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Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods. (English) Zbl 1409.35225

Summary: The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher’s equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.{
©2019 American Institute of Physics}

MSC:

35R11 Fractional partial differential equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Khader, M. M.; Saad, K. M., A numerical approach for solving the problem of biological invasion (fractional Fisher equation) using Chebyshev spectral collocation method, Chaos Solitons Fractals, 110, 169-177 (2018) · Zbl 1448.65185 · doi:10.1016/j.chaos.2018.03.018
[2] Khader, M. M.; Saad, K. M., On the numerical evaluation for studying the fractional KdV, KdV-Burger’s and Burger’s equations, Eur. Phys. J. Plus, 133, 1-13 (2018) · doi:10.1140/epjp/i2018-12191-x
[3] Kilbas, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993) · Zbl 0818.26003
[4] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6 (1997) · Zbl 0890.65071
[5] Khader, M. M., On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16, 2535-2542 (2011) · Zbl 1221.65263 · doi:10.1016/j.cnsns.2010.09.007
[6] Khader, M. M.; Babatin, M. M., Numerical treatment for solving fractional SIRC model and influenza A, Comput. Appl. Math., 33, 3, 543-556 (2014) · Zbl 1328.92077 · doi:10.1007/s40314-013-0079-6
[7] Podlubny, I., Fractional Differential Equations (1999) · Zbl 0918.34010
[8] Abdeljawad, T., A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequalities Appl., 130, 1-11 (2017) · Zbl 1368.26003 · doi:10.1186/s13660-017-1400-5
[9] Abdeljawad, T.; Baleanu, D., Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 9, 1098-1107 (2017) · Zbl 1412.47086 · doi:10.22436/jnsa
[10] Abdulhameed, M.; Vieru, D.; Roslan, R., Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel, Physica A, 484, 233-252 (2017) · Zbl 1499.76136 · doi:10.1016/j.physa.2017.05.001
[11] Ablowitz, M. J.; Zeppetella, A., Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math. Biol., 41, 835-840 (1979) · Zbl 0423.35079 · doi:10.1007/BF02462380
[12] Gómez-Aguilar, J. F., Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Physica A, 20, 562-572 (2017) · Zbl 1400.82228 · doi:10.1016/j.physa.2016.08.072
[13] Bhrawy, A. H.; Alghamdi, M. A., Approximate solutions of Fisher’s type equations with variable coefficients, Abstr. Appl. Anal., 1, 1-16 (2013) · Zbl 1297.65126 · doi:10.1155/2013/176730
[14] Atangana, A., Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505, 688-706 (2018) · Zbl 1514.34009 · doi:10.1016/j.physa.2018.03.056
[15] Atangana, A.; Gómez-Aguilar, J. F., Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133, 166, 1-23 (2018) · doi:10.1140/epjp/i2018-12021-3
[16] Atangana, A.; Baleanu, D., New fractional derivative with non-local and non-singular kernel, Therm. Sci., 20, 2, 757-763 (2016) · doi:10.2298/TSCI160111018A
[17] Mason, J. C.; Handscomb, D. C., Chebyshev Polynomials (2003)
[18] Tadjeran, C.; Meerschaert, M. M., A second-order accurate numerical method for the two dimensional fractional diffusion equation, J. Comput. Phys., 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
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