This manuscript investigates a computational method based on fractional-order Fibonacci functions (FFFs) for solving distributed-order (DO) fractional differential equations and DO time-fractional diffusion equations. Extra DO fractional derivative operator and pseudo-operational matrix of fractional integration for FFFs are proposed. To evaluate the unknown coefficients in the FFF expansion, utilizing the matrices, an optimization problem relating to considered equations is formulated. This approach converts the original problems into a system of algebraic equations. The approximation error is proposed. Several problems are proposed to investigate the applicability and computational efficiency of the scheme. The approximations achieved by some existing schemes are also tested conforming to the efficiency of the present method. Also, the model of the motion of the DO fractional oscillator is solved, numerically.
[1] M. Abbaszadeh and M. Dehghan, Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection-diffusion equation, Eng. Comput., (2019), 1–17.
[2] T. M. Atanackovic, L. Oparnica, and S. Pilipovic, On a nonlinear distributed order fractional differential equation, J. Math. Anal. Appl., 328(1) (2007), 590–608.
[3] R. L. Bagley and P. J. Torvik, On the existence of the order domain and the solution of distributed-order equations, Int. J. Appl. Math., 1(7) (2000), 865–882.
[4] A. T. Benjamin, and J. J. Quinn, Proofs that really count: the art of combinatorial proof, Mathematical Association of America, 2003.
[5] D. Boyadzhiev, H. Kiskinov, M. Veselinova, and A. Zahariev, Stability analysis of linear distributed order fractional systems with distributed delays, Fract. Calc. Appl. Anal., 20(4) (2017), 914.
[6] M. Caputo, Distributed order differential equations modelling dielectric induction and diffuion, Fract. Calc. Appl. Anal., 4 (2001), 421–442.
[7] M. Caputo, Elasticita e dissipazione, Zanichelli, Bologna, Italy, 1969.
[8] M. Caputo, Mean fractional-order-derivatives differential equations and filters, Ann. dell’Universita di Ferrara, 41(1) (1995), 73–84.
[9] M. Caputo, and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Prog. Fract. Differ. Appl., 2(1) (2016), 1–11.
[10] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, fundamentals in single domains, Springer, Berlin, 2006.
[11] T. Eftekhari and S. M. Hosseini, A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order, Comput. Appl. Math., 41(6) (2022), 1–22.
[12] T. Eftekhari and J. Rashidinia, A new operational vector approach for timefractional subdiffusion equations of distributed order based on hybrid functions, Math. Methods Appl. Sci., (2022).
[13] T. Eftekhari, J. Rashidinia, and K. Maleknejad, Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space, Adv. Differ. Equ., 2021(1) (2021), 1–22. https://doi.org/10.1186/s13662-021-03617-0
[14] 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.
[15] S. Guo, L. Mei, Z. Zhang, and Y. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation, Appl. Math. Lett., 85 (2018), 157–163.
[16] Z. Jiao, Y. Chen, and I. Podlubny, Distributed-order dynamic system: stability, simulation, applications and perspectives, Springer, New York, 2012.
[17] E. Kryszing, Introductory functional analysis with applications, Wiley, New York, 1978.
[18] J. Li, F. Liu, L. Feng, and I. Turner, A novel finite volume method for the Riesz space distributed-order diffusion equation, Comput. Math. Appl., 74(4) (2017), 772–783.
[19] K. Maleknejad, J. Rashidinia, and T. Eftekhari, Numerical solutions of distributed order fractional differential equations in the time domain using the Mn¨tz-Legendre wavelets approach, Numer. Methods Partial Differ. Equ., 37(1) (2021), 707–731.
[20] M. Morgado, M. Rebelo, L. Ferras, and N. Ford, Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method, Appl. Numer. Math. 114 (2017), 108–123.
[21] H. S. Najafi, A.Refahi Sheikhani, and A. Ansari, Stability analysis of distributed order fractional differential equa- tions, Abstr. Appl. Anal., 2011 (2011).
[22] Z. Odibat and S.Momani Application of variational iteration method to nonlinear differential equations of frac- tional order, Int. J. Nonlinear Sci. Numer. Simul., 1 (2006), 15–27.
[23] M. Pourbabaee and A. Saadatmandi, A novel Legendre operational matrix for distributed order fractional differ- ential equations, Appl. Math. Comput., 361 (2019), 215–231.
[24] P. Rahimkhani, Y. Ordokhani, and P. M. Lima, An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets, Appl. Numer. Math., 145 (2019), 1–27.
[25] J. Rashidinia, T. Eftekhari, and K. Maleknejad, A novel operational vector for solving the general form of dis- tributed order fractional differential equations in the time domain based on the second kind Chebyshev wavelets, Numer. Algorithms, 88(4) (2021), 1617–1639.
[26] S. Sabermahani and Y. Ordokhani, Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis, J. Vib. Control, 27(15-16) (2021), 1778–1792.
[27] S. Sabermahani, Y. Ordokhani, and P. Rahimkhani, Application of Two-Dimensional Fibonacci Wavelets in Fractional Partial Differential Equations Arising in the Financial Market, Int. J. Appl. Comput., 8(3) (2022), 1–20.
[28] S. Sabermahani, Y. Ordokhani, and S.A. Yousefi, Fibonacci wavelets and their applications for solving two classes of time-varing delay problems, Optim. Control Appl. Methods, 41(2) (2020), 395–416.
[29] S. Sabermahani, Y. Ordokhani, and S.A. Yousefi, Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations, Eng. Comput., 36 (2020), 795–806.
[30] X. Wang, F. Liu, and X. Chen, Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys., 2015 (2015).
[31] B. Yuttanan and M.Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364.
[32] X. Zheng, H. Liu, H. Wang, et al., An Efficient Finite Volume Method for Nonlinear Distributed-Order Space- Fractional Diffusion Equations in Three Space Dimensions, J. Sci. Comput., 80 (2019), 1395–1418.
)
