An efficient numerical approach for solving the variable-order time fractional diffusion equation using chebyshev spectral collocation method

Sun, HongGuang, Yong Zhang, Dumitru Baleanu, Wen Chen, and YangQuan Chen. A new collection of real  world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation 64 (2018): 213-231.

Zamani, Majid, Masoud Karimi-Ghartemani, and Nasser Sadati. FOPID controller design for robust performance using particle swarm optimization. Fractional Calculus and Applied Analysis 10, no. 2 (2007): 169-187.

Oustaloup, Alain, Francois Levron, Benoit Mathieu, and Florence M. Nanot. Frequency-band complex noninteger dierentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47, no. 1 (2000): 25-39.

Coimbra, Carlos FM. Mechanics with variableorder dierential operators. Annalen der Physik 12, no. 1112 (2003): 692-703.

Cooper, G. R. J., and D. R. Cowan. Filtering using variable order vertical derivatives. Computers & Geosciences 30, no. 5 (2004): 455-459.
Tseng, Chien-Cheng. Design of variable and adaptive fractional order FIR dierentiators. Signal Processing 86, no. 10 (2006): 2554-2566.

Shen, Shujun, Fawang Liu, J. Chen, I. Turner, and Vo Anh. Numerical techniques for the variable order time fractional diusion equation. Applied Mathematics and Computation 218, no. 22 (2012): 10861-10870.

Zhang, Hongmei, Fawang Liu, Mantha S. Phanikumar, and Mark M. Meerschaert. A novel numerical method for the time variable fractional order mobile-immobile advectiondispersion model. Computers & Mathematics with Applications 66, no. 5 (2013): 693-701.
Lin, Ran, Fawang Liu, Vo Anh, and I. Turner. Stability and convergence of a new explicit nite-dierence approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics and Computation 212, no. 2 (2009): 435-445.

Zhuang, Pinghui, Fawang Liu, Vo Anh, and Ian Turner. Numerical methods for the variable-order fractional advection-diusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis 47, no. 3 (2009): 1760-1781.

Sweilam, N. H., M. M. Khader, and H. M. Almarwm. Numerical studies for the variable-order nonlinear fractional wave equation. Fractional Calculus and Applied Analysis 15, no. 4 (2012): 669-683.

Zhang, Hongmei, Fawang Liu, Pinghui Zhuang, Ian Turner, and Vo Anh. Numerical analysis of a new spacetime variable fractional order advectiondispersion equation. Applied mathematics and computation 242 (2014): 541-550.

Chen, Chang-Ming, Fawang Liu, Vo Anh, and Ian Turner. Numerical schemes with high spatial accuracy for a variable-order anomalous subdiusion equation. SIAM Journal on Scientic Computing 32, no. 4 (2010): 1740-1760.

Cao, Jianxiong, Yanan Qiu, and Guojie Song. A compact nite dierence scheme for variable order subdiusion equation.Communications in Nonlinear Science and Numerical Simulation 48 (2017): 140-149.

Zhao, Xuan, Zhi-zhong Sun, and George Em Karniadakis. Second-order approximations for variable order fractional derivatives: algorithms and applications. Journal of Computational Physics 293 (2015): 184-200.

Bhrawy, A. H., and MA3520157 Zaky. Numerical algorithm for the variable-order Caputo fractional functional dierential equation. Nonlinear Dynamics 85, no. 3 (2016): 1815-1823.
Zayernouri, Mohsen, and George Em Karniadakis. Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Journal of Computational Physics 293 (2015): 312-338.

Bhrawy, Ali H., and Mahmoud A. Zaky. Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrdinger equations. Computers & Mathematics with Applications 73, no. 6 (2017): 1100-1117.

Chen, Yi-Ming, Yan-Qiao Wei, Da-Yan Liu, and Hao Yu. Numerical solution for a class of nonlinear variable order fractional dierential equations with Legendre wavelets. Applied Mathematics Letters 46 (2015): 83-88.

Chen, Yiming, Liqing Liu, Baofeng Li, and Yannan Sun. Numerical solution for the variable order linear cable equation with Bernstein polynomials. Applied Mathematics and Computation 238 (2014): 329-341.

Li, Xiuying, and Boying Wu. A numerical technique for variable fractional functional boundary value problems. Applied Mathematics Letters 43 (2015): 108-113.

Li, Xiuying, and Boying Wu. A new reproducing kernel method for variable order fractional boundary value problems for functional dierential equations. Journal of Computational and Applied Mathematics 311 (2017): 387-393.

Samko, Stefan G., and Bertram Ross. Integration and dierentiation to a variable fractional order. Integral transforms and special functions 1, no. 4 (1993): 277-300.

Samko, Stefan G. Fractional integration and dierentiation of variable order. Analysis Mathematica 21, no. 3 (1995): 213-236.

Samko, Stefan G., Anatoly A. Kilbas, and Oleg I. Marichev. Fractional integrals and derivatives. Vol. 1. Yverdon-les-Bains, Switzerland: Gordon and Breach Science Publishers, Yverdon, 1993.

Caputo, Michele. Linear models of dissipation whose Q is almost frequency independent. Annals of Geophysics 19, no. 4 (1966): 383-393.

Guo, Benyu. Spectral methods and their applications. World Scientic, 1998.

Canuto, Claudio, M. Yousu Hussaini, Alo Quarteroni, and A. Thomas Jr. Spectral methods in
uid dynamics. Springer Science & Business Media, 2012.

Tang, Tao. Spectral and high-order methods with applications. Beijing: Science Press, 2006.

Shen, Jie, Tao Tang, and Li-Lian Wang. Spectral methods: algorithms, analysis and applications. Vol. 41. Springer Science & Business Media, 2011.

Steeb, Willi-Hans, and Tan Kiat Shi. Matrix calculus and Kronecker product with applications and C++ programs. World Scientic, 1997.

Steeb, Willi-Hans, and Yorick Hardy. Problems and solutions in introductory and advanced matrix calculus. World Scientic Publishing Company, 2016.

Zhang, Kai, JingShi Li, YongCun Song, and XiaoShen Wang. An alternating direction method of multipliers for elliptic equation constrained optimization problem. Science China Mathematics 60, no. 2 (2017): 361-378.

Liu, Wenjie, Boying Wu, and Jiebao Sun. Spacetime spectral collocation method for the onedimensional sineG ordon equation. Numerical Methods for Partial Dierential Equations 31, no. 3 (2015): 670-690.

Sun, Hongguang, Wen Chen, Changpin Li, and Yangquan Chen. Finite dierence schemes for variable-order time fractional diusion equation. International Journal of Bifurcation and Chaos 22, no. 04 (2012): 1250085.

Srivastava, V., and K. N. Rai. A multi-term fractional diusion equation for oxygen delivery through a capillary to tissues. Mathematical and Computer Modelling 51, no. 5-6 (2010): 616-624.

Luchko, Yury. Initial-boundary-value problems for the generalized multi-term time-fractional diusion equation. Journal of Mathematical Analysis and Applications 374, no. 2 (2011): 538-548.

Jin, Bangti, Raytcho Lazarov, Yikan Liu, and Zhi Zhou. The Galerkin nite element method for a multi-term time-fractional diusion equation. Journal of Computational Physics 281 (2015): 825-843