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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. 

Faculty of Science, Department of Mathematics-Computer, Necmettin Erbakan University, Konya, 42090, Turkey

* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received  June 2019 Revised  July 2019 Published  January 2020

In this paper, we have investigated some analytical, numerical and approximate-analytical methods by considering the time-fractional nonlinear fractional Burger–Fisher equation (FBFE). (1/G$ ' $)-expansion method, finite difference method and Laplace perturbation method have been considered to solve the FBFE. Firstly, we have obtained the analytical solution of the mentioned problem via (1/G$ ' $)-expansion method. Also, we have compared the numerical method solutions and have obtained that which method is more effective and accurate. 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, doi: 10.3934/dcdss.2020258
References:
[1]

K. A. Abro and J. Gómez-Aguilar, A comparison of heat and mass transfer on a waltersb fluid via caputo-fabrizio versus atangana-baleanu fractional derivatives using the fox-h function, The European Physical Journal Plus, 2019 (2019), 101.   Google Scholar

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A. Allwright and A. Atangana, Augmented Upwind Numerical Schemes for a Fractional Advection-Dispersion Equation in Fractured Groundwater Systems, Discrete & Continuous Dynamical Systems-S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025.  Google Scholar

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A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

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E. BalcıI. Öztürk and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos, Solitons and Fractals, 123 (2019), 43-51.   Google Scholar

[7]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216.  Google Scholar

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E. Bas, B. Acay and R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 023110, 7pp. doi: 10.1063/1.5082390.  Google Scholar

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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

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Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal MESA, 5 (2014), 489-498.   Google Scholar

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

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H. N. Ismail and A. A. A. Rabboh, A Restrictive Pade Approximation for the Solution of the Generalized Fisher and Burger isher Equations, Applied Mathematics and Computation, 154 (2004), 203-210.  doi: 10.1016/S0096-3003(03)00703-3.  Google Scholar

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D. Kaya and S. M. El-Sayed, A numerical simulation and explicit solutions of the generalized burgers–fisher equation, Applied Mathematics and Computation, 152 (2004), 403-413.  doi: 10.1016/S0096-3003(03)00565-4.  Google Scholar

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

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show all references

References:
[1]

K. A. Abro and J. Gómez-Aguilar, A comparison of heat and mass transfer on a waltersb fluid via caputo-fabrizio versus atangana-baleanu fractional derivatives using the fox-h function, The European Physical Journal Plus, 2019 (2019), 101.   Google Scholar

[2]

A. Allwright and A. Atangana, Augmented Upwind Numerical Schemes for a Fractional Advection-Dispersion Equation in Fractured Groundwater Systems, Discrete & Continuous Dynamical Systems-S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025.  Google Scholar

[3]

J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A gronwall inequality via the generalized proportional fractional derivative with applications, Journal of Inequalities and Applications, 2019 (2019), Paper No. 101, 12 pp. doi: 10.1186/s13660-019-2052-4.  Google Scholar

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[5]

D. AvciB. B. Iskender Eroglu and N. Özdemir, Conformable heat equation on a radial symmetric plate, Thermal Science, 21 (2017), 819-826.  doi: 10.2298/TSCI160427302A.  Google Scholar

[6]

E. BalcıI. Öztürk and S. Kartal, Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos, Solitons and Fractals, 123 (2019), 43-51.   Google Scholar

[7]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012. doi: 10.1142/9789814355216.  Google Scholar

[8]

E. Bas, B. Acay and R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 023110, 7pp. doi: 10.1063/1.5082390.  Google Scholar

[9]

E. Bonyah, A. Atangana and M. A. Khan, Modeling the spread of computer virus via caputo fractional derivative and the beta-derivative, Asia Pacific Journal on Computational Engineering, 4 (2017), Article number 1. doi: 10.1186/s40540-016-0019-1.  Google Scholar

[10]

A. Bratsos and A. Khaliq, An exponential time differencing method of lines for Burgers–Fisher and coupled Burgers equations, Journal of Computational and Applied Mathematics, 356 (2019), 182-197.  doi: 10.1016/j.cam.2019.01.028.  Google Scholar

[11]

M. Caputo, Linear models of dissipation whose q is almost frequency independent Ⅱ, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x.  Google Scholar

[12]

M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel., Progress in Fractional Differentiation and Applications, 1 (2015), 1-13.   Google Scholar

[13]

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, 2014. Google Scholar

[14]

V. ChandrakerA. Awasthi and S. Jayaraj, Numerical Treatment of Burger-Fisher Equation, Procedia Technology, 25 (2016), 1217-1225.   Google Scholar

[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]

F. Evirgen and M. Yavuz, An alternative approach for nonlinear optimization problem with caputo-fabrizio derivative, ITM Web of Conferences: EDP Sciences, 22 (2018), 01009.  doi: 10.1051/itmconf/20182201009.  Google Scholar

[17]

A. Ghorbani, Beyond adomian polynomials: He polynomials, Chaos, Solitons & Fractals, 39 (2009), 1486-1492.  doi: 10.1016/j.chaos.2007.06.034.  Google Scholar

[18]

Z. Hammouch and T. Mekkaoui, Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives, Journal MESA, 5 (2014), 489-498.   Google Scholar

[19]

Z. Hammouch and T. Mekkaoui, Approximate analytical and numerical solutions to fractional KPP-like equations, Gen, 14 (2013), 1-9.   Google Scholar

[20]

Z. HammouchT. Mekkaoui and F. B. Belgacem, Numerical simulations for a variable order fractional Schnakenberg model, In 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. Ismail and A. A. A. Rabboh, A Restrictive Pade Approximation for the Solution of the Generalized Fisher and Burger isher Equations, Applied Mathematics and Computation, 154 (2004), 203-210.  doi: 10.1016/S0096-3003(03)00703-3.  Google Scholar

[23]

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

[24]

F. Jarad and T. Abdeljawad, Generalized Fractional Derivatives and Laplace Transform, Discrete & Continuous Dynamical Systems-S, 13 (2019), 709-722.  doi: 10.3934/dcdss.2020039.  Google Scholar

[25]

D. Kaya, S. Gulbahar, A. Yokus and M. Gulbahar, Solutions of the fractional combined kdv-mkdv equation with collocation method using radial basis function and their geometrical obstructions, Advances in Difference Equations, 2018 (2018), Paper No. 77, 16 pp. doi: 10.1186/s13662-018-1531-0.  Google Scholar

[26]

D. Kaya and S. M. El-Sayed, A numerical simulation and explicit solutions of the generalized burgers–fisher equation, Applied Mathematics and Computation, 152 (2004), 403-413.  doi: 10.1016/S0096-3003(03)00565-4.  Google Scholar

[27]

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

[28]

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

[29]

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, Advances in Difference Equations, 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6.  Google Scholar

[30]

I. Podlubny, Fractional Differential Equation: An Introduction To Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Elsevier, 198, 1999.  Google Scholar

[31]

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 & Continuous Dynamical Systems-S, 12 (2019), 645-664.   Google Scholar

[32]

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

[33]

N. Sene, Stokes first problem for heated flat plate with atangana–baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 68-75.  doi: 10.1016/j.chaos.2018.10.014.  Google Scholar

[34]

N. A. SheikhF. AliM. SaqibI. KhanS. A. A. JanA. S. Alshomrani and M. S. Alghamdi, Comparison and analysis of the atangana aleanu and caputo- abrizio fractional derivatives for generalized casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), 789-800.   Google Scholar

[35]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[36]

R. SubashiniC. RavichandranK. Jothimani and H. M. Baskonus, Existence results of hilfer integro-differential equations with fractional order, Discrete & Continuous Dynamical Systems-S, 13 (2020), 911-923.  doi: 10.3934/dcdss.2020053.  Google Scholar

[37]

T. A. Sulaiman, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous burgers equation involving mittag-leffler kernel, Physica A: Statistical Mechanics and its Applications, 527 (2019), 121126, 20pp. doi: 10.1016/j.physa.2019.121126.  Google Scholar

[38]

A. TouchentZ. HammouchT. Mekkaoui and F. Belgacem, Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional PDEs, Fractal and Fractional, 2 (2018), 22.  doi: 10.3390/fractalfract2030022.  Google Scholar

[39]

S. UcarE. UcarN. Özdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with atangana aleanu derivative, Chaos, Solitons & Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar

[40]

F. Usta and Z. Sarikaya, On generalization of pachpatte type inequalities for conformable fractional integral, TWMS Journal of Applied and Engineering Mathematics, 8 (2018), 106-113.   Google Scholar

[41]

F. Usta and M. Z. Sarikaya, The analytical solution of van der pol and lienard differential equations within conformable fractional operator by retarded integral inequalities, Demonstratio Mathematica, 52 (2019), 204-212.  doi: 10.1515/dema-2019-0017.  Google Scholar

[42]

A.-M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations, Applied Mathematics and Computation, 169 (2005), 321-338.  doi: 10.1016/j.amc.2004.09.054.  Google Scholar

[43]

X. Wang and Y. Lu, Exact solutions of the extended Burgers–Fisher equation, Chinese Physics Letters, 7 (1990), 145-147.  doi: 10.1088/0256-307X/7/4/001.  Google Scholar

[44]

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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. (37)
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. (38)
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. (39)
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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