# 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:

show all references

##### References:
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] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [2] Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793 [3] 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 [4] Stephanie Flores, Elijah Hight, Everardo Olivares-Vargas, Tamer Oraby, Jose Palacio, Erwin Suazo, Jasang Yoon. Exact and numerical solution of stochastic Burgers equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2735-2750. doi: 10.3934/dcdss.2020224 [5] Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080 [6] 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 [7] Jingwei Hu, Jie Shen, Yingwei Wang. A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions. Kinetic & Related Models, 2020, 13 (4) : 677-702. doi: 10.3934/krm.2020023 [8] Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295 [9] Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2020 Impact Factor: 2.425