Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations

[1] M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed order time fractional diffusion-wave equation with error estimate, Numer. Algor., 75 (2017), 173-211.
[2] O. Abdulaziz, I. Hashim, and S. Momani, Application of homotopy perturbation method to fractional IVPs, J. Comput. Appl. Math., 216 (2008), 574-584.
[3] M. Ahmadi Darani and A. Saadatmandi, The operational matrix of fractional derivative of the fractional order Chebyshev functions and its applications, Comput. Methods Differ. Equ., 5 (2017), 67-87.
[4] H. Aminikhah, A. H. Refahi Sheikhani, T. Houlari, and H. Rezazadeh, Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation, IEEE/CAA J. Autom. Sin., 6 (2019), 760-765.
[5] T. M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a viscoelastic body, Acta Mech., 159 (2002), 77-86.
[6] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part I, Int. J. Appl. Math., 2 (2000), 865-882.
[7] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed order equations-part II, Int. J. Appl. Math.,2 (2000), 965-987.
[8] A. Baseri, S. Abbasbandy, and E. Babolian, A collocation method for fractional diffusion equation in a long time with Chebyshev functions, Appl. Math. Comput., 322 (2018), 55–65.
[9] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.
[10] M. Caputo, Mean fractional-order-derivative differential equation and filters, Ann. Univ. Ferrara Sez. VII (N.S.), 41 (1995), 73-84.
[11] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003) 259-279.
[12] A. V. Chechkin, J. Klafter, and I. M. Sokolov, Fractional fokker-Planck equation for ultraslow kinetics, Europhys. Lett., 63 (2003), 326-332.
[13] M. Dehghan and M. Abbaszadeh, A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation, Comput. Math. Appl., 75 (2018), 2903-2914.
[14] M. Dehghan and M. Abbaszadeh, An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl. Numer. Math., 131 (2018) 190-206.
[15] J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.
[16] K. Diethelm and N. J. Ford, Numerical analysis for distributed order differential equations, J. Comput. Appl. Math., 225 (2009), 96-104.
[17] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62 (2011), 2364-2373.
[18] N. J. Ford, M. L. Morgado, and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time, Electron. Trans. Numer. Anal., 44 (2015), 289-305.
[19] F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956.
[20] Z. Jiao, Y. Q. Chen, and I. Podlubny, Distributed-Order Dynamic System Stability, Simulation, Applications and and Perspective, Springer, London, 2012.

[21] N. Jibenja, B. Yuttanan, and M. Razzaghi, An efficient method for numerical solutions of distributed-order fractional differential equations, J. Comput. Nonlinear Dynam., 13 (2018), 111003.
[22] J. T. Katsikadelis, Numerical solution of distributed order fractional differential equations, J. Comput. Phys., 259 (2014), 11-22.
[23] J. T. Katsikadelis, The fractional distributed order oscillator: A numerical solution, J. Serb. Soc. Comput. Mech., 6 (2012), 148-159.
[24] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.
[25] Y. Li, H. Sheng, and Y. Q. Chen, On distributed order integrator/differentiator, signal processing, 91 (2011), 1079-1084.
[26] S. Mashayekhi and M. Razzaghi, Numerical solution of distributed order fractional differential equations by hybrid functions, J. Comput. Phys., 315 (2016), 169-181.
[27] S. Mashayekhi and M. Razzaghi, Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation, Math. Meth. Appl. Sci., 39 (2016), 353-365.
[28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley 1993.
[29] A. Mohebbi, Analysis of a numerical method for the solution of time fractional Burgers equation, Bull. Iranian Math. Soc., 44 (2018), 457-480.
[30] A. Mohebbi, On the split-step method for the solution of nonlinear Schr¨odinger equation with the Riesz space fractional derivative, Comput. Methods Differ. Eq., 4 (2016) 54-69.
[31] K. B. Oldham and J. Spanier, The Fractional Calculus. New York: Academic Press 1974.
[32] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
[33] I. Podlubny, T. Skovranek, B. M. Vinagre Jara, I. Petras, V. Verbitsky, and Y. Q. Chen, Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013).
[34] M. Pourbabaee and A. Saadatmandi, A novel Legendre operational matrix for distributed order fractional differential equations, Appl. Math. Comput., 361 (2019), 215-231.
[35] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional order differential equations, Comput. Math. Appl., 59 (2010), 1326-1336.
[36] A. Saadatmandi, A. Khani, and M. R. Azizi, A sinc-Gauss-Jacobi collocation method for solving Volterra’s population growth model with fractional order, Tbilisi Math. J., 11 (2018), 123-137.
[37] J. Shen, T. Tang, and L. L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer-Verlag Berlin Heidelberg 2011.
[38] P. L. Trung Duong, E. Kwok, and M. Lee, Deterministic analysis of distributed order systems using operational matrix, Appl. Math. Model., 40 (2016), 1929-1940.
[39] Y. Yang, Y. Ma, and L. Wang, Legendre polynomials operational matrix method for solving fractional partial differential equations with variable coefficients, Math. Probl. Eng. (2015), Art. ID 915195.
[40] B. Yuttanan and M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364.
[41] M. A. Zaky and J. A. Tenreiro Machado, On the formulation and numerical simulation of distributed order fractional optimal control, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 177-189.
[42] F. Zhou, Y. Zhao, Y. Li, and Y. Q. Chen, Design, implementation and application of distributed order PI control, ISA Trans., 52 (2013), 429-437.