#### Abstract

The generalized time-fractional, one-dimensional, nonlinear Burgers equation with time-variable coefficients is numerically investigated. The classical Burgers equation is generalized by considering the generalized Atangana-Baleanu time-fractional derivative. The studied model contains as particular cases the Burgers equation with Atangana-Baleanu, Caputo-Fabrizio, and Caputo time-fractional derivatives. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed. Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

#### 1. Introduction

The nonlinear convective–diffusive partial differential equations describe various mathematical models in important fields such as heat and mass transfer, fluid mechanics, and engineering. During the last years, various solution methods of ordinary differential equations and partial differential equations have been elaborated.

Burgers’ equation is one of the most important equations involving both nonlinear propagation effects and diffusive effects. A particular form of Burgers’ equation describes the nonlinear wave propagation (the inviscid Burgers’ equation).

Burgers’ equation is a suitable tool for analysis in various fields such as turbulent flows, gas dynamics, shock wave theory, nonlinear wave propagation, longitudinal elastic waves in isotropic solids, sedimentation of polydispersive suspensions and colloids, growth of molecular interfaces, traffic flow, and cosmology [1].

By studying Burgers’ equation with random initial conditions or random forcing, Bec and Khanin [2] explained Burgers’ turbulence. The study of random Lagrangian systems, stochastic partial differential equations, the applications of field theory to the understanding of dissipative anomalies, and of multiscaling in hydrodynamic turbulence are some fields that have significantly benefited from the progress in Burgers’ turbulence. Yu [3] analytically studied the stability and density waves for traffic flow using the perturbation method and shown that the triangular shock waves, soliton wave, and kink wave appear for the density waves.

In the last years, researchers have proved that many phenomena in engineering, bioengineering, physics, and chemistry can be successfully described by mathematical models that use mathematical tools from fractional calculus, i.e., the theory of derivatives and integrals of noninteger order.

Models of viscoelastic materials, Caputo and Mainardi [4]; the signal processing, Marks and Hall [5]; diffusion problems, Olmstead and Handelsman [6]; viscoplastic materials modeling, Diethelm and Freed [7]; mechanical systems subject to damping, Gaul et al. [8]; relaxation and reaction kinetics of polymers, Glockle and Nonnenmacher [9]; and heat conduction, Hristov [10, 11] are some of the important problems modeled with the help of fractional differential operators.

A very useful collection of numerical algorithms for Caputo-type derivatives, Riemann-Liouville integral operator, and Mittag-Leffler functions is that of Diethelm et al. [12].

In the literature, there are articles in which solutions of Burgers’ equation with different time-fractional derivatives have been determined. We recall some of them. The effects of fractional-order of derivatives on the wave solutions of the generalized Zakharov-Kuznetsov-Burgers equation have been investigated by Faraz et al. [13]. The analytical approximate wave solutions are obtained using the homotopy analysis method and the time-fractional Caputo’s derivative, while exact solutions are determined with the help of the first integral method and the fractional derivative in Jumarie’s modified Riemann–Liouville sense. Bira et al. [14] studied a nonlinear time-fractional system of Boussinesq-Burgers equations. Using Lie group analysis, the authors derived the infinitesimal groups of transformations, the system of optimal algebras for the symmetry group of transformations, and the similarity variables that reduce the system of fractional partial differential equations to a system of fractional ordinary differential equations. The exact solutions and the physical significance of the solutions are obtained under the invariance condition. Saad et al. [15] have extended the model of the Burgers equation to generalized models based on Liouville-Caputo, Caputo-Fabrizio, and Mittag-Leffler time-fractional derivatives. Using the homotopy analysis transform method, the authors obtained approximate solutions of the newly proposed models. Baleanu and Shiri [16] numerically solved a system of fractional differential equations involving nonsingular Mittag-Leffler kernel using the collocation methods on discontinuous piecewise polynomial space. The existence and regularity of solutions and convergence of the introduced methods are derived.

Recently, Vieru et al. [17] have generalized the time-fractional Atangana-Baleanu derivative. The newly proposed definition contains as particular cases the time-fractional Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives.

It is important to note that the operators of fractional derivatives are nonlocal in time and therefore have the advantage of modeling phenomena with memory. Caputo fractional derivatives are nonlocal operators but their kernel is singular. This weakness could have a negative effect when modeling real-world problems.

The fractional derivative operators with the Mittag-Leffler kernel have all the benefits of Caputo operators; in addition, the kernel is nonsingular. Also, their fractional integral operators are the fractional average of the Riemann–Liouville fractional integral of the given function and the function itself. Caputo derivative was conceived for a description of linear short time elastic responses of deformed solids. It was consequently applied to the field of linear viscoelasticity where the Riemann-Liouville derivative was already applied to describe viscoelastic effects. It is known that the asymptotic behaviors of derivative operators with Mittag-Leffler kernel match the power-law behavior. The new fractional operators based on Mittag-Leffler functions have stronger and complex memories allowing capturing behaviors combining simultaneously (crossover) classical diffusion and anomalous behavior. Therefore, to model more complex and nonlinear phenomena, the new operators could be useful tools [11].

In this paper, a nonlinear, one-dimensional, generalized Burgers equation with time-variable coefficients is numerically studied. The generalization consists of considering the fractional differential Burgers’ equation with the generalized time-fractional Atangana-Baleanu fractional derivative with Mittag-Leffler kernel.

A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed along with the consistency, stability, and convergence of the proposed method.

It is important to point out that the studied generalized model can be customized to generate solutions to the problems described by the time-fractional Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractional derivatives.

Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

#### 2. Preliminary Mathematics

In this section, we present the basic mathematical elements regarding the two-parametric Mittag-Leffler functions and the generalized time-fractional Atangana-Baleanu derivatives. These mathematical notions are necessary for the next sections of this paper.

##### 2.1. One-Parametric and Two-Parametric Mittag-Leffler Functions

The classical one-parametric Mittag-Leffler function is defined as [18, 19] where is the Euler integral of the second kind.

The two-parametric Mittag-Leffler function generalizes the function and is defined by

It is easy to notice that function (1) is a particular case of function (2), so we have

Let us recall some properties of Mittag-Leffler functions.

The following special form of the one-parametric Mittag-Leffler function [12]: along with its derivative: has applications in the theory of fractional-order viscoelasticity and in some problems described by fractional differential equations with constant coefficients.

Some numerical algorithms for determining numerical values of the Mittag-Leffler functions have been presented in the reference [12]. These algorithms are based on the integral representations of the Mittag-Leffler functions. We will use in this paper the following integral representations:

If , then

The integral representation along with the definition of the Laplace transform of a function , give the following relationship:

In the particular case , Equation (12) becomes

##### 2.2. Generalized Atangana-Baleanu Time-Fractional Derivative

The function is called the generalized Atangana-Baleanu kernel.

The Laplace transform of the kernel (14) is given by

Using the Laplace transform, the following properties of the generalized Atangana-Baleanu kernel (14) are found: therefore,

In the above relations, functions , and are, respectively, Caputo kernel, Caputo-Fabrizio kernel, Atangana-Baleanu kernel, and the Dirac’s distribution.

*Definition 1. *The generalized Atangana-Baleanu fractional derivative in Caputo sense.

If the generalized Atangana-Baleanu fractional derivative in Caputo sense, of order of the function , is defined by the relation

Using Equations (17) and (18), we obtain the following properties of the generalized Atangana-Baleanu time-fractional derivative:

where denotes the time-fractional Caputo derivative, is time-fractional Caputo-Fabrizio derivative, and denotes the time-fractional Atangana-Baleanu derivative.

Associated with the generalized Atangana-Baleanu derivative, we define the following fractional integral operator:

where the kernel is defined as

It is observed that ; therefore,

Using property (22), the fractional integral operator can be defined for .

The fractional integral operator (20) has the following properties:

Regarding the generalized Atangana-Baleanu derivative and associated fractional integral operator, we remember the foloowingproposition.

Proposition 2. *The following relationships are fulfilled:
*

The demonstration of the above proposition can be found in the reference [17].

The generalized fractional integral operator (20) contains the following particular cases:

i.e., the well-known Riemann-Liouville fractional integral operator. that is, the integral operator associated to the Caputo-Fabrizio derivative.

that is, the fractional integral operator associated with the Atangana-Baleanu fractional derivative.

#### 3. Problem Formulation

The classical one-dimensional Burgers equation with variable coefficients, defined for , is [20]

where and are differentiable and bounded functions of the variable . For , Rizun and Engel’brekht [20] have determined the analytical solution of Equation (28).

In the present paper, we consider a generalized form of Equation (28), namely,

where is the generalized Atangana-Baleanu derivative defined by Equation (18). Along with Equation (29), we consider the initial and boundary conditions

In the following, we will elaborate a numerical scheme for determining the solution of problem (29)–(31). The proposed scheme is based on the finite-difference method and on the properties of the Mittag-Leffler functions.

Let us consider the discrete set of spatial nodes , respectively, the discrete set of the time nodes , where , are the increment steps of and , respectively.

##### 3.1. Numerical Evaluation of the Generalized Atangana-Baleanu Derivative

Using Equation (18), we have

The first-order time derivative is approximated by [21].

Using approximation (33), Equation (32) becomes where , and is the truncation error.

Now, using the properties of Mittag-Leffler functions given in Equation (7), we obtain where

Introducing notations where , Equation (35) can be written in the equivalent form

The truncation error is defined as where is a constant coming from the bounded of the Mittag-Leffler function. The inequality (39) ensures the consistency of the proposed method because, assuming that the function is sufficiently smooth on its domain of definition, the truncation error tends to zero if the time step tends to zero. So, at point , the generalized Atangana-Baleanu time-fractional derivative is approximated by

In the above relations, the numerical values of the Mittag-Leffler functions are evaluated using the integral representation (10).

An important property of the coefficients is given in the following.

Lemma 3. *Coefficients given by (37) have negative values for . For , and if .**Proof. Let a function defined as
*

Using the definition of Mittag-Leffler function, can be written as

Using the formula , the derivative of function is given by

It is known that Mittag-Leffler function is an increasing function. Because, in this study, parameter , it results that function is a decreasing function.

The coefficients are written as

Since, the function is decreasing we obtain that .

. Since , we obtain ; therefore, for . Using the asymptotic expansion of Mittag-Leffler function [22], we have that proves the property in Lemma 3.

In the following, two examples of the application of formula (40) are presented.

*Example 1. *The generalized Atangana-Baleanu derivative of function .

The time-fractional generalized Atangana-Baleanu derivative of function is given by

Table 1 gives the values of this derivative determined with the analytical expression (45), respectively, with the numerical formula (40) for ,.

Numerical results obtained by Equations (40) and (45) are graphically illustrated in Figure 1.

It can be seen in Table 1 and Figure 1 that there is a very good accuracy of the numerical method given by equation (40).

*Example 2. *Find the solution of the fractional equation

Using the Laplace transform, it is found that the analytical solution of Equation (46) is given by

Table 2 gives the values of the solution of Equation (46) determined with the analytical expression (47), respectively, with the numerical formula (40) for ,.

Numerical results for the solution of Equation (46), obtained by Equations (40) and (47) are graphically illustrated in Figure 2.

It can be seen in Table 2 and Figure 2 that there is a very good accuracy of the numerical method given by Equation (40).

##### 3.2. Particular Cases

Note that the expression (40) of the generalized Atangana-Baleanu time-fractional derivative can be easily customized in the following cases: (a)If , it is obtained the expression of the time-fractional Atangana-Baleanu derivative(b)If and , it is obtained the expression of the time-fractional Caputo-Fabrizio derivative(c)If and , it is obtained the expression of the time-fractional Caputo derivative

We must note that in the first two cases, the formulas (34)–(40) which determine the numerical values of the fractional derivative remain valid, obviously with the corresponding particularizations of the parameters and .

In the third case, for , there is an indeterminacy because .

To eliminate this indeterminacy, we use the following asymptotic expansion of the Mittag-Leffler function [22]:

Using Equation (44) into (34), we obtain the following relation:

As expected, the integrand in (45) is the Caputo kernel . Replacing (45) into (34), we obtain the following approximate formula for the time-fractional Caputo derivative:

where

##### 3.3. Numerical Solution to the Burgers Equation with Generalized Atangana-Baleanu Time-Fractional Derivative

To determine the numerical solution of Equation (29), we approximate the first- and second-order derivative with respect to by [23–25].

The nonlinear term is replaced by the equivalent term

For simplicity of calculations and proving the stability of the scheme, the nonlinear term is approximated by [30, 31]

Using Equations (40), (52), and (54), we obtain the following numerical scheme for fractional Burgers equation (29):

The initial and boundary conditions (30) and (31) are transformed in following discrete relationships:

Using (57) and (58), the numerical scheme (55) is written in the following metrical form: where the matrix is the following tridiagonal matrix:

with where

In the following, we assume that .

Let be the exact solution of Equation (29). The *local truncation error* of the numerical scheme (55) is

According to the Taylor expansion, it is found that exists a constant such that ; therefore, the discrete operator (with finite differences) converges towards the continuous operator (with derivatives) for (vanishing truncation error, so, *the numerical scheme is consistent*).

Let us introduce the notations:

The following boundedness theorem will be proved.

Theorem 6. *Assume that . There exists a constant such that the numerical solution derived by the finite difference scheme (55) satisfies inequality .*

*Proof. *Equation (55) can be written in the equivalent form
Multiplying Equation (65) by , summing for , and using Equation (64), we obtain
where
☐

A straightforward calculus leads to

Based on the boundary conditions (57) and (58), it results that .

Using the property

from equality (66) and Lemma 3, we obtain

Using (70) and mathematical induction, we obtain that for . Therefore, we have

Equation (71) implies that

therefore,

But, there exist a constant such that , respectively, . Theorem 6 shows that for a given function , the numerical solution of the problem is bounded as time increases.

In order to prove a stability result, we consider a perturbed problem, i.e., the fractional equation (29) with a different initial condition .

We say that the numerical method is *globally stable* if there exists a constant such that , where and are solutions corresponding to initial conditions and , respectively. Using the boundedness theorem one can prove that the n