# American Institute of Mathematical Sciences

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. [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. [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. [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. [5] I. Daubechies, Ten Lectures on Wavelet, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104. [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. [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. [8] M. H. Heydari, M. 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. [9] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic, New York, 1974. [10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [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. [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. [13] S. G. Rubin and R. A. Graves, Cubic spline approximation for problems in fluid mechanics, NASA TR R-436, Washington, DC, 1975. [14] B. S. T. Alkahtani, A. 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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [5] I. Daubechies, Ten Lectures on Wavelet, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970104. [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. [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. [8] M. H. Heydari, M. 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. [9] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic, New York, 1974. [10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [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. [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. [13] S. G. Rubin and R. A. Graves, Cubic spline approximation for problems in fluid mechanics, NASA TR R-436, Washington, DC, 1975. [14] B. S. T. Alkahtani, A. 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. [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. [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. [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. [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. [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. [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. [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.
Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0025$, $m' = 10$ and $\nu = 1$ at $t = 1$
Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0005$ and $\nu = 1$ at $t = 0.1$
Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.005$ and $\nu = 1$ at $t = 0.5$
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 $N=40$ $m'=10$ $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.096733 0.075146 0.090053 0.073586 $L_{\infty}\times10^{3}$ 0.272943 0.106340 0.258623 0.104141 $\alpha=0.75$ [6] Present $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.035448 0.069536 $L_{\infty}\times10^{3}$ 0.124569 0.098312
 $\alpha=0.1$ $\alpha=0.25$ [6] Present [6] Present $N=40$ $m'=10$ $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.096733 0.075146 0.090053 0.073586 $L_{\infty}\times10^{3}$ 0.272943 0.106340 0.258623 0.104141 $\alpha=0.75$ [6] Present $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.035448 0.069536 $L_{\infty}\times10^{3}$ 0.124569 0.098312
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 $N=40$ $m'=10$ $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.434586 0.570509 0.176195 0.284035 $L_{\infty}\times10^{3}$ 0.642003 0.807275 0.265419 0.401953 $\Delta t=0.0005$ [6] Present $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.068869 0.141630 $L_{\infty}\times10^{3}$ 0.211883 0.200442
 $\Delta t=0.002$ $\Delta t=0.001$ [6] Present [6] Present $N=40$ $m'=10$ $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.434586 0.570509 0.176195 0.284035 $L_{\infty}\times10^{3}$ 0.642003 0.807275 0.265419 0.401953 $\Delta t=0.0005$ [6] Present $N=40$ $m'=10$ $L_{2}\times10^{3}$ 0.068869 0.141630 $L_{\infty}\times10^{3}$ 0.211883 0.200442
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 $N=80$ $m'=10$ $N=80$ $m'=10$ $L_{2}\times10^{3}$ 0.006528 0.006980 0.005835 0.006492 $L_{\infty}\times10^{3}$ 0.009164 0.009547 0.008250 0.008854 $\nu=0.1$ [6] Present $N=80$ $m'=10$ $L_{2}\times10^{3}$ 0.003105 0.004288 $L_{\infty}\times10^{3}$ 0.004847 0.005714
 $\nu=1$ $\nu=0.5$ [6] Present [6] Present $N=80$ $m'=10$ $N=80$ $m'=10$ $L_{2}\times10^{3}$ 0.006528 0.006980 0.005835 0.006492 $L_{\infty}\times10^{3}$ 0.009164 0.009547 0.008250 0.008854 $\nu=0.1$ [6] Present $N=80$ $m'=10$ $L_{2}\times10^{3}$ 0.003105 0.004288 $L_{\infty}\times10^{3}$ 0.004847 0.005714
Error norms for various collocation points and for $\Delta t = 0.00025$, $\alpha = 0.5$ at $t = 1$
 [6] Present [6] Present $N=10$ $m'=10$ $N=20$ $m'=20$ $L_{2}\times10^{3}$ 1.787278 0.024252 0.440305 0.024212 $L_{\infty}\times10^{3}$ 2.415589 0.032824 0.583583 0.033666 [6] Present $N=40$ $m'=40$ $L_{2}\times10^{3}$ 0.092735 0.024210 $L_{\infty}\times10^{3}$ 0.120495 0.033727
 [6] Present [6] Present $N=10$ $m'=10$ $N=20$ $m'=20$ $L_{2}\times10^{3}$ 1.787278 0.024252 0.440305 0.024212 $L_{\infty}\times10^{3}$ 2.415589 0.032824 0.583583 0.033666 [6] Present $N=40$ $m'=40$ $L_{2}\times10^{3}$ 0.092735 0.024210 $L_{\infty}\times10^{3}$ 0.120495 0.033727
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 $N=120$ $m'=16$ $N=120$ $m'=16$ $L_{2}\times10^{3}$ 1.220123 1.153760 0.532436 0.466776 $L_{\infty}\times10^{3}$ 1.725765 1.563758 0.753171 0.609456 $\Delta t=0.0005$ [6] Present $N=120$ $m'=16$ $L_{2}\times10^{3}$ 0.188710 0.126335 $L_{\infty}\times10^{3}$ 0.267546 0.180767
 $\Delta t=0.002$ $\Delta t=0.001$ [6] Present [6] Present $N=120$ $m'=16$ $N=120$ $m'=16$ $L_{2}\times10^{3}$ 1.220123 1.153760 0.532436 0.466776 $L_{\infty}\times10^{3}$ 1.725765 1.563758 0.753171 0.609456 $\Delta t=0.0005$ [6] Present $N=120$ $m'=16$ $L_{2}\times10^{3}$ 0.188710 0.126335 $L_{\infty}\times10^{3}$ 0.267546 0.180767
 [1] Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793 [2] Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951 [3] Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429 [4] Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365 [5] Zhaosheng Feng, Qingguo Meng. Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 285-291. doi: 10.3934/dcdsb.2007.7.285 [6] Jianzhong Wang. Wavelet approach to numerical differentiation of noisy functions. Communications on Pure & Applied Analysis, 2007, 6 (3) : 873-897. doi: 10.3934/cpaa.2007.6.873 [7] Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121 [8] Taposh Kumar Das, Óscar López Pouso. New insights into the numerical solution of the Boltzmann transport equation for photons. Kinetic & Related Models, 2014, 7 (3) : 433-461. doi: 10.3934/krm.2014.7.433 [9] T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277 [10] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [11] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [12] Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 [13] Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503 [14] Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737 [15] Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173 [16] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [17] Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25 [18] Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022 [19] Fausto Cavalli, Giovanni Naldi. A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation. Kinetic & Related Models, 2010, 3 (1) : 123-142. doi: 10.3934/krm.2010.3.123 [20] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571

2018 Impact Factor: 0.545