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International Journal of Differential Equations
Volume 2010 (2010), Article ID 846107, 16 pages
On the Selection and Meaning of Variable Order Operators for Dynamic Modeling
School of Engineering, University of California, P.O. Box 2039, Merced, CA 95344, USA
Received 4 August 2009; Accepted 8 October 2009
Academic Editor: Nikolai Leonenko
Copyright © 2010 Lynnette E. S. Ramirez and Carlos F. M. Coimbra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Citations to this Article [25 citations]
The following is the list of published articles that have cited the current article.
- Chang-Ming Chen, and K. Burrage, “Numerical analysis for a variable-order nonlinear cable equation,” Journal of Computational and Applied Mathematics, vol. 236, no. 2, pp. 209–224, 2011.
- Hu Sheng, Hongguang Sun, YangQuan Chen, and TianShuang Qiu, “Synthesis of multifractional Gaussian noises based on variable-order fractional operators,” Signal Processing, vol. 91, no. 7, pp. 1645–1650, 2011.
- Teodor M. Atanackovic, and Stevan Pilipovic, “Hamilton's Principle With Variable Order Fractional Derivatives,” Fractional Calculus and Applied Analysis, vol. 14, no. 1, pp. 94–109, 2011.
- Chang-Ming Chen, F. Liu, V. Anh, and I. Turner, “Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5729–5742, 2011.
- S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh, “Numerical techniques for the variable order time fractional diffusion equation,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 10861–10870, 2012.
- Chang-Ming Chen, F. Liu, I. Turner, V. Anh, and Y. Chen, “Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation,” Numerical Algorithms, 2012.
- Rami Ahmad El-Nabulsi, “Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent,” Computational and Applied Mathematics, 2013.
- Deshun Yin, Yixin Wang, Yanqing Li, and Chen Cheng, “Variable-order fractional mean square displacement function with evolution of diffusibility,” Physica A: Statistical Mechanics and its Applications, 2013.
- Tatiana Odzijewicz, Agnieszka B. Malinowska, and Delfim F. M. Torres, “Noether’s theorem for fractional variational problems of variable order,” Central European Journal of Physics, 2013.
- A. G. Butkovskii, S. S. Postnov, and E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation,” Automation and Remote Control, vol. 74, no. 4, pp. 543–574, 2013.
- H. Zhang, F. Liu, Mantha S. Phanikumar, and Mark M. Meerschaert, “A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model,” Computers & Mathematics with Applications, 2013.
- Hongmei Zhang, and Shujun Shen, “The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equations,” Numerical Mathematics-Theory Methods and Applications, vol. 6, no. 4, pp. 571–585, 2013.
- Marko Janev, Stevan Pilipovic, and Dusan Zorica, “An expansion formula for fractional derivatives of variable order,” Central European Journal of Physics, vol. 11, no. 10, pp. 1350–1360, 2013.
- Abdon Atangana, and Adem Kiliçman, “A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order,” Mathematical Problems in Engineering, vol. 2013, pp. 1–6, 2013.
- Ricardo Almeida, and Delfim F. M. Torres, “An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order,” The Scientific World Journal, vol. 2013, pp. 1–11, 2013.
- H. Zhang, F. Liu, P. Zhuang, I. Turner, and V. Anh, “Numerical analysis of a new space–time variable fractional order advection–dispersion equation,” Applied Mathematics and Computation, vol. 242, pp. 541–550, 2014.
- Xuan Zhao, Zhi-zhong Sun, and George Em Karniadakis, “Second-order approximations for variable order fractional derivatives: Algorithms and applications,” Journal of Computational Physics, 2014.
- Edmundo Capelas de Oliveira, and José António Tenreiro Machado, “A Review of Definitions for Fractional Derivatives and Integral,” Mathematical Problems in Engineering, vol. 2014, pp. 1–6, 2014.
- Abdon Atangana, “On the stability and convergence of the time-fractional variable order telegraph equation,” Journal of Computational Physics, 2015.
- Shole Yaghoobi, Behrouz Parsa Moghaddam, and Karim Ivaz, “An efficient cubic spline approximation for variable-order fractional differential equations with time delay,” Nonlinear Dynamics, 2016.
- Emile Franc Doungmo Goufo, “Stability and convergence analysis of a variable order replicator–mutator process in a moving medium,” Journal of Theoretical Biology, vol. 403, pp. 178–187, 2016.
- J. Orosco, and C. F. M. Coimbra, “On the control and stability of variable-order mechanical systems,” Nonlinear Dynamics, 2016.
- Dominik Sierociuk, Michal Macias, Wiktor Malesza, and Grzegorz Sarwas, “Dual Estimation of Fractional Variable Order Based on the Unscented Fractional Order Kalman Filter for Direct and Networked Measurements,” Circuits, Systems, and Signal Processing, 2016.
- Behrouz Parsa Moghaddam, and José António Tenreiro Machado, “Extended Algorithms for Approximating Variable Order Fractional Derivatives with Applications,” Journal of Scientific Computing, 2016.
- A.H. Bhrawy, and M.A. Zaky, “Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations,” Computers & Mathematics with Applications, 2016.