Abstract

In this article, we solve nonlinear systems of third order KdV Equations and the systems of coupled Burgers equations in one and two dimensions with the help of two different methods. The suggested techniques in addition with Laplace transform and Atangana–Baleanu fractional derivative operator are implemented to solve four systems. The obtained results by implementing the proposed methods are compared with exact solution. The convergence of the method is successfully presented and mathematically proved. The results we get are compared with exact solution through graphs and tables which confirms the effectiveness of the suggested techniques. In addition, the results obtained by employing the proposed approaches at different fractional orders are compared, confirming that as the value goes from fractional order to integer order, the result gets closer to the exact solution. Moreover, suggested techniques are interesting, easy, and highly accurate which confirm that these methods are suitable methods for solving any partial differential equations or systems of partial differential equations as well.

1. Introduction

In recent years, fractional calculus has surpassed ordinary calculus in popularity. The standard calculus has reached its pinnacle in terms of discovery. Mathematicians and engineers require fractional calculus as a solution. This permits a more accurate description of real-world phenomena than the traditional “integer” order. Numerous mathematicians, including Fourier, Laplace, Riesz, and others, were engaged and made substantial contribution to the subject. Modern definitions of fractional order derivatives and integrals, such as the Atangana–Baleanu fractional integral [1], the Caputo fractional derivative [2], and the Caputo–Fabrizio fractional derivative [3], have ushered in a new era in the literature of fractional derivatives. Models based on fractional calculus can accurately depict numerous engineering, physics, and chemistry processes, among others [4]. In addition, fractional calculus is used to simulate the frequency-dependent damping behavior of a variety of viscoelastic materials [5], the dynamics of interfaces between nanoparticles and substrates [6], economics [7], and numerous other applications [811].

Finding the actual or approximate solutions of FDEs is crucial in all of these areas of study, but because we lack a method for obtaining the precise solution of these sorts of FDEs, we must focus on approximating the exact solution. Determining the exact answer to such FDEs and other scientific applications is a difficult task in mathematics. Unlike the approximate answer [12], the exact solution enables us to comprehend the problem’s mechanism and complexity. Obtaining exact analytical expressions to FDEs is exceedingly difficult, if not impossible, due to the complexity of computation involved in these equations. As a result, it is necessary to seek out some useful approximations and numerical techniques, such as the homotopy perturbation method [13], variation iteration method [14], residual power series method [15], approximate-analytical method [16], Elzaki transform decomposition method [17], Iterative Laplace transform method [18], Adomian decomposition method [19], reduced differential transform method, and others [2023].

In this paper, we provided two analytic approaches in conjunction with the Laplace transformation and fractional derivatives in Antagana–Baleanu solution to satisfy fractional-order problems [24]. The first method is the mixing of Laplace transform (LT) and variational iteration method known as variational iteration transform method (VITM) which was first developed by He [25] and is an effective solution for a broad variety of problems in scientific fields [26, 27]. The second significant methodology for solving nonlinear functional equations is the combination of the Adomian decomposition method and Laplace transform, which was first developed by George Adomian (1923–1996) in the 1980s. The technique depends on the decomposition of a nonlinear equation result into a series of functions. A polynomial produced by a power series expansion of an analytic function returns each series term. This method for solving several nonlinear fractional-order differential equations is interesting, straightforward, and accurate.

Harry Bateman introduced the Burgers equation in 1915 [28], and it was subsequently dubbed by the Burgers equation. The Burgers equation has many applications in science and engineering, especially when dealing with nonlinear problems. Burgers equation applications have grown in prominence and attention among mathematical scientists and researchers. This equation is acknowledged to represent a range of phenomena, such as dynamic modeling, heat conduction, acoustic waves, and turbulence [2931]. In 1895, Korteweg and Vries initially derived the Korteweg-De Vries (KDV) equation. The KDV equation is used to predict long waves, tides, solitary waves, and wave propagation in a shallow canal. The KDV equation is utilized in several disciplines, including fluid mechanics, signal processing, hydrology, viscoelasticity, and fractional kinetics.

2. Preleminaries

In this section, we presented some basic definitions of fractional calculus related to our present work.

Definition 1. The derivative by Caputo having order fraction is defined as

Definition 2. The derivative by Caputo having order fraction with the aid of Laplace transform is defined as

Definition 3. The fractional Atangana–Baleanu derivative in terms of Caputo manner is defined aswhere normalization function is denoted by with and is the Mittag–Leffler function.

Definition 4. In terms of Riemann–Liouville, the Atangana–Baleanu fractional derivative is defined as

Definition 5. The Atangana–Baleanu operator in connection with Laplace transform is given by

Definition 6. Consider , and h is a function of order , then the fractional integral operator for is defined as

3. Idea of LTDM

Here, we discuss the methodology of LTDM for solving fractional-order partial differential equations.having initial terms

Here, the fractional-order AB operator is indicated from having order , are linear and nonlinear operator, and represent the source term.

On taking the Laplace transform of (7), we get

Using the differentiation property of LT, we obtainwhere .

Now, using the inverse Laplace transform, we getwhere shows the term that come from the source term. LTDM generates the result of the infinite series of and decomposing the nonlinear operator aswhere the Adomian polynomials are represented by and putting equations (12) and (14) into (11), we get

The terms listed below are defined as

As a result, all of the components for are determined as

4. VITM Formulation

Here, we discuss the methodology of VITM for solving fractional-order partial differential equations.having initial term

Here, the fractional-order AB operator is indicated from . are linear and non,linear operator and represent the source term.

On taking the Laplace transform of (18), we get

Using the differentiation property of LT, we obtain

The method of iteration for the equation (21) is Lagrange multiplier and

Equation (22) series form solution is obtained by using the inverse Laplace transform.

5. Applications

To show the validity and capability of the suggested techniques, we implemented proposed methods for solving four nonlinear systems.

5.1. Problem 1

Consider system of homogeneous KdV equation having order threewith initial source

On taking the Laplace transform of (25), we get

We obtain when we use the Laplace inverse transform

Assume that the solution, and in series form aswhere , , and are Adomian polynomials that characterize the nonlinear terms, and so equation (28) is rewritten as

The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (14),

As a result, when comparing the two sides of equation (30)

For ,

For ,

The approximate solution to the series is written as

We achieve the exact solution by putting

5.1.1. VITM Analytical Results

For Equation (25), we have the iteration formula:where

For ,

We achieve the exact solution by putting

The analytical solution and exact solution is shown in Figures 1(a) and 1(b) at and . Figure 1(c) shows the absolute error, and Figure 1(d) gives the solution at various fractional-order graph for . The behavior of the exact solution and analytical solution for is seen in Figures 2(a) and 2(b). Tables 1 and 2 show the comparison of the exact and our methods solution in addition with the absolute error at different fractional-order. From the figure and tables, it is clear that our methods solution is in good agreement with the exact solution.

5.2. Problem 2

Consider the generalized coupled Hirota Satsuma KdV systemwith initial source

On taking the Laplace transform of (41), we get

We obtain when we use the Laplace inverse transform

Assume that the solution, , , and in series form aswhere , , , and are Adomian polynomials that characterize the nonlinear terms, and so equation (30) is rewritten as

The decomposition of nonlinear terms by Adomian polynomials is defined as in equation (7),

As a result, when comparing the two sides of (46),

For ,

For ,

The approximate solution to the series is written as

We achieve the exact solution by putting ,

5.2.1. VITM Analytical Results

For equation (41), we have the iteration formulawhere

For ,