Abstract
This paper is aimed at constructing fractional power series (FPS) solutions of fractional Burgers-Huxley equations using residual power series method (RPSM). RPSM is combining Taylor’s formula series with residual error function. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme are reliable and powerful in finding the numerical solutions of fractional Burgers-Huxley equations. The numerical results reveal that the RPSM is very effective, convenient and quite accurate to time dependence kind of nonlinear equations. It is predicted that the RPSM can be found widely applicable in engineering.
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References
K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, New York, 1974).
A. M. Lopes, J. A. T. Machado, C. M. A. Pinto, and A. M. S. F. Galhano, “Fractional dynamics and MDS visualization of earthquake phenomena,” Comput. Math. Appl. 66, 647–658 (2013).
H. Beyer and S. Kempfle, “Definition of physical consistent damping laws with fractional derivatives,” Z. Angew. Math. Mech. 75, 623–635 (1995).
J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Sci. Technol. Soc. 15, 86–90 (1999).
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent-II,” Geophys. J. Int. 13, 529–539 (1967).
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods (World Scientific, Singapore, 2009).
G. M. Zaslavsky, “Chaos fractional kinetics and anomalous transport,” Phys. Rep. 371, 461–580 (2002).
R. Hirota, “Exact enve lope-soliton solutions of a non linear wave,” J. Math. Phys. 14, 805–809 (1973).
S. Kumar, D. Kumar, and J. Singh, “Numerical computation of fractional Black-Scholes equation arising in nancial market,” Egypt. J. Basic Appl. Sci. 1, 177–183 (2014).
W. Maliet, “The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations,” J. Comput. Appl. Math. 164, 529–541 (2004).
V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Commun. Nonlin. Sci. Numer. Simul. 13, 1642–1654 (2008).
Z. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Comput. Math. Appl. 59, 1462–1472 (2010).
Q. Wang, “Homotopy perturbation method for fractional order KdV equation,” Appl. Math. Comput. 190, 1795–1802 (2007).
S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Phys. Lett. A 365, 345–350 (2007).
M. Zurigat, “Solving nonlinear fractional differential equation using a multi-step Laplace-Adomian decomposition method,” Ann. Univ. Craiova, Math. Comput. Sci. Ser. 39, 162–172 (2012).
S. M. El-Sayed and D. Kaya, “Exact and numerical traveling wave solutions of Whitham Broer-Kaup equations,” Appl. Math. Comput. 167, 1339–1349 (2005).
M. Rafei and H. Daniali, “Application of the variational iteration method to the Whitham Broer-Kaup equations,” Comput. Math. Appl. 54, 1079–1085 (2007).
A. A. Freihat, M. Zurigat, and A. H. Handam, “The multi-step homotopy analysis method for modified epidemiological model for computer viruses,” Afrika Math. 26, 585–596 (2015).
A. H. Handam, A. A. Freihat, and M. Zurigat, “The multi-step homotopy analysis method for solving fractional-order model for HIV infection of CD4+ T cells,” Proyecc. J. Math. 34, 307–322 (2015).
M. Zurigat, A. A. Freihat, and A. H. Handam, “The multi-step homotopy analysis method for solving the Jaulent-Miodek equations,” Proyecc. J. Math. 34, 45–54 (2015).
S. Haq and M. Ishaq, “Solution of coupled Whitham-Broer-Kaup equations using optimal homotopy asymptotic method,” Ocean Eng. 84, 81–88 (2014).
R. B. Albadarneh, I. M. Batiha, and M. Zurigat, “Numerical solutions for linear fractional differential equations of order using finite difference method (FFDM),” J. Math. Comput. Sci. 16, 103–111 (2016).
O. Abu Arqub, “Series solution of fuzzy differential equations under strongly generalized differentiability,” J. Adv. Res. Appl. Math. 5, 31–52 (2013).
M. Alquran, “Analytical solutions of fractional foam drainage equation by residual power series method,” Math. Sci. 8, 153–160 (2014).
M. Alquran, “Analytical solutions of time-fractional two-component evolutionary system of order 2 by residual power series method,” J. Appl. Anal. Comput. 5, 589–599 (2015).
O. Abu Arqub, A. EI-Ajou, Z. Al Zhour, and S. Momani, “Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique,” Entropy 16, 471–493 (2014).
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Freihet, A.A., Zuriqat, M. Analytical Solution of Fractional Burgers-Huxley Equations via Residual Power Series Method. Lobachevskii J Math 40, 174–182 (2019). https://doi.org/10.1134/S1995080219020082
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DOI: https://doi.org/10.1134/S1995080219020082