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**An explicit numerical method for the fractional cable equation.**
*(English)*
Zbl 1237.65097

Summary: An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to the fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicts very accurately the conditions under which the present explicit method is stable. This is thoroughly checked by means of extensive numerical integrations.

### MSC:

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

35L10 | Second-order hyperbolic equations |

35R11 | Fractional partial differential equations |

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

### Keywords:

fractional cable equation; Riemann-Liouville derivatives; difference scheme; Grünwald-Letnikov formular; stability; convergence; von Neumann method### Software:

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\textit{J. Quintana-Murillo} and \textit{S. B. Yuste}, Int. J. Differ. Equ. 2011, Article ID 231920, 12 p. (2011; Zbl 1237.65097)

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