# 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
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##### 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
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