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

Asif Yokus; Mehmet Yavuz

doi:10.3934/dcdss.2020258

Article Contents

2021, Volume 14, Issue 7: 2591-2606. Doi: 10.3934/dcdss.2020258

This issue Previous Article Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives Next Article

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

Asif Yokus^1, and
Mehmet Yavuz^2, ,

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
^* Corresponding author: mehmetyavuz@erbakan.edu.tr

Received: May 31, 2019

Revised: August 31, 2020

Early access: May 2021

Published: July 2021

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Abstract

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.

Keywords:

Nonlinear time–fractional Burger–Fisher equation,
finite difference method,
Laplace perturbation method,
linear stability,
analytical solution,
Caputo fractional derivative.

Mathematics Subject Classification: Primary: 26A33, 35R11, 65M06; Secondary: 65H20.

Citation:

Full Text(HTML)

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)

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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)

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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)

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Figure 4. 2D Numerical and exact travelling wave solution and absolute error of the Eq. (1) for Finite Difference and Laplace Perturbation Methods

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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} $

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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

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