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Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 101–116 | Cite as

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

  • A. H. Bhrawy
  • M. A. Zaky
Original Paper

Abstract

The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

Keywords

One-dimensional cable equation  Two-dimensional variable-order nonlinear cable equation Collocation method Jacobi polynomials Operational matrix of fractional derivative   Variable-order derivative 

Notes

Acknowledgments

The authors thank the referees for constructive comments and suggestions which have improved the paper.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceBeni-Suef UniversityBeni SuefEgypt
  3. 3.Department of Applied MathematicsNational Research CenterCairoEgypt

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