June  2019, 12(3): 533-542. doi: 10.3934/dcdss.2019035

A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation

1. 

Eǧil Vocational and Technical Anatolian High School, Diyarbakır, Turkey

2. 

İnonu University, Department of Mathematics, Malatya, Turkey

3. 

İnonu University, Department of Physics, Malatya, Turkey

* Corresponding author: Tel.:+904223773745

Received  February 2017 Revised  September 2017 Published  September 2018

In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.

Citation: Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035
References:
[1]

A. Atangana and J. F. Gez-Aguilar, A new derivative with normal distribution kernel: Theory, methods and applications, Physica A: Statistical Mechanics and its Applications, 476 (2017), 1-14.  doi: 10.1016/j.physa.2017.02.016.  Google Scholar

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I. Celik, Chebyshev Wavelet collocation method for solving generalized Burgers- Huxley equation, Mathematical Methods in the Applied Sciences, 39 (2016), 366-377.  doi: 10.1002/mma.3487.  Google Scholar

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I. Daubechies, Ten Lectures on Wavelet, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

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A. Esen and O. Tasbozan, Numerical solution of time fractional burgers equation by cubic b-spline finite elements, Mediterranean Journal of Mathematics, 13 (2016), 1325-1337.  doi: 10.1007/s00009-015-0555-x.  Google Scholar

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A. K. Gupta and S. Saha Ray, Travelling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena, AIP Adv., 4 (2014), http://dx.doi.org/10.1063/1.4895910. 097120-1-11. Google Scholar

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M. H. HeydariM. R. Hooshmandasl and F. M. Maalek Ghaini, A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Applied Mathematical Modelling, 38 (2014), 1597-1606.  doi: 10.1016/j.apm.2013.09.013.  Google Scholar

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K. B. Oldham and J. Spanier, The Fractional Calculus, Academic, New York, 1974.  Google Scholar

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I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

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M. Razzaghi and S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and Computers in Simulation, 53 (2000), 185-192.  doi: 10.1016/S0378-4754(00)00170-1.  Google Scholar

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M. Razzaghi and S. Yousefi, Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32 (2001), 495-502.  doi: 10.1080/00207720120227.  Google Scholar

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S. G. Rubin and R. A. Graves, Cubic spline approximation for problems in fluid mechanics, NASA TR R-436, Washington, DC, 1975. Google Scholar

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B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, Journal of Nonlinear Sciences and Applications, 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

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J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. Google Scholar

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P. K. Sahu and S. Saha Ray, Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system, Appl. Math. Comput., 256 (2015), 715-723.  doi: 10.1016/j.amc.2015.01.063.  Google Scholar

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P. K. Sahu and S. Saha Ray, Two dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations, J. Intell. Fuzzy Syst., 28 (2015), 1271-1279.   Google Scholar

[18]

Y. Wang and Q. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218 (2012), 8592-8601.  doi: 10.1016/j.amc.2012.02.022.  Google Scholar

[19]

C. Yang and J. Hou, Chebyshev wavelets method for solving Bratu's problem, Boundary Value Problems, 142 (2013), 1-9.  doi: 10.1186/1687-2770-2013-142.  Google Scholar

[20]

F. Zhou and X. Xu, Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets, Applied Mathematics and Computation, 247 (2014), 353-367.  doi: 10.1016/j.amc.2014.08.091.  Google Scholar

[21]

L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. Nonlinear Sci Numer. Simul., 17 (2012), 2333-2341.  doi: 10.1016/j.cnsns.2011.10.014.  Google Scholar

show all references

References:
[1]

A. Atangana and J. F. Gez-Aguilar, A new derivative with normal distribution kernel: Theory, methods and applications, Physica A: Statistical Mechanics and its Applications, 476 (2017), 1-14.  doi: 10.1016/j.physa.2017.02.016.  Google Scholar

[2]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons & Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.  Google Scholar

[3]

E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, 188 (2007), 417-426.  doi: 10.1016/j.amc.2006.10.008.  Google Scholar

[4]

I. Celik, Chebyshev Wavelet collocation method for solving generalized Burgers- Huxley equation, Mathematical Methods in the Applied Sciences, 39 (2016), 366-377.  doi: 10.1002/mma.3487.  Google Scholar

[5]

I. Daubechies, Ten Lectures on Wavelet, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[6]

A. Esen and O. Tasbozan, Numerical solution of time fractional burgers equation by cubic b-spline finite elements, Mediterranean Journal of Mathematics, 13 (2016), 1325-1337.  doi: 10.1007/s00009-015-0555-x.  Google Scholar

[7]

A. K. Gupta and S. Saha Ray, Travelling wave solution of fractional KdV-Burger-Kuramoto equation describing nonlinear physical phenomena, AIP Adv., 4 (2014), http://dx.doi.org/10.1063/1.4895910. 097120-1-11. Google Scholar

[8]

M. H. HeydariM. R. Hooshmandasl and F. M. Maalek Ghaini, A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Applied Mathematical Modelling, 38 (2014), 1597-1606.  doi: 10.1016/j.apm.2013.09.013.  Google Scholar

[9]

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic, New York, 1974.  Google Scholar

[10]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[11]

M. Razzaghi and S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and Computers in Simulation, 53 (2000), 185-192.  doi: 10.1016/S0378-4754(00)00170-1.  Google Scholar

[12]

M. Razzaghi and S. Yousefi, Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32 (2001), 495-502.  doi: 10.1080/00207720120227.  Google Scholar

[13]

S. G. Rubin and R. A. Graves, Cubic spline approximation for problems in fluid mechanics, NASA TR R-436, Washington, DC, 1975. Google Scholar

[14]

B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, Journal of Nonlinear Sciences and Applications, 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

[15]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. Google Scholar

[16]

P. K. Sahu and S. Saha Ray, Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system, Appl. Math. Comput., 256 (2015), 715-723.  doi: 10.1016/j.amc.2015.01.063.  Google Scholar

[17]

P. K. Sahu and S. Saha Ray, Two dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations, J. Intell. Fuzzy Syst., 28 (2015), 1271-1279.   Google Scholar

[18]

Y. Wang and Q. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218 (2012), 8592-8601.  doi: 10.1016/j.amc.2012.02.022.  Google Scholar

[19]

C. Yang and J. Hou, Chebyshev wavelets method for solving Bratu's problem, Boundary Value Problems, 142 (2013), 1-9.  doi: 10.1186/1687-2770-2013-142.  Google Scholar

[20]

F. Zhou and X. Xu, Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets, Applied Mathematics and Computation, 247 (2014), 353-367.  doi: 10.1016/j.amc.2014.08.091.  Google Scholar

[21]

L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. Nonlinear Sci Numer. Simul., 17 (2012), 2333-2341.  doi: 10.1016/j.cnsns.2011.10.014.  Google Scholar

Figure 1.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0025$, $m' = 10$ and $\nu = 1$ at $t = 1$
Figure 2.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0005$ and $\nu = 1$ at $t = 0.1$
Figure 3.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.005$ and $\nu = 1$ at $t = 0.5$
Table 1.  Error norms for various values of $\alpha$ and for $\Delta t = 0.00025$ at $t = 1$
$\alpha=0.1$ $\alpha=0.25$
[6]Present[6]Present
$N=40$ $m'=10$ $N=40$ $m'=10$
$L_{2}\times10^{3}$0.0967330.0751460.0900530.073586
$L_{\infty}\times10^{3}$0.2729430.1063400.2586230.104141
$\alpha=0.75$
[6]Present
$N=40$ $m'=10$
$L_{2}\times10^{3}$0.0354480.069536
$L_{\infty}\times10^{3}$0.1245690.098312
$\alpha=0.1$ $\alpha=0.25$
[6]Present[6]Present
$N=40$ $m'=10$ $N=40$ $m'=10$
$L_{2}\times10^{3}$0.0967330.0751460.0900530.073586
$L_{\infty}\times10^{3}$0.2729430.1063400.2586230.104141
$\alpha=0.75$
[6]Present
$N=40$ $m'=10$
$L_{2}\times10^{3}$0.0354480.069536
$L_{\infty}\times10^{3}$0.1245690.098312
Table 2.  Error norms for various values of $\Delta t$ and for $\nu = 1$, $\alpha = 0.5$ at $t = 1$
$\Delta t=0.002$ $\Delta t=0.001$
[6]Present[6]Present
$N=40$ $m'=10$ $N=40$ $m'=10$
$L_{2}\times10^{3}$0.4345860.5705090.1761950.284035
$L_{\infty}\times10^{3}$ 0.6420030.8072750.2654190.401953
$\Delta t=0.0005$
[6]Present
$N=40$ $m'=10$
$L_{2}\times10^{3}$0.0688690.141630
$L_{\infty}\times10^{3}$0.2118830.200442
$\Delta t=0.002$ $\Delta t=0.001$
[6]Present[6]Present
$N=40$ $m'=10$ $N=40$ $m'=10$
$L_{2}\times10^{3}$0.4345860.5705090.1761950.284035
$L_{\infty}\times10^{3}$ 0.6420030.8072750.2654190.401953
$\Delta t=0.0005$
[6]Present
$N=40$ $m'=10$
$L_{2}\times10^{3}$0.0688690.141630
$L_{\infty}\times10^{3}$0.2118830.200442
Table 3.  Error norms for various values of $\nu$ and for $\Delta t = 0.0005$, $\alpha = 0.5$ at $t = 0.1$
$\nu=1$ $\nu=0.5$
[6]Present[6]Present
$N=80$ $m'=10$ $N=80$ $m'=10$
$L_{2}\times10^{3}$0.0065280.0069800.0058350.006492
$L_{\infty}\times10^{3}$ 0.0091640.0095470.0082500.008854
$\nu=0.1$
[6]Present
$N=80$ $m'=10$
$L_{2}\times10^{3}$ 0.0031050.004288
$L_{\infty}\times10^{3}$ 0.0048470.005714
$\nu=1$ $\nu=0.5$
[6]Present[6]Present
$N=80$ $m'=10$ $N=80$ $m'=10$
$L_{2}\times10^{3}$0.0065280.0069800.0058350.006492
$L_{\infty}\times10^{3}$ 0.0091640.0095470.0082500.008854
$\nu=0.1$
[6]Present
$N=80$ $m'=10$
$L_{2}\times10^{3}$ 0.0031050.004288
$L_{\infty}\times10^{3}$ 0.0048470.005714
Table 4.  Error norms for various collocation points and for $\Delta t = 0.00025$, $\alpha = 0.5$ at $t = 1$
[6] Present [6] Present
$N=10$ $m'=10$ $N=20$ $m'=20$
$L_{2}\times10^{3}$1.7872780.0242520.4403050.024212
$L_{\infty}\times10^{3}$2.4155890.0328240.5835830.033666
[6]Present
$N=40$ $m'=40$
$L_{2}\times10^{3}$0.0927350.024210
$L_{\infty}\times10^{3}$0.1204950.033727
[6] Present [6] Present
$N=10$ $m'=10$ $N=20$ $m'=20$
$L_{2}\times10^{3}$1.7872780.0242520.4403050.024212
$L_{\infty}\times10^{3}$2.4155890.0328240.5835830.033666
[6]Present
$N=40$ $m'=40$
$L_{2}\times10^{3}$0.0927350.024210
$L_{\infty}\times10^{3}$0.1204950.033727
Table 5.  Error norms for various values of $\Delta t$ and for $\nu = 1$, $\alpha = 0.5$ at $t = 1$
$\Delta t=0.002$ $\Delta t=0.001$
[6]Present[6]Present
$N=120$ $m'=16$ $N=120$ $m'=16$
$L_{2}\times10^{3}$1.2201231.1537600.5324360.466776
$L_{\infty}\times10^{3}$1.7257651.5637580.7531710.609456
$\Delta t=0.0005$
[6]Present
$N=120$ $m'=16$
$L_{2}\times10^{3}$0.1887100.126335
$L_{\infty}\times10^{3}$0.2675460.180767
$\Delta t=0.002$ $\Delta t=0.001$
[6]Present[6]Present
$N=120$ $m'=16$ $N=120$ $m'=16$
$L_{2}\times10^{3}$1.2201231.1537600.5324360.466776
$L_{\infty}\times10^{3}$1.7257651.5637580.7531710.609456
$\Delta t=0.0005$
[6]Present
$N=120$ $m'=16$
$L_{2}\times10^{3}$0.1887100.126335
$L_{\infty}\times10^{3}$0.2675460.180767
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