, Xiaomin
  • Dou, Fangfang
  • Karageorghis, Andreas
  • Chen, C.S.

Abstract

The method of fundamental solutions (MFS) is a simple and efficient numerical technique for solving certain homogenous partial differential equations (PDEs) which can be extended to solving inhomogeneous equations through the method of particular solutions (MPS). In this paper, radial basis functions (RBFs) are considered as the basis functions for the construction of a particular solution of the inhomogeneous equation. A hybrid method coupling these two methods using both fundamental solutions and RBFs as basis functions has been effective for solving a large class of PDEs. In this paper, we propose an improved fictitious points method in which the centres of the RBFs are distributed inside and outside the physical domain of the problem and which considerably improves the performance of the MPS–MFS. We also describe various techniques to deal with the several parameters present in the proposed method, such as the location of the fictitious points, the source location in the MFS, and the estimation of a good value of the RBF shape parameter. Five numerical examples in 2D/3D and for second/fourth–order PDEs are presented and the performance of the proposed method is compared with that of the traditional MPS–MFS.

Suggested Citation

  • Zhu, Xiaomin & Dou, Fangfang & Karageorghis, Andreas & Chen, C.S., 2020. "A fictitious points one–step MPS–MFS technique," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    • Handle: RePEc:eee:apmaco:v:382:y:2020:i:c:s0096300320302988
    DOI: 10.1016/j.amc.2020.125332
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300320302988
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2020.125332?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(5), pages 687-698, October.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    4. Lin, Ji & Zhao, Yuxiang & Watson, Daniel & Chen, C.S., 2020. "The radial basis function differential quadrature method with ghost points," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 173(C), pages 105-114.
    5. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(3), pages 381-386, June.
    6. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(4), pages 525-537, August.
    7. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(2), pages 285-292, April.
    8. ,, 1998. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 14(1), pages 151-159, February.
    9. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    10. Wu, Hui-Yuan & Duan, Yong, 2016. "Multi-quadric quasi-interpolation method coupled with FDM for the Degasperis–Procesi equation," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 83-92.
    11. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lin, Ji & Zhao, Yuxiang & Watson, Daniel & Chen, C.S., 2020. "The radial basis function differential quadrature method with ghost points," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 173(C), pages 105-114.
    2. Chein-Shan Liu & Zhuojia Fu & Chung-Lun Kuo, 2017. "Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 9(6), pages 112-123, December.
    3. Bin-Mohsin, B. & Lesnic, D., 2012. "Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1445-1458.
    4. Karageorghis, Andreas & Tappoura, Demetriana & Chen, C.S., 2021. "The Kansa RBF method with auxiliary boundary centres for fourth order boundary value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 581-597.
    5. Marin, Liviu & Cipu, Corina, 2017. "Non-iterative regularized MFS solution of inverse boundary value problems in linear elasticity: A numerical study," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 265-286.
    6. Marc Peeters & Zeger Degraeve, 2004. "The Co-Printing Problem: A Packing Problem with a Color Constraint," Operations Research, INFORMS, vol. 52(4), pages 623-638, August.
    7. Tallys H. Yunes & Arnaldo V. Moura & Cid C. de Souza, 2005. "Hybrid Column Generation Approaches for Urban Transit Crew Management Problems," Transportation Science, INFORMS, vol. 39(2), pages 273-288, May.
    8. Ihor Borachok & Roman Chapko & B. Tomas Johansson, 2022. "A method of fundamental solutions with time-discretisation for wave motion from lateral Cauchy data," Partial Differential Equations and Applications, Springer, vol. 3(3), pages 1-13, June.
    9. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    10. Hamacher, Horst W. & Pedersen, Christian Roed & Ruzika, Stefan, 2007. "Multiple objective minimum cost flow problems: A review," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1404-1422, February.
    11. Zeger Degraeve & Marc Peeters, 2003. "Optimal Integer Solutions to Industrial Cutting-Stock Problems: Part 2, Benchmark Results," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 58-81, February.
    12. Viktoria Spaiser & David J. T. Sumpter, 2016. "Revising the Human Development Sequence Theory Using an Agent-Based Approach and Data," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 19(3), pages 1-1.
    13. Dou, Fangfang & Li, Zi-Cai & Chen, C.S. & Tian, Zhaolu, 2018. "Analysis on the method of fundamental solutions for biharmonic equations," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 346-366.
    14. Allen C. Goodman & Miron Stano, 2000. "Hmos and Health Externalities: A Local Public Good Perspective," Public Finance Review, , vol. 28(3), pages 247-269, May.
    15. Bettina Campedelli & Andrea Guerrina & Giulia Romano & Chiara Leardini, 2014. "La performance della rete ospedaliera pubblica della regione Veneto. L?impatto delle variabili ambientali e operative sull?efficienza," MECOSAN, FrancoAngeli Editore, vol. 2014(92), pages 119-142.
    16. Penn Loh & Zoë Ackerman & Joceline Fidalgo & Rebecca Tumposky, 2022. "Co-Education/Co-Research Partnership: A Critical Approach to Co-Learning between Dudley Street Neighborhood Initiative and Tufts University," Social Sciences, MDPI, vol. 11(2), pages 1-17, February.
    17. O'Brien, Raymond & Patacchini, Eleonora, 2003. "Testing the exogeneity assumption in panel data models with "non classical" disturbances," Discussion Paper Series In Economics And Econometrics 0302, Economics Division, School of Social Sciences, University of Southampton.
    18. YongSeog Kim & W. Nick Street & Gary J. Russell & Filippo Menczer, 2005. "Customer Targeting: A Neural Network Approach Guided by Genetic Algorithms," Management Science, INFORMS, vol. 51(2), pages 264-276, February.
    19. Yanling Li & Zita Oravecz & Shuai Zhou & Yosef Bodovski & Ian J. Barnett & Guangqing Chi & Yuan Zhou & Naomi P. Friedman & Scott I. Vrieze & Sy-Miin Chow, 2022. "Bayesian Forecasting with a Regime-Switching Zero-Inflated Multilevel Poisson Regression Model: An Application to Adolescent Alcohol Use with Spatial Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(2), pages 376-402, June.
    20. Oscar J. Cacho & Robyn L. Hean & Russell M. Wise, 2003. "Carbon‐accounting methods and reforestation incentives," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 47(2), pages 153-179, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:382:y:2020:i:c:s0096300320302988. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.