Approximation solution of fractional partial differential equations by neural networks.

*(English)*Zbl 1236.65110Summary: Neural networks with radial basis functions (RBFs) method are used to solve a class of initial boundary value of fractional partial differential equations (PDEs) with variable coefficients on a finite domain. It takes the case where a left-handed or right-handed fractional spatial derivative may be present in the partial differential equations. Convergence of this method is discussed. A numerical example using neural networks RBF method for a two-sided fractional PDE also is presented and compared with other methods.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35R11 | Fractional partial differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

##### Keywords:

neural networks; radial basis functions; initial boundary value of fractional partial differential equation; variable coefficients; convergence; numerical example
PDF
BibTeX
XML
Cite

\textit{A. A. S. Almarashi}, Adv. Numer. Anal. 2012, Article ID 912810, 10 p. (2012; Zbl 1236.65110)

Full Text:
DOI

##### References:

[1] | M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” NSFgrant DMC, pp. 563-573, 2004. · Zbl 1126.76346 |

[2] | A. S. Chaves, “A fractional diffusion equation to describe Lévy flights,” Physics Letters A, vol. 239, no. 1-2, pp. 13-16, 1998. · Zbl 1026.82524 · doi:10.1016/S0375-9601(97)00947-X |

[3] | D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Levy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413-1423, 2000. · doi:10.1029/2000WR900032 |

[4] | F. Liu, V. Aon, and I. Turner, “Numerical solution of the fractional advection-dispersion equation,” 2002, http://academic.research.microsoft.com/Publication/3471879. |

[5] | K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1993. · Zbl 0789.26002 |

[6] | R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto, “Fractional calculus and continuous-time finance. III. The diffusion limit,” in Mathematical Finance, Trends Math., pp. 171-180, Birkhäuser, Basel, Switzerland, 2001. · Zbl 1138.91444 |

[7] | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Swizerland, 1993. · Zbl 0818.26003 |

[8] | G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017-1033, 2004. · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003 |

[9] | S. Haykin, Neural Networks, Prntice-Hall, 2006. · Zbl 0828.68103 |

[10] | K.-I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks, vol. 2, no. 3, pp. 183-192, 1989. |

[11] | A. Al-Marashi and K. Al-Wagih, “Approximation solution of boundary values of partial differential equations by using neural networks,” Thamar University Journal, no. 6, pp. 121-136, 2007. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.