<abstract xmlns="http://www.w3.org/1999/xhtml">In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms <italic>L</italic>2 and <italic>L</italic>∞ have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.</abstract>

In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms L2 and L∞ have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"A Crank-Nicolson Approximation for the time Fractional Burgers Equation"}

In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation....

	N = 10	N = 20	N = 40	M = 80

L₂ × 10³	22.64087326	5.44483424	1.22007333	0.16846258
L_∞ × 10³	30.49445082	7.70051177	1.72552915	0.23826679

	N = 10	N = 20	N = 40	N = 80

L₂ × 10³	0.81667090	0.23594859	0.09215464	0.05901177
L_∞ × 10³	1.13678094	0.32086237	0.12245871	0.09790576


	N = 40		N = 80		N = 100
	Present	[9]	Present	[9]	Present	[9]
L₂ × 10³	1.22007333	1.224329	0.16846258	0.177703	0.04239382	0.052299
L_∞ × 10³	1.72552915	1.730469	0.23826679	0.253053	0.05996900	0.076541

	γ = 0.1	γ = 0.25	γ = 0.75	γ = 0.9

L₂ × 10³	0.02411976	0.02490518	0.02669547	0.02579288
L_∞ × 10³	0.03409905	0.03521115	0.03774592	0.03646791

	Δt = 0.0001	Δt = 0.00025	Δt = 0.0005	Δt = 0.001

L₂ × 10³	0.08344454	0.22252258	0.45497994	0.92040447
L_∞ × 10³	0.23648323	0.59722584	1.19848941	2.40107923

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Published Online: Aug 20, 2020

Page range: 177 - 184

Received: Jul 27, 2019

Accepted: Jul 29, 2019

DOI: https://doi.org/10.2478/amns.2020.2.00023

Keywords
Fractional order derivatives, Crank Nicolson Finite Difference methods, Fractional Burgers equation

© 2020 M. Onal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

The error norms L2 and L∞ of the time fractional Burgers equation problem using the Crank Nicolson finite difference method with ν = 1.0, Δt = 0.00025 and tf =1 for various values of M

Comparison of results at tf = 1.0 for γ = 0.5, Δt = 0.0001, ν = 1.0 and various mesh sizes

A comparison of the errors for Example 1 at tf =1

The error norms L2 and L∞ of Example 1 for N=120, tf = 1.0, Δt = 0.00025 for different values of γ

Comparison of results at tf = 1.0 for ν = 1, Δx = 0.025 and various time steps

The error norms L2 and L∞ of Example 1 for γ=0.5, Δt=0.00025, tf =1.0, N=40 and various values of ν

Comparison of results at tf = 1.0 for ν=1, N = 40 and various time steps

	ν = 1	ν = 0.5	ν = 0.1	ν = 0.01	ν = 0.001

L₂ × 10³	0.41761157	0.51259161	1.03928112	2.13880314	6.46231244
L_∞ × 10³	0.59040780	0.72342731	1.51280185	4.65707275	22.95302718

	Δt = 0.0001	Δt = 0.00025	Δt = 0.0005	Δt = 0.001	Δt = 0.0025	Δt = 0.005

L₂ × 10³	0.09215464	0.16142590	0.27906906	0.51583267	1.22775834	2.41506007
L_∞ × 10³	0.12245871	0.24577898	0.47478371	0.93280810	2.30696715	4.59744900

A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Published Online: Aug 20, 2020

Page range: 177 - 184

Received: Jul 27, 2019

Accepted: Jul 29, 2019

DOI: https://doi.org/10.2478/amns.2020.2.00023

KeywordsFractional order derivatives, Crank Nicolson Finite Difference methods, Fractional Burgers equation

© 2020 M. Onal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

The error norms L2 and L∞ of the time fractional Burgers equation problem using the Crank Nicolson finite difference method with ν = 1.0, Δt = 0.00025 and tf =1 for various values of M

Comparison of results at tf = 1.0 for γ = 0.5, Δt = 0.0001, ν = 1.0 and various mesh sizes

A comparison of the errors for Example 1 at tf =1

The error norms L2 and L∞ of Example 1 for N=120, tf = 1.0, Δt = 0.00025 for different values of γ

Comparison of results at tf = 1.0 for ν = 1, Δx = 0.025 and various time steps

The error norms L2 and L∞ of Example 1 for γ=0.5, Δt=0.00025, tf =1.0, N=40 and various values of ν

Comparison of results at tf = 1.0 for ν=1, N = 40 and various time steps

Keywords
Fractional order derivatives, Crank Nicolson Finite Difference methods, Fractional Burgers equation