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

Ömer Oruç; Alaattin Esen; Fatih Bulut

doi:10.3934/dcdss.2019035

Article Contents

2019, Volume 12, Issue 3: 533-542. Doi: 10.3934/dcdss.2019035

This issue Previous Article Weak Galerkin mixed finite element methods for parabolic equations with memory Next Article Numerical analysis and pattern formation process for space-fractional superdiffusive systems

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
* Corresponding author: Tel.:+904223773745

Received: February 2017

Revised: September 2017

Published: October 2018

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 65T60, 65N35; Secondary: 35R11.

Citation:

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Figure 1. Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0025$, $m' = 10$ and $\nu = 1$ at $t = 1$

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Figure 2. Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0005$ and $\nu = 1$ at $t = 0.1$

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Figure 3. Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.005$ and $\nu = 1$ at $t = 0.5$

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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
	$N=40$	$m'=10$	$N=40$	$m'=10$
$L_{2}\times10^{3}$	0.096733	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

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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
	$N=40$	$m'=10$	$N=40$	$m'=10$
$L_{2}\times10^{3}$	0.434586	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

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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
	$N=80$	$m'=10$	$N=80$	$m'=10$
$L_{2}\times10^{3}$	0.006528	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

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Table 4. Error norms for various collocation points and for $\Delta t = 0.00025$, $\alpha = 0.5$ at $t = 1$

	[6]	Present	[6]	Present
	$N=10$	$m'=10$	$N=20$	$m'=20$
$L_{2}\times10^{3}$	1.787278	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

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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
	$N=120$	$m'=16$	$N=120$	$m'=16$
$L_{2}\times10^{3}$	1.220123	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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