Abstract

This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg–de Vries equations with the Atangana–Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg–de Vries and the modified system of Korteweg–de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.

1. Introduction

Many researchers have been working on various aspects of fractional derivatives in recent years. Caputo and Fabrizio modified the existing Caputo derivative to develop the Caputo–Fabrizio fractional derivative [1–5] based on a nonsingular kernel. Because of its advantages, numerous researchers utilized this operator to investigate various types of fractional-order partial differential equations [6–9]. To address this issue, Atangana and Baleanu proposed a new fractional operator called the Atangana–Baleanu derivative, which combines Caputo and Riemann–Liouville derivatives. Because of the existence of the Mittag–Leffler kernel, which is a generalization of the exponential kernel, this new Atangana–Baleanu derivative has a long memory. Moreover, the Atangana–Baleanu operator outperforms other operators, and different scientific models have been successfully solved. Many advances have been made in fractional calculus over the last few years by borrowing ideas from classical calculus, but it does not remain easy. Scholars have the main concern to obtain a numerical solution; for this, numerous efficient methodologies have been constructed for fractional differential equations, such as the Adomian decomposition transform method [10], variational iteration transform method [11, 12], optimal homotopy asymptotic method [13], and homotopy perturbation method [14, 15].

Korteweg and de Vries introduced the Korteweg–de Vries equation in 1895 to model shallow water waves in a canal [16]. The suggested system Korteweg–de Vries equations play a crucial role in diverse engineering and applied sciences such as plasma physics, water waves, hydrodynamics, and theory of the quantum field. The Korteweg–de Vries equations are usually investigated in the analysis of nonlinear dispersive waves [17]. They define the interactions among two long waves with various dispersion relations. Many researchers have been interested in these schemes, and a lot of works have been done. For example, Ghoreishi et al. applied the homotopy analysis method to achieve numerical results of a modified system of Korteweg–de Vries equations [18]. Kaya and Inan in [19] achieved traveling wave results of the system of Korteweg–de Vries and modified system of Korteweg–de Vries equations. The fractional-order system of Korteweg–de Vries equations is defined as follows:where is the fractional-order derivative of and , , and are constants, respectively. The functions and are considered as important functions of time and space, disappearing for and respectively. The other method eliminates to the conventional coupled Korteweg–de Vries equations since is implemented.

A classic model in this hierarchy is the modified coupled Korteweg–de Vries system. The following nonlinear partial differential equations govern this model [20]:

The modified Korteweg–de Vries equation in its standard type is simplified by the modified couple Korteweg–de Vries equation (2), with . Korteweg–de Vries models are a source of nonevolution equations with a wide range of implementations in science and engineering. The Korteweg–de Vries models, for instance, generate ion-acoustic result in fluid mechanics [21, 22]. Long waves characterise geophysical fluid dynamics in shallow and deep oceans [23, 24]. Various studies have suggested numerous systems to overcome the fractional-order Korteweg–de Vries equation employing various methodologies, such as the differential transform method [25], Adomian decomposition method [26], natural decomposition method [27], homotopy analysis method [28], Elzaki projected differential transform method [29], variational iteration method [30], new iterative method [31], modified tanh technique [32], and Lie symmetry analysis [33]. Analogously, same solutions for (2) have been suggested by Inc and Cavlak [34], Fan [35], Lin et al. [36], Inc et al. [37], and Ghoreishi et al. [18].

Daftardar-Gejji and Jafari [38] proposed an innovative iterative method of solving functional equations with approximation solutions. The new iterative approach is constructed on the justification of disappearing the nonlinear functions is identified as the iterative transformation technique [39]. This procedure is quick and accurate, and it avoids the utilization of complicated integrals, unconditioned matrix, and infinite series forms. This technique does not require any expressive parameters for the model. Numerous researchers have analyzed new iterative transformation methods to solve partial differential equations, such as the Fornberg–Whitham equation [40], KdV equation [31], and Klein–Gordon equation [41].

The Adomian decomposition method was firstly introduced by Adomian in 1980 and implemented by several investigators. In recent decades, numerous researchers have investigated the solutions of integral and differential equations by different techniques with the mixed Laplace transform. The Adomian decomposition method was modified with many integral transformations, such as Laplace, -Laplace, Elzaki, Aboodh, and Mohand. Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson formula [42] for solving nonlinear integrodifferential and Volterra integral equations based on the Newton–Raphson method, discrete Adomian decomposition technique [43] applied for investigating the fractional-order Navier–Stokes model, Laplace–Adomian decomposition method [44] study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative.

2. Basic Definitions

Definition 1. The fractional-order Caputo derivative is defined by

Definition 2. The Laplace transformation connected with fractional Caputo derivative is expressed by

Definition 3. In the Caputo sense, the Atangana–Baleanu derivative is defined aswhere is a normalization function such that , and represents the Mittag–Leffler function.

Definition 4. The Atangana–Baleanu derivative in the Riemann–Liouville sense is defined as

Definition 5. The Laplace transform connected with the Atangana–Baleanu operator is defined as

Definition 6. Consider , and f is a function of ; then, the fractional-order integral operator of is given as

3. The General Implementation of the Modified Decomposition Method

Suppose the nonlinear fractional partial differential equationswith the conditionwhere show the fractional-order Caputo derivative operator with , while is linear, are nonlinear functions, and defines the source term.

Applying the Laplace transformation to (9), we get

Taking the Laplace transformation differentiation, we find

The inverse Laplace transformation of (12) gives

The adomain decomposition method series form solution is defined as

Thus, the nonlinear function can be calculated by the Adomian polynomials defined aswhere

Putting (14) and (15) into (13), we have

Lastly, the iterative methodology for (17) is achieved as

4. The General Discussion of the New Iterative Transformation Method

Let us assume the following general fractional partial differential equationwith the conditionwhere and are linear and nonlinear terms and shows the source term.

Using the Laplace transformation to (19), we get

Taking the Laplace transformation differentiation property, we get

The inverse Laplace transformation of (22) gives

From the iterative connection, we achieve

Also, the linear operator is ; therefore,and defines the nonlinear term as in [38].where

By putting (24), (25), and (??) into (23), we obtain

As a result, we determine the next iteration

Finally, (19) and (20) yield the -term result in the series form, defined as

5. Uniqueness and Existence Solutions for the Modified Decomposition Method

Theorem 1 (uniqueness theorem). The unique result of equation (9) provide space whenever where .

Proof. Assume that represents all continuous mappings on the Banach space, defined on having the norm . For this, we introduce a mapping , and we havewhen and . Suppose that and are also Lipschitzian with and where and are Lipschitz constants, respectively, and are various values of the mapping.The mapping is a contraction under the assumption . As a result of the Banach contraction fixed point theorem, there is a unique solution to (9). As a result, the proof is complete.

Theorem 2 (convergence analysis). The solution general form of (9) will be convergent.

Proof. Suppose is the partial sum; that is, . Firstly, we define that is a Banach space Cauchy sequence in . Using into consideration of Adomian polynomials, we achieveNow,Consider ; then,where . Similarly, we have the triangular inequalityand since , we get ; then,However, (since is bounded). Thus, as , . Hence, is a Cauchy sequence in . As a solution, the series converges, and this completes the proof.