Advertisement

Fuzzy multiquadric radial basis functions for solving fuzzy partial differential equations

  • M. Dirbaz
  • T. AllahviranlooEmail author
Article
  • 25 Downloads

Abstract

In this paper, first we define the fuzzy multiquadric radial basis functions (FMQRBF). In the following, using the (FMQRBF) as the basis functions on the fuzzy interpolation expansion, we introduce the fuzzy multiquadric radial basis functions interpolation. Moreover, by considering (FMQRBF) and our obtained fuzzy method based on generalized Hukuhara difference (modified Euler’s) (Dirbaz and Allahviranloo in Fuzzy Sets Syst 2016:1–24, 2016), we present an algorithm of the fuzzy meshless method of lines for solving fuzzy partial differential equations. Finally, by the proposed fuzzy method we solve some numerical examples and analyze the errors in details.

Keywords

Fuzzy multi quadric radial basic functions Fuzzy interpolation Fuzzy modified Euler’s method Fuzzy partial differential equation 

Mathematics Subject Classification

35A08 35A35 

Notes

References

  1. Abdukhalikov KS, Tulenbaev MS, Umirbaev UU (1994) On fuzzy bases of vector space. Fuzzy Sets Syst 63:201–206MathSciNetCrossRefGoogle Scholar
  2. Allahviranloo T (2004) Numerical method for fuzzy system of linear equations. Appl Math Comput 2004:493–502MathSciNetzbMATHGoogle Scholar
  3. Allahviranloo T, Gouyande Z, Armand A, Hasnoglu A (2015b) On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets Syst 265:1–23MathSciNetCrossRefGoogle Scholar
  4. Allahviranloo T, Gouyandeh Z, Armand A (2015a) A method for solving fuzzy differential equation based on fuzzy taylor expansion. IOS press, Amsterdam, pp 1–16zbMATHGoogle Scholar
  5. Allahviranloo T, Salahshour S, Khezerloo M (2011) Maximal and minimal symmetric solutions of fully fuzzy linear systems. J Comput Appl Math 2011:235MathSciNetzbMATHGoogle Scholar
  6. Ameri R, Dehghan OR (2010) Fuzzy basis of fuzzy hyperbolictor space. Iran J Fuzzy Syst 2010:97–113zbMATHGoogle Scholar
  7. Baxter BJC (1992) The interpolation theory of radial basis functions, a dissertation presented in fulfillment of the requirements for degree of doctor of philosophy. Cambridge University Press, CambridgeGoogle Scholar
  8. Bayona V, Moscos M, Kindelan M (2011) Optimal constant shape parameter for multi quadric based RBF-FD method. Elsevier, AmsterdamzbMATHGoogle Scholar
  9. Bede B, Stefanini L (2013) Generalized differentiability of fuzzy valued functions. Fuzzy Sets Syst 230:119–141MathSciNetCrossRefGoogle Scholar
  10. Buckley JJ, Feuring T (1999) Introduction to fuzzy partial differential equations. Fuzzy Sets Syst 1999:241–248MathSciNetCrossRefGoogle Scholar
  11. Chenoweth ME (2012) A local radial basis function method for the numerical solution of partial differential equations. Numer Anal Comput Commons 2012:243Google Scholar
  12. Dirbaz M, Allahviranloo T (2016) A new algorithm for solving impulsive fuzzy initial value problem based on fuzzy methods. Fuzzy Sets Syst 2016:1–24Google Scholar
  13. Dirbaz M, Dirbaz F (2016) Numerical solution of impulsive fuzzy initial value problem by modified Euler’s method. J Fuzzy Set valued Anal 1:50–57CrossRefGoogle Scholar
  14. Fornberg B, Flyer N (2005) Accuracy of radial basis function interpolation and derivative approximations on 1-d infinite grids. Adv Comput Math 23:37–55MathSciNetCrossRefGoogle Scholar
  15. Fornberg B, Wright G (2004) Stable computation of multi quadratic interpolants for all values of the shape parameter. Comput Math Appl 47:497–523CrossRefGoogle Scholar
  16. Haq S, Hussain A, Uddin M (2011) RBFs meshless method of lines for the numerical solution of time-dependent nonlinear coupled partial differential equations. Appl Math 2:414MathSciNetCrossRefGoogle Scholar
  17. Khosropour F, Eslami E, Buckley JJ (2002) Fuzzy vector analysis. J Fuzzy Math 10(875):884MathSciNetzbMATHGoogle Scholar
  18. Majdisova Z, Skala V (2017) Radial basis function approximation comparsion and applications. Appl Math Model 51:728–743MathSciNetCrossRefGoogle Scholar
  19. Malik DS, Mordeson JN (1991) Fuzzy vector space. Inf Sci 55:271–281MathSciNetCrossRefGoogle Scholar
  20. Mitra S, Basak J (2001) A fuzzy radial basis function network. Neural Comput Appl 2001:244–252CrossRefGoogle Scholar
  21. Mongillo M (2011) Choosing basis functions and shape parameters for radial basis function methods. SIAM 2011:20Google Scholar
  22. Oberkampf WL, Deland SM, Rutherford BM, Diegert KV, Alvin KF (2002) Error and uncertainty in modeling and simulation. Reliab Eng Syst Saf 2002:1–25Google Scholar
  23. Puri M, Ralescu D (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558MathSciNetCrossRefGoogle Scholar
  24. Sarra SA (2004) Adaptive radial basis function methods for time dependent partial differential equations. Appl Numer Math 54:21MathSciNetGoogle Scholar
  25. Shen Q (2009) A meshless method of lines for the numerical solution of KDV equation using Radial basis functions. Eng Anal Bound Elements 2009:1171–1180MathSciNetCrossRefGoogle Scholar
  26. Stefanini L (2008) A generalization of Hukuhara difference for interval and fuzzy arithmetic. Econ Math Stat 2008:1–13Google Scholar
  27. Stefanini L (2010) A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst 161:1564–1584MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Faculty of Engineering and Natural SciencesBahcesehir UniversityIstanbulTurkey

Personalised recommendations