July  2021, 14(7): 2591-2606. doi: 10.3934/dcdss.2020258

Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation

1. 

Faculty of Science, Department of Actuary, Firat University, Elazig 23200, Turkey

2. 

Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya 42090, Turkey

* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received  June 2019 Revised  September 2020 Published  May 2021

In this paper, we investigate some analytical, numerical and approximate analytical methods by considering time-fractional nonlinear Burger–Fisher equation (FBFE). (1/G$ ' $)-expansion method, finite difference method (FDM) and Laplace perturbation method (LPM) are considered to solve the FBFE. Firstly, we obtain the analytical solution of the mentioned problem via (1/G$ ' $)-expansion method. Also, we compare the numerical method solutions and point out which method is more effective and accurate. We study truncation error, convergence, Von Neumann's stability principle and analysis of linear stability of the FDM. Moreover, we investigate the $ L_{2} $ and $ L_\infty $ norm errors for the FDM. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace perturbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.

Citation: Asif Yokus, Mehmet Yavuz. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2591-2606. doi: 10.3934/dcdss.2020258
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show all references

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[12]

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[13]

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[14]

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[15]

W. ChenL. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl., 59 (2010), 1614-1620.  doi: 10.1016/j.camwa.2009.08.004.  Google Scholar

[16]

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[17]

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[18]

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[19]

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[20]

Z. HammouchT. Mekkaoui and F. B. Belgacem, Numerical simulations for a variable order fractional Schnakenberg model, AIP Conference Proceedings, 1637 (2014), 1450-1455.  doi: 10.1063/1.4907312.  Google Scholar

[21]

J. Hristov, Space-fractional diffusion with a potential power-law coefficient: Transient approximate solution, Progress in Fractional Differentiation and Applications, 3 (2017), 19-39.   Google Scholar

[22]

H. N. A. Ismail and A. A. A. Rabboh, A restrictive padé approximation for the solution of the generalized Fisher and Burger-Fisher equations, Appl. Math. Comput., 154 (2004), 203-210.  doi: 10.1016/S0096-3003(03)00703-3.  Google Scholar

[23]

F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 709-722.  doi: 10.3934/dcdss.2020039.  Google Scholar

[24]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar

[25]

D. Kaya and S. El-Sayed, A numerical simulation and explicit solutions of the generalized Burgers–Fisher equation, Appl. Math. Comput., 152 (2004), 403-413.  doi: 10.1016/S0096-3003(03)00565-4.  Google Scholar

[26]

D. Kaya, S. Gülbahar, A. Yokuş and M. Gülbahar, Solutions of the fractional combined KdV-mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Difference Equ., 2018 (2018), Paper No. 77, 16 pp. doi: 10.1186/s13662-018-1531-0.  Google Scholar

[27]

D. Kaya, A. Yokus and U. Demiroglu, Comparison of exact and numerical solutions for the Sharma-Tasso-Olver equation, In Numerical Solutions of Realistic Nonlinear Phenomena, Springer, Cham, (2020), 53–65. Google Scholar

[28]

A. Keten, M. Yavuz and D. Baleanu, Nonlocal cauchy problem via a fractional operator involving power kernel in banach spaces, Fractal Fract., 3 (2019), 27. doi: 10.3390/fractalfract3020027.  Google Scholar

[29]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[30]

D. KumarJ. SinghH. M. Baskonus and H. Bulut, An effective computational approach for solving local fractional telegraph equations, Nonlinear Sci. Lett. A: Math. Phys. Mech, 8 (2017), 200-206.   Google Scholar

[31]

V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Difference Equ., 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6.  Google Scholar

[32]

P. A. NaikM. YavuzS. QureshiJ. Zu and S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, The European Physical Journal Plus, 135 (2020), 1-42.   Google Scholar

[33]

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M. B. RiazN. A. AsifA. Atangana and M. I. Asjad, Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 645-664.  doi: 10.3934/dcdss.2019041.  Google Scholar

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K. M. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.  Google Scholar

[36]

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Figure 1.  Traveling wave solution $ {u_1}(x,t) $ of Eq. (1) by substituting the values $ \mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;\delta = - 2, $ in Eq. (41)
Figure 2.  Traveling wave solution $ {u_2}(x,t) $ of Eq. (1) by substituting the values $ \mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;\delta = - 2, $ in Eq. (42)
Figure 3.  Traveling wave solution $ {u_3}(x,t) $ of Eq. (1) by substituting the values $ \mu = 1,\;\lambda = - 0.5,\;{c_1} = 1,\;\alpha = 0.8,\;v = 1, $ in Eq. (43)
Figure 4.  2D Numerical and exact travelling wave solution and absolute error of the Eq. (1) for Finite Difference and Laplace Perturbation Methods
Table 1.  Exact solution, numerical results and absolute error of FDM and LPM for Eq. $ (1) $ at $ \Delta x = \Delta t = 0.01 $
FDM LPM FDM LPM
$ x_{i} $ $ t_{j} $ Numerical Numerical Exact Errors Errors
$ 0.00 $ $ 0.01 $ -0.182279 -0.182996 -0.182350 7.09713$ \times $10$ ^{-5} $ 6.45583$ \times $10$ ^{-4} $
$ 0.01 $ $ 0.01 $ -0.182296 -0.183012 -0.182367 7.09118$ \times $10$ ^{-5} $ 6.45497$ \times $10$ ^{-4} $
$ 0.02 $ $ 0.01 $ -0.182312 -0.183027 -0.182383 7.09118$ \times $10$ ^{-5} $ 6.44425$ \times $10$ ^{-4} $
$ 0.03 $ $ 0.01 $ -0.182328 -0.183043 -0.182399 7.07930$ \times $10$ ^{-5} $ 6.44366$ \times $10$ ^{-4} $
$ 0.04 $ $ 0.01 $ -0.182344 -0.183058 -0.182415 7.07337$ \times $10$ ^{-5} $ 6.4332$ \times $10$ ^{-4} $
$ 0.05 $ $ 0.01 $ -0.182360 -0.183074 -0.182431 7.06744$ \times $10$ ^{-5} $ 6.4329$ \times $10$ ^{-4} $
FDM LPM FDM LPM
$ x_{i} $ $ t_{j} $ Numerical Numerical Exact Errors Errors
$ 0.00 $ $ 0.01 $ -0.182279 -0.182996 -0.182350 7.09713$ \times $10$ ^{-5} $ 6.45583$ \times $10$ ^{-4} $
$ 0.01 $ $ 0.01 $ -0.182296 -0.183012 -0.182367 7.09118$ \times $10$ ^{-5} $ 6.45497$ \times $10$ ^{-4} $
$ 0.02 $ $ 0.01 $ -0.182312 -0.183027 -0.182383 7.09118$ \times $10$ ^{-5} $ 6.44425$ \times $10$ ^{-4} $
$ 0.03 $ $ 0.01 $ -0.182328 -0.183043 -0.182399 7.07930$ \times $10$ ^{-5} $ 6.44366$ \times $10$ ^{-4} $
$ 0.04 $ $ 0.01 $ -0.182344 -0.183058 -0.182415 7.07337$ \times $10$ ^{-5} $ 6.4332$ \times $10$ ^{-4} $
$ 0.05 $ $ 0.01 $ -0.182360 -0.183074 -0.182431 7.06744$ \times $10$ ^{-5} $ 6.4329$ \times $10$ ^{-4} $
Table 2.  $ L_2 $ and $ L_\infty $ error norm when $ 0\leq \Delta x = \Delta t\leq 1 $
$ \Delta x=\Delta t $ $ L_2 $ $ L_\infty $
$ 0.1 $ 0.000444585 0.000216239
$ 0.05 $ 0.000343115 0.000190017
$ 0.02 $ 0.000148088 0.000114257
$ 0.01 $ 0.000067741 0.000070912
$ 0.002 $ 9.27231$ \times $10$ ^{-6} $ 0.0000210066
$ 0.001 $ 3.82314$ \times $10$ ^{-6} $ 0.0000121997
$ \Delta x=\Delta t $ $ L_2 $ $ L_\infty $
$ 0.1 $ 0.000444585 0.000216239
$ 0.05 $ 0.000343115 0.000190017
$ 0.02 $ 0.000148088 0.000114257
$ 0.01 $ 0.000067741 0.000070912
$ 0.002 $ 9.27231$ \times $10$ ^{-6} $ 0.0000210066
$ 0.001 $ 3.82314$ \times $10$ ^{-6} $ 0.0000121997
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