A composite collocation method based on the fractional Chelyshkov wavelets for distributed-order fractional mobile-immobile advection-dispersion equation
Abstract
In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.
Keyword : mobile-immobile advection-dispersion, convergence, Chelyshkov wavelet, distributed order, composite collocation method
How to Cite
Marasi, H., & Derakhshan, M. (2022). A composite collocation method based on the fractional Chelyshkov wavelets for distributed-order fractional mobile-immobile advection-dispersion equation. Mathematical Modelling and Analysis, 27(4), 590–609. https://doi.org/10.3846/mma.2022.15311
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Caputo. Elasticitàe Dissipazione. Zanichelli, 1969.
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K. Diethelm and N.J. Ford. Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math., 225(1):96–104, 2009. https://doi.org/10.1016/j.cam.2008.07.018
S.S. Ezz-Eldien. New quadrature approach based on operational matrix for solving a class of fractional variational problems. J. Comput. phys., 317:362– 381, 2016. https://doi.org/10.1016/j.jcp.2016.04.045
N.J. Ford and M.L. Morgado. Distributed order equations as boundary value problems. Comput. Math. Appl., 64(10):2973–2981, 2012. https://doi.org/10.1016/j.jcp.2016.04.045
G.H. Gao, H.W. Sun and Z.Z. Sun. Some high-order difference schemes for the distributed-order differential equations. J. Comput. Phys., 298:337–359, 2015. https://doi.org/10.1016/j.jcp.2015.05.047
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M. Hamid, M. Usman, R.U. Haq and Z. Tian. A spectral approach to analyze the nonlinear oscillatory fractional-order differential equations. Chaos. Solitons. Fractals., 146:110921, 2021. https://doi.org/10.1016/j.chaos.2021.110921
M. Hamid, M. Usman, R.U. Haq, Z. Tian and W. Wang. Linearized stable spectral method to analyze two-dimensional nonlinear evolutionary and reactiondiffusion models. Numer. Methods. Partial. Differ. Equ., 38(2):243–261, 2020. https://doi.org/10.1002/num.22659
M. Hamid, M. Usman, R.U. Haq and W. Wang. A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Physica. A., 551:124227, 2020. https://doi.org/10.1016/j.physa.2020.124227
M. Hamid, M. Usman, W. Wang and Z. Tian. A stable computational approach to analyze semi-relativistic behavior of fractional evolutionary problems. Numer. Methods. Partial. Differ. Equ., 38(2):122–136, 2020. https://doi.org/10.1002/num.22617
M. Hamid, M. Usman, W. Wang and Z. Tian. Hybrid fully spectral linearized scheme for time-fractional evolutionary equations. Math. Methods. Appl. Sci., 44(5):3890–3912, 2021. https://doi.org/10.1002/mma.6996
M. Hamid, M. Usman, T. Zubair, R.U. Haq and W. Wang. Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative. Eur. Phys. J. Plus., 134(10):1–17, 2019. https://doi.org/10.1140/epjp/i2019-12871-y
L. Hormander. The analysis of linear partial differential operators. 1990.
Z. Jiao, Y.Q. Chen and I. Podlubny. Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Springer London, 2012.
Y. Luchko. Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal., 12(4):409–422, 2009.
L. Moradi, F. Mohammadi and D. Baleanu. A direct numerical solution of timedelay fractional optimal control problems by using Chelyshkov wavelets. J. Vib. Control., 25(2):310–324, 2019. https://doi.org/10.1177/1077546318777338
M. Pourbabaee and A. Saadatmandi. A novel Legendre operational matrix for distributed order fractional differential equations. Appl. Math. Comput., 361:215–231, 2019. https://doi.org/10.1016/j.amc.2019.05.030
P. Rahimkhani, Y. Ordokhani and E. Babolian. Fractional-order Bernoulli wavelets and their applications. Appl. Math. Model., 40(17-18):8087–8107, 2016. https://doi.org/10.1016/j.apm.2016.04.026
T. Srokowski. L´evy flights in nonhomogeneous media: Distributedorder fractional equation approach. Phys. Rev. E., 78(3):031135, 2008. https://doi.org/10.1103/PhysRevE.78.031135
H. Ye, F. Liu and V. Anh. Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys., 298:652–660, 2015. https://doi.org/10.1016/j.jcp.2015.06.025
S. Yuzbasi. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Appl. Math. Comput., 219(11):6328– 6343, 2013. https://doi.org/10.1016/j.amc.2012.12.006