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 _{2} and _{∞} 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.

#### Keywords

- Fractional order derivatives
- Crank Nicolson Finite Difference methods
- Fractional Burgers equation

#### MSC 2010

- 35R11
- 65N06

Fractional order integral and derivative are the generalizations of classical integral and derivative concepts which are examined in detail by Leibniz and Newton. The concepts of fractional integral and derivative are as old as integer order integral and derivative concepts, and the fractional derivative expression is first mentioned by Leibniz's letter to L’Hospital in 1695 [1]. In the letter, Leibniz's question was ’Can the integer order derivatives be generalized to fractional order derivative’. This is known as the first emergence of the concept of fractional differential. In addition to Leibniz, many scientists such as Liouville, Riemann, Weyl, Lagrange, Laplace, Fourier, Euler and Abel have worked on the same subject [2]. Many definitions are given in the literature for fractional derivative. Some of these are Riemann-Liouville, Caputo, Grünwald-Letnikov, Wely, Riesy fractional derivatives [3]. Some studies have shown that these definitions are equivalent under certain conditions. There is more than one derivative definition in the fractional analysis, making it possible to use the most suitable one according to the problem and thus to get the best solution for this problem.

However, if the derivative is described as how the derivative of the fractional order is defined, the expression that when the order is selected being equal to the integer is the same as the integer order of the derivative.

The description of Caputo fractional derivative was first introduced by the Italian mathematician M.Caputo in the 1960s to eliminate the problem of the calculation of Riemann-Liouville definition of initial values in Laplace transform applications. The fundamental advantage of the Caputo approach is the fact that the appropriate initial conditions defined for Caputo fractional differential equations are identical. Therefore, in recent studies in the literature, in the exact and approximate solutions of fractional differential equations, instead of Riemann-Liouville fractional derivative operator, Caputo fractional derivative operator has been more preferred. Recently, studies on the solution of fractional differential equations have increased. There are several studies about fractional problems and their computational accuracy in the literature [YoussefI. K.El DewaikM. H.

Finite difference methods are widely used in the solution of many linear and nonlinear partial differential equations. In general, the following way is followed in applying a finite difference method to a partial differential equation:

The given solution area of the problem is divided into meshes with geometric shapes and approximate solution for the problem is calculated on the nodes of each mesh. Proper finite difference approaches are obtained by using Taylor series instead of derivatives in differential equations. Thus, the present problem of solution of the differential equation has been converted into the problem of the solution of an algebraic system of equation consisting of difference equations. Thus the algebraic equation system obtained now may be solved easily by one of the direct or iterative methods.

In this manuscript, we will deal with the nonlinear time fractional Burgers equation with the given initial and boundary conditions as a test problem presented as

Throughout this manuscript, in order to contribute to literature, we will consider numerical solutions of the time fractional Burgers equations using finite difference approach. For the presented numerical solutions, to get a Crank Nicolson finite difference scheme to solve the time fractional Burgers equation as utilized in explicit difference method in Ref. [17], we will also discretize the derivative respect to time using the widely-known L1 formula [18]

Let's assume the fact that the solution domain for the present problem 0 ≤ _{j}_{n}_{x}_{xx}

We are able to get the following system of algebraic equations
^{γ}

Numerical results obtained for the Eq. (1) are got using the Crank Nicolson finite difference methods. The efficiency of the present methods are tested using the error norm _{2}_{∞}

_{f}_{2}_{∞}

The error norms _{2} and _{∞} of the time fractional Burgers equation problem using the Crank Nicolson finite difference method with _{f}

_{2} × 10^{3} | 22.64087326 | 5.44483424 | 1.22007333 | 0.16846258 |

_{∞} × 10^{3} | 30.49445082 | 7.70051177 | 1.72552915 | 0.23826679 |

A comparison of the errors for Example 1 at _{f}

Present | [9] | Present | [9] | Present | [9] | |
---|---|---|---|---|---|---|

_{2} × 10^{3} | 1.22007333 | 1.224329 | 0.16846258 | 0.177703 | 0.04239382 | 0.052299 |

_{∞} × 10^{3} | 1.72552915 | 1.730469 | 0.23826679 | 0.253053 | 0.05996900 | 0.076541 |

_{f}

The error norms _{2} and _{∞} of Example 1 for _{f}

_{2} × 10^{3} | 0.41761157 | 0.51259161 | 1.03928112 | 2.13880314 | 6.46231244 |

_{∞} × 10^{3} | 0.59040780 | 0.72342731 | 1.51280185 | 4.65707275 | 22.95302718 |

The error norms _{2} and _{∞} of Example 1 for _{f}

_{2} × 10^{3} | 0.02411976 | 0.02490518 | 0.02669547 | 0.02579288 |

_{∞} × 10^{3} | 0.03409905 | 0.03521115 | 0.03774592 | 0.03646791 |

Comparison of results at _{f}

_{2} × 10^{3} | 0.81667090 | 0.23594859 | 0.09215464 | 0.05901177 |

_{∞} × 10^{3} | 1.13678094 | 0.32086237 | 0.12245871 | 0.09790576 |

Comparison of results at _{f}

Δ | Δ | Δ | Δ | Δ | Δ | |
---|---|---|---|---|---|---|

_{2} × 10^{3} | 0.09215464 | 0.16142590 | 0.27906906 | 0.51583267 | 1.22775834 | 2.41506007 |

_{∞} × 10^{3} | 0.12245871 | 0.24577898 | 0.47478371 | 0.93280810 | 2.30696715 | 4.59744900 |

_{f}

Comparison of results at _{f}

Δ | Δ | Δ | Δ | |
---|---|---|---|---|

_{2} × 10^{3} | 0.08344454 | 0.22252258 | 0.45497994 | 0.92040447 |

_{∞} × 10^{3} | 0.23648323 | 0.59722584 | 1.19848941 | 2.40107923 |

As a conclusion, in the present study the numerical solutions of the time fractional Burgers equation have been obtained by using finite difference method based on Crank-Nicolson discretization. The obtained results are compared with analytic and some of the numerical results available in the literature. This comparison has shown that the presented method is efficient and effective and can also be used for a wide range of physical and scientific applications. Moreover to illustrate the accuracy of the present method the error norms _{2} and _{∞} are computed and given in tables. Two test problems have been used to show the accuracy of the present scheme for various values of parameters in the problem. Tables and figures show the results of these various tests and also comparisons with some available results together with the error norms _{2} and _{∞}. Finally the present method has been shown to be applicable for more widely used fractional differential equations.

#### 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

_{2} × 10^{3} | 22.64087326 | 5.44483424 | 1.22007333 | 0.16846258 |

_{∞} × 10^{3} | 30.49445082 | 7.70051177 | 1.72552915 | 0.23826679 |

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

_{2} × 10^{3} | 0.81667090 | 0.23594859 | 0.09215464 | 0.05901177 |

_{∞} × 10^{3} | 1.13678094 | 0.32086237 | 0.12245871 | 0.09790576 |

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

Present | [ | Present | [ | Present | [ | |
---|---|---|---|---|---|---|

_{2} × 10^{3} | 1.22007333 | 1.224329 | 0.16846258 | 0.177703 | 0.04239382 | 0.052299 |

_{∞} × 10^{3} | 1.72552915 | 1.730469 | 0.23826679 | 0.253053 | 0.05996900 | 0.076541 |

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

_{2} × 10^{3} | 0.02411976 | 0.02490518 | 0.02669547 | 0.02579288 |

_{∞} × 10^{3} | 0.03409905 | 0.03521115 | 0.03774592 | 0.03646791 |

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

Δ | Δ | Δ | Δ | |
---|---|---|---|---|

_{2} × 10^{3} | 0.08344454 | 0.22252258 | 0.45497994 | 0.92040447 |

_{∞} × 10^{3} | 0.23648323 | 0.59722584 | 1.19848941 | 2.40107923 |

#### 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

_{2} × 10^{3} | 0.41761157 | 0.51259161 | 1.03928112 | 2.13880314 | 6.46231244 |

_{∞} × 10^{3} | 0.59040780 | 0.72342731 | 1.51280185 | 4.65707275 | 22.95302718 |

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

Δ | Δ | Δ | Δ | Δ | Δ | |
---|---|---|---|---|---|---|

_{2} × 10^{3} | 0.09215464 | 0.16142590 | 0.27906906 | 0.51583267 | 1.22775834 | 2.41506007 |

_{∞} × 10^{3} | 0.12245871 | 0.24577898 | 0.47478371 | 0.93280810 | 2.30696715 | 4.59744900 |

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