On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.


Introduction
The idea of a fractional derivative was initially put forth by renowned mathematician Leibniz in 1695.Leibniz solved the equations including integrals with non-integer order or derivatives of fractional order.Researchers in fractional calculus have been interested in it recently because of its many applications and the fact that fractional calculus is widely used in a variety of nonlinear complex systems found in fluid mechanics [1].The author investigated the dynamics of the motion of an accelerating mass-spring system using fractional calculus.They derive the classical equations of motion and generate the corresponding Lagrangian using the integer-order Euler-Lagrange equations.Furthermore, the fractional Euler-Lagrange equations are created and solved numerically [2], and the generalised Lagrangian is introduced via the so-called fractional derivative operators.The author [3] explored the fundamental role of the liver by creating mathematical models that properly represent their function.
The study proposed a new model of the human liver using the exponential kernel and the Caputo-Fabrizio fractional derivative.The symmetry analysis of several fractional KdV type equations has been examined in [4,5].The fractional order Zika virus model is explored in [6], and the COVID-19 mathematical model is studied in [7].The existence and stability analysis along with numerical simulations of fractal-fractional tuberculosis model is explored in [8] and other hybrid boundary value issues of fractional order are studied in [9,10].
Notable fractional operators that are used are the Caputo [11,12], Riemann-Liouville [13,14], Caputo-Fabrizo [15], and Atangana-Baleanu [16] operators.These operators are flexible tools for investigating chaotic and fractal events with non-local kernels [17].The features, benefits, and drawbacks of these operators are covered in detail in [18], which also offers some interesting outcomes from investigations on a fractional optimum control issue in systems with time delay arguments.Likewise, the findings presented in [19] about fractional optimum control for variable-order differential systems offer encouraging perspectives regarding the effectiveness of employing these derivatives in mathematical modelling.On the other hand, these categories come with built-in limitations.The Caputo derivative has addressed this limitation, but it is still insufficient to fully describe the phenomena of index law.The Riemann-Liouville derivative is not enough for understanding the significance of the initial requirements.In 2015, Caputo and Fabrizio created a new fractional derivative that they termed the CF derivative.Atangana and Baleanu put in a lot of effort to resolve local problems.They developed the modified Mittag-Leffler function's Liouville-Caputo and Riemann-Liouville derivatives.Lately, the Atangana-Baleanu (AB) [20] and Caputo-Fabrizio (CF) [21] operators have become popular tools for investigating DEs.Both the exponential and Mittag-Leffler kernels are necessary for these operators to operate properly.The CF and AB operators have multiple applications in the literature.For example, the CF operator has been used to study HIV-1 infection in [22].Ahmad et al used the CF operator to study the fractional-order Ambartsumian problem [23].Rahman et al [24] studied the f 4 -model with the CF and AB operators.Other applications are listed in the literature [25][26][27].
A mathematical model of waves on shallow water surfaces is known as the KdV-Burgers equation.It is especially noteworthy because it is the fundamental representation of a perfectly solvable model, that is, a nonlinear partial differential equation (PDE) with an exact and precise solution.Over the last few decades, there has been a lot of interest in the KdV-Burgers model, which arises in several practical situations: the transport of liquids carrying gas bubbles [28], the turbulence of undulating bores in shallow water [29], weakly nonlinear plasma waves with particular dissipative properties [30], and the waves passing through an elastic pipe filled with a viscous liquid [31].In the areas of circuit theory, turbulence, ferroelectricity theory, and other subjects, it may also be applied as a nonlinear model [32,33].The standard form of the equation KdV-Burger is 1 It is generally recognized that equation (1) combines the Burgers [35] and KdV [34] models.The radial disturbance of the pipe wall is proportional to ( ) , where x and t are the temporal and characterized variables.Johnson [36] noted that the suggested model emerged at a certain limit of the matter in the region of wave propagation in fluid-filled elastic pipes.The model (1) was accurate in the far-field in an initially linear (small amplitude) near-field solution.Nonlinearity ( ( )( ( )) ) , and dissipation ( ( )) x t , xx  all appear in this equation, which represents a wave model in its simplest form.We address the fractional KdV-Burger's equation in the following form: where  is a function of x and t, the space variable is denoted by x, the time variable by t, and the positive constant is c.Equation (2) is initially written in the ABC sense, then subsequently it is examined in the CF sense.
The ADM is a sophisticated analytical method that was used to resolve an ODE in a physical application [37][38][39].After that, the Natural transformation is used to improve ADM.As a result, we can use this combination of ADM and Natural transformation, also known as the NTDM, to obtain an appropriate analyticapproximate solution of nonlinear differential equations.The primary reason for using NTDM in our paper is its ease of understanding and follow-up, as well as its higher convergent nature towards the exact solution as compared to ADM (see [40] for more details).The recommended approach generate reliable outcomes that provide accurate solutions to the required problem.In the numerical example, our methods resulted in infinite series.The series provides an exact solution to the relevant problem when written in closed form.This study can be used as a basic reference by researchers to explore this approach and implement it in many applications to obtain precise and approximate outcomes in a few simple steps.Numerous PDEs and FPDEs have been resolved by NTDM, like fractional Caudrey-Dodd-Gibbon equations [41], fractional-order Kaup-Kupershmidt equation [42], fractional-order Gardner and Cahn-Hilliard equations [43].
This study is structured as below: In section 2, we provide some essential fundamental data regarding FC theory that enables us to arrive at our outcomes.The key concepts of the methods that are suggested to solve a general formulation for FPDEs are presented in section 3. Section 4 presents the convergence analysis of the proposed method.Section 5 presents the application of NTDM on certain attractive case of KdV-Burgers to obtain useful solutions.In order to demonstrate the effectiveness and precision of the suggested method, we also display the graphical simulations of the projected solutions with a range of σ values as well as the outcome with the precise solution for σ = 1.Section 5 presents our conclusions.

Preliminaries
This section offers some basic definitions and properties of fractional derivatives with nonsingular kernels.
Definition 2.5.The Natural transform (NT) of the function ( ) t  is defined by

  
with ( ) H t showing the Heaviside function.
Definition 2.6.The inverse NT of the function ( )   l, is defined by , then with c 1 and c 2 are constants.
with c 1 and c 2 are constants.

Description of the proposed method
Consider the general fractional differential equation given as below.
here , , h(x, t) are linear, nonlinear and the source terms respectively.

Case I (NTDM CF )
Taking the NT of equation (15) in Caputo-Fabrizio sense, we have Implementing inverse NT to equation (17), we have representing the nonlinear term as The series form solution of ( ) x t ,  is taken as By using equations (21)-( 22) in (20), we obtain Lastly the NTDM CF solution to (15) is carried out by using (24) into (22) as Taking the NT of equation (15) in Atangana-Baleanu Caputo sense, we have Implementing inverse NT to equation (26), we have  with ( ( )) x t ,   representing the nonlinear term as with A i illustrates the Adomian polynomials [45,46].The series form solution of ( ) x t ,  is taken as By using equations ( 29)-( 30) into (28), we obtain Lastly the NTDM ABC solution to (15) is carried out by using (32) into (30) as

Convergence analysis
The uniqueness and convergence analysis for NTDM CF and NTDM ABC is presented as below.
Theorem 4.1.The NTDM CF result for ( 15) is unique when is Banach space,∀continuous function on J.
is a non-linear mapping, where  are are two different function values and P 1 ,P 2 are Lipschitz constants.
( ) I is contraction as . From Banach fixed point theorem the result of ( 15) is unique .
Theorem 4.2.The NTDM ABC result for ( 15) is unique when Proof: Since this proof is the same as that of theorem 1, it has been omitted.
. Similarly, we have . Therefore, As a result, m  is a Cauchy sequence in H, implying that the series m  is convergent.
Theorem 4.4.NTDM ABC solution of ( 15) is convergent.Proof: Since this proof is the same as that of theorem 3, it has been omitted.

Numerical applications 5.1. Application
Let us assume equation (2) in the following form: x, 0


Implementing NT to equation (39), we obtain Implementing inverse NT, we have

Application of NTDM CF
The series form solution of ( ) x t ,  is taken as The nonlinear term ( )( ( )) x m m 0   can be evaluated by the Adomian polynomials prescribed as

  
The first components are illustrated as Lastly, the recursive relation for equation (44) is established as


The approximate solution for NTDM CF is expressed as:  The series form solution of ( ) x t ,  is taken as x m m 0   can be evaluated by the Adomian polynomials prescribed as

  
The first components are illustrated as Lastly, the recursive relation for equation (49) is established as


The approximate solution for NTDM ABC is expressed as:

Numerical simulation studies
This study employs two unique methodologies to examine the approximate solution of nonlinear fractional logistic differential equation.The Caputo fractional derivative operator at any order for variable values of space    The 3D variation in fractional order σ for the solution employing the suggested method is displayed in figure 5.The 2D variation in fractional order σ for the solution employing the suggested method is displayed in figure 6. Figure 7 illustrates the nature of absolute error for the given problem using both operators.The comparison of the exact and suggested approach solution at different fractional orders in ABC manner is presented in table 1.The comparison of the exact and suggested approach solution at different fractional orders in CF manner is presented in table 2. The graph illustrates how different fractional orders impact the behaviour of the solution and shows how variations in σ impact the solution with respect to the given parameters.Also, the numerical results are briefly specified in the table, which enables a direct comparison of the performance of the solution at various fractional orders.It need to be noted that throughout the calculations, we used different approximations and that using accurate results to the problem gave us a better estimate.By increasing the order of the approximation, which adds more terms to the solution, we could have been able to get approximation solutions that were more accurate.

Conclusion
This paper examines the fractional KdV-Burger's model with initial guess under two nonlocal operators: Mittag-Leffler and exponential kernels.For the particular case σ = 1, the NTDM was used to generate a series solution that leads to the exact values.The suggested approach is easy to adopt and straightforward.The approach is applied to KdV Burger's equation with success.Thus, we obtained extremely precise calculated solutions to the fractional KdV Burger's equation.The NTDM is extremely reliable and helpful for achieving analytical approximations, as seen by the 3D and 2D surface solutions.The results demonstrate both the wide applicability of the technique to fractional nonlinear evolution problems and its consistent accuracy.Numerical simulation of the solutions shows rapid convergence.As a result, it provides more realistic series solutions, which in real physical situations usually converge fast.As seen in the figures, changing the fractional order has a progressive effect on the predicted solutions behavior for the problems under consideration.Researchers studying mathematical physics may find the current results useful in understanding more complex behavior related to similar challenges.Numerous more high dimensional FPDEs that are frequently encountered in engineering, applied science, and other scientific domains can be studied using the current methodology.

Figure 1 .
Figure 1.Nature of the accurate solution, analytical solution in terms of CF and ABC derivatives at σ = 1.

Figure 2 .Figure 3 .
Figure 2. Nature of the analytical solution of CF and ABC derivatives at σ = 0.75.

Figure 4 .
Figure 4. Nature of the analytical solution of CF and ABC derivatives at σ = 0.25.

Figure 5 .
Figure 5. 3D nature of the analytical solution of CF and ABC derivatives at numerous orders of σ.

Figure 6 .
Figure 6.2D nature of the analytical solution of CF and ABC derivatives at numerous orders of σ.

Figure 7 .
Figure 7.Comparison of the analytical solution of CF and ABC derivatives in terms of absolute error.

Table 1 .
Nature of our method solution at different orders of σ in ABC derivative manner as well as exact solution.

Table 2 .
Behavior of our method solution at different orders of σ in CF derivative manner as well as exact solution.