Analytical Approximate Solutions of Caputo Fractional KdV-Burgers Equations Using Laplace Residual Power Series Technique

Burqan, Aliaa; Khandaqji, Mona; Al-Zhour, Zeyad; El-Ajou, Ahmad; Alrahamneh, Tasneem

doi:https://doi.org/10.1155/2024/7835548

Journal of Applied Mathematics

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Abstract Introduction Results Conclusion Abbreviations Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 7835548 | https://doi.org/10.1155/2024/7835548

Analytical Approximate Solutions of Caputo Fractional KdV-Burgers Equations Using Laplace Residual Power Series Technique

Aliaa Burqan,¹Mona Khandaqji,²Zeyad Al-Zhour,³Ahmad El-Ajou,⁴and Tasneem Alrahamneh⁴

Academic Editor: Fernando Simoes

Received17 Feb 2023

Revised23 Jan 2024

Accepted22 Feb 2024

Published08 Mar 2024

Abstract

The KdV-Burgers equation is one of the most important partial differential equations, established by Korteweg and de Vries to describe the behavior of nonlinear waves and many physical phenomena. In this paper, we reformulate this problem in the sense of Caputo fractional derivative, whose physical meanings, in this case, are very evident by describing the whole time domain of physical processing. The main aim of this work is to present the analytical approximate series for the nonlinear Caputo fractional KdV-Burgers equation by applying the Laplace residual power series method. The main tools of this method are the Laplace transform, Laurent series, and residual function. Moreover, four attractive and satisfying applications are given and solved to elucidate the mechanism of our proposed method. The analytical approximate series solution via this sweet technique shows excellent agreement with the solution obtained from other methods in simple and understandable steps. Finally, graphical and numerical comparison results at different values of are provided with residual and relative errors to illustrate the behaviors of the approximate results and the effectiveness of the proposed method.

1. Introduction

Many events in chemistry, physics, biology, and other majors may be efficiently described using fractional calculus because accurate modeling of physical phenomena depends not only on immediate time but also on past time [1]. At this time, fractional calculus is growing in scale because it has unique and versatile properties [2]. The theory and applications of fractional F-PDEs have previously received many notable contributions [3]. These equations can be used, by researchers, more effectively to analyze and describe many phenomena in an assortment of fields, including mechanical and dynamical systems [4], pantograph equations [5], Navier-Stokes equations [6], and water wave propagation phenomena [7]. However, due to the complexity of the fractional calculus involving these equations, no approach provides an exact solution to the F-PDEs.

A variety of techniques is employed to solve several F-PDEs. The L-RPSM was created in 2020 by Eriqat et al. [5] to obtain the analytical approximate series solutions of the linear and nonlinear neutral fractional pantograph equations, and this method was subsequently used to investigate the exact and approximate (solitary, vector) solutions for various linear and nonlinear time-F-PDEs [8, 9]. The L-RPSM is constructed based on the LT and RPSM by transforming the differential equations to the Laplace space and then using an appropriate expansion to solve the new equation. The L-RPSM does not rely on the fractional derivation to determine the coefficients of the series as in RPSM but depends on the concept of the limit, so few calculations generate the coefficients compared to RPSM. The current technique is quick, requires little computer memory, and is not influenced by computational round-off errors. Furthermore, this technique computes the coefficients of the power series using a chain of equations with more than one variable, indicating that the present method has a rapid convergence.

In the past two years, many works have employed the L-RPSM in providing accurate and approximate solutions to many F-PDEs, for example, fractional Fisher’s equation and logistic system model [10], nonlinear fractional reaction-diffusion for bacteria growth model [11], fractional Lane-Emden equations [12], fuzzy quadratic Riccati FDE [13], a hyperbolic system of Caputo time-F-PDEs with variable coefficients [9], time-fractional Navier-Stokes equations [6], and time-fractional nonlinear water wave partial differential [7].

The KdV-BE was developed by Korteweg and de Vries [14], derived by Su and Gardner [15], and is used to describe nonlinear waves and many physical phenomena. It is also used to model problems established in many applied mathematics fields, including heat conduction, acoustic waves, gas dynamics, and traffic flow [16]. Nonlinear F-PDEs, such as fractional Burgers equations [17], F-KdV-BE [18], and fractional Schrödinger-KdV-BE [19], have also recently been proposed to explain some significant events and dynamic physics processes.

The homogeneous balancing method in [20], the truncated expansion method in [21], and the exponential function method in [22] were just a few of the several techniques used to obtain the exact solution of the KdV-BEs. Recently, many researchers have given more attention to the investigation of numerical techniques for solving KdV-BEs, such as the RPSM [23], Adomian decomposition method [24, 25], element-free Galerkin method [26], and explicit restrictive Taylor method [27].

Nonlinear fractional KdV-BEs are most commonly used in space-fractional or time-fractional derivative applications, according to studies [28, 29]. There are no applications for nonlinear fractional KdV-BEs that address the space-time-fractional state, and it is difficult to solve them using the established method, as described in the literature [28, 29], since it requires additional execution time to complete the approximation task.

The main objective of this work is to predict and create the ASSs for the following initial value Caputo FKdV-BE by using the L-RPSM: where and are constants and and refer to the order of time-Caputo and space-CFD, respectively.

One of the main advantages of our proposed method (L-RPSM) is an efficient simple technique for finding exact and approximate series solutions to the linear and nonlinear F-PDEs compared with others such as residual power series, two-stage order-one Runge-Kutta, one-leg , variational iterative, Chebyshev polynomials method, Laguerre wavelet, Bernoulli wavelet, Boubaker polynomials, Hermit wavelet, and price-wise fractional-order Taylor methods. It must be noted that the study of the model’s existence, uniqueness, and stability analysis of model (1) has been previously studied. The reader can refer to references [30, 31].

We arrange this paper as follows: In Section 2, we review some definitions and properties of some basic concepts and results related to the CFD, LT, and fractional Taylor and Laurent expansions, which are essential in constructing the L-RPS solutions for the FDEs. Section 3 gives an analytical solution of the nonlinear Caputo FKdV-BE based on the L-RPSM approach. In Section 4, the efficiency, effectiveness, and applicability of the L-RPSM are demonstrated by testing four nonlinear Caputo FKdV-BEs and comparing them with the results obtained in [23]. Finally, we summarize the outcomes of this paper in Conclusion.

2. Basic Concepts and Results

This section reviews some basic concepts and results related to the Caputo fractional operator, Laplace transform, and fractional Taylor and Laurent expansions which are essential in constructing the L-RPSM solutions for the FPDEs.

Definition 1. The fractional derivative of order for the real-valued function in the time-Caputo sense is denoted by and defined as where , , is an interval, and is the time-Riemann-Liouville fractional integral of order of the function that defines as Note that the space-CFD is denoted and defined by the expression .

Definition 2 ([32]). Let be a piecewise continuous function on and of exponential order The LT of the is denoted and given by whereas the inverse LT of the function is defined as The following Lemma gives the critical properties of the Laplace and inverse Laplace transformations required in this work.

Lemma 3 ([32]). If and are piecewise continuous functions on the region and of exponential orders and , respectively, where . Suppose that , , and are constants, then (i)(ii)(iii)(iv)

Theorem 4 (see [10]). Let be a piecewise continuous function on and of exponential order . Suppose that the function has the following fractional power series: Then, .
For more details regarding the convergence conditions for the fractional expansions, see [23].

Theorem 5 (see [10]). Let be a piecewise continuous function on and of exponential order and can be represented as the fractional expansion in Theorem 4. If on where , then the remainder of the fractional expansion as in Eq. (6) satisfies the following inequality:

3. L-RPSM Solutions to the Caputo FKdV-BE

This section provides an analytical solution for the Caputo FKdV-BE in Eq. (1) based on the L-RPS approach. Before we start with that and without losing the other cases, we will set the value of to be 2, leaving the reader to repeat the same steps to reach the construction required for the solution in the other two cases, and . Part (iv) of Lemma 3 produces the following equation when the LT is applied to both sides concerning : where .

According to Theorem 4, we assume the L-RPS approach dictates that the series solution of Eq. (8) takes the following fractional expansion:

Since , the th truncated series solution of takes the following form:

We define the Laplace residual function and the th Laplace residual function, respectively, of Eq. (8) as follows:

The following are some relevant facts to determine the ASS according to the L-RPS technique, see [8]:

Therefore, to obtain the unknown coefficients , we must first substitute the th truncated series solution for into the th Laplace residual function in Eq. (12). Then, we solve and gather the acquired coefficients in terms of fractional expansion series . After that, we apply the inverse LT to both sides of the resulting Laplace series solution to get the ASS of the original problem in Eq. (1).

Now, is substituted into the first-residual function to get

The first coefficient can be obtained by multiplying both sides of Eq. (14) by and then by solving . Then, we have

Thus, the 1st truncated series solution can be expressed as

Similar to how was calculated, is likewise calculated by inserting the 2nd truncated series into Eq. (12) to obtain

Next, multiply both sides of Eq. (17) by , and then by solving , one can get

So, the 2nd truncated series solution of Eq. (8) is given as

The following results are also easily obtained for by following the same procedure as above:

In fact, the 3rd truncated series solution of Eq. (8), based on the prior findings of , , and , is provided by

One may derive for by processing the preceding stages and utilizing the solution of the system . Then, we get the th truncated series solution as follows:

In the following step, we operate the inverse LT to both sides of Eq. (22) to get the th ASS of the initial value problem of the Caputo FKdV-BE of Eq. (1) as follows:

Finally, the ASS of the Caputo FKdV-BE of Eq. (1) is given by

Using the same procedure as before, we can discover ASSs to the same Caputo FKdV-BE of Eq. (1) in the case of and as we mentioned above.

When , the initial coefficients of the th truncated series solution to Eq. (1) are:

Whereas for , the first four coefficients will be as follows:

In the next section, we utilize the obtained coefficients in Eqs. (23), (25), and (27) to introduce ASSs to four examples of time, space, and time-space Caputo FKdV-BE.

4. Some Attractive Applications

In this section, the efficiency, effectiveness, and applicability of the L-RPSM are demonstrated by testing four nonlinear Caputo FKdV-BE initial value problems and comparing them with the results in [23]. It should be noted that the MATHEMATICA 12 software package is used for all numerical and symbolic operations.

Application 6. Consider the following nonlinear time-Caputo FKdV-BE Comparing Eq. (28) with Eq. (1), we find that , , , , and with the initial condition , where and are constants. As a result, using the methodology described in Section 3, the LT of Eq. (28) is the ASS of Eq. (29) is and consequently the th truncated series of will be According to the results in Eqs. (15), (18), and (20), we derive for , 2, and 3 as: Then, the 3rd truncated series in Eq. (31) becomes Consequently, the 3rd approximate L-RPS solution of Eq. (28) takes the following expansions: The authors introduced in [23] an approximate RPSM solution for Eq. (28), where the obtained 3rd ASS coincided with in Eq. (34).

Remark 7. The solution of Eq. (28) has a general form that coincides with the following exact solution when . To demonstrate the geometric behaviors of the L-RPS solution of Eq. (28) and the range of its accuracy, a comparison between the approximate solution and exact solution is given in 3D for a different value of and , see Figure 1. From Figure 1, it is clear that the behaviors of the subfigures are almost identical and coincidental, especially for Figures 1(c) and 1(d) where . These subfigures appear identical and in perfect agreement in terms of accuracy with each other. As a result, compared to the exact solution, computing a few terms can yield an outstanding approximation. Consequently, the overall error can vanish by finding a larger number of the series terms.

(a)

(b)

(c)

(d)

To confirm this, numerical values for the actual and the relative errors of the approximation are calculated in Table 1 when . Table 2 shows the residual error of the approximation when and 0.90. The numerical results show that the obtained approximation is acceptable mathematically. The actual and relative errors are defined, respectively, as follows: while the residual error for the problem in Eq. (28) is given by:

Application 8. Consider the nonlinear time-Caputo FKdV-BE As we discussed in the previous application, comparing Eq. (38) with Eq. (1), we find that , , and with the initial condition , where is a constant. Therefore, according to the working mechanism of the L-RPSM, we transfer Eq. (1) to the Laplace space as where , assume the solution of Eq. (39) in the following form: and the th truncated series of will be According to the results in Eqs. (15), (18), and (20), we set , as follows: Using the same procedures as in Application 6, we get the 3rd approximate L-RPS solution of Eq. (38) as follows: It is worth mentioning that the solution in Eq. (43) matches the solution obtained in RPSM [23].

Remark 9. For , a little focus while simplifying the series solution in Eq. (43) leads us to the exact solution of Eq. (38) in the following closed form: To confirm the efficiency of the L-RPS method and the accuracy of the ASS obtained with it, Figure 2 graphically shows the behavior of the solution and its compatibility with the exact solution in the case of . It also shows the consistency of the solution’s behavior with different values of .

(a)

(b)

(c)

(d)

For more analysis of the results, numerical values for the actual and the relative errors of the approximation are calculated in Table 3 when the . Table 4 shows the residual error of the approximation when and 0.90. The numerical results show that the obtained approximation is acceptable mathematically.

Application 10. Consider the nonlinear space-Caputo FKdV-BE In this application, the parameter values of Eq. (1) are while is arbitrary in and the initial condition . Depending on the steps of the L-RPSM, the form of Eq. (1) in Laplace space is the series solution of Eq. (46) is and the th truncated series of is According to the results in Eqs. (15), (18), and (20), we compute for , 2, and 3 considering as where Consequently, by considering the values of in Eq. (50) and the form of the coefficients in Eq. (49), the 3rd approximate L-RPS solution of Eq. (45) is given by Figure 3 shows a simulation of the solution of Eq. (45) on in 3D space. Additionally, it displays the approximate solution on the domain for several values of , , , and .
We noticed from Figures 3(a)–3(d) that while each surface almost agrees well in its behavior, the representations of the surface graph solutions decrease steadily as the values of and increase in the definite domain. Also, the space-CFD plays a consistent role in the solutions.

(a)

(b)

(c)

(d)

Application 11. Consider the following nonlinear time-space-Caputo FKdV-BE Comparing Eq. (52) with Eq. (1), we find that with the initial condition . As a result, using the methodology described in Section 3, the LT of Eq. (52) is the series solution of Eq. (53) is and consequently the th truncated series of will be According to the results in Eqs. (15), (18), and (20), we derive for , 2,and 3 as where Consequently, by considering the values of in Eq. (57), the 3rd approximate L-RPS solution of Eq. (52) is given as The advantage of the L-RPSM is that it is possible to pick any point in the integration domain and as well the ASS and all its time-space-CFDs will be applicable. In other words, it is possible to find a continuous ASS in which the ASS is continuously dependent on the time-space-CFD. Our next objective is to graphically illustrate the mathematical behavior of the approximate solution of Eq. (52) and its time-space-CFD geometrically. To do so, we plot the surface graph of the approximate solution when , , , and on the domain as shown in Figure 4.

(a)

(b)

(c)

(d)

5. Conclusion

The L-RPS technique has been used in this study to find an ASS to the Caputo FKdV-BE. Three well-known physical applications are examined to confirm the applicability and superiority of the proposed strategy. In this technique, the FRPS is modified by coupling it to the LT operator. The advantage of using the L-RPSM is that it provides a more accurate convergence of the McLaurin series and requires a few computations without discretization, perturbation, or other physical restriction conditions. We explained the obtained ASS using graphics and numerical simulation, and they were compared with other well-known techniques in the literature. Consequently, the results confirm that the L-RPSM approach is straightforward for handling a variety of nonlinear FPDEs that might occur in engineering and scientific problems.

The main algorithm of L-RPSM for solving FKdV-BE of Eq. (1) can be summarized by the following steps: (i)Applying the Laplace transform to the FKdV-BE(ii)Using the Laurent series expansion to represent the solution of Laplace FKdV-BE in a new space. The coefficients of this expansion are determined in a similar way to the RPS method but with a new vision and a new analysis(iii)Applying the inverse Laplace transform in step (ii), then we obtain a solution to this problem in the original space

The main advantages of LRPSM are as follows: (i)It enables us to use the Laplace transform to solve nonlinear equations, while that was limited to linear equations only(ii)It simplifies the processing of fractional differential equations by converting them to algebraic equations(iii)Iterations can be calculated simply by functions built into any mathematical software using the concept of limit at infinity, and unlike RPS, they do not require the derivative at each iteration(iv)It does not require modeling assumptions, such as linearization, perturbation, or discretization

Thus, in future studies, the L-RPSM approach may be extended to find exact ASSs to other equations or model types that represent real-life phenomena, such as fractional integral equations, algebraic equations, and differential equations with boundary conditions instead of initial conditions, a new modified fractional Nagumo equation [33], spatiotemporal dynamic systems of interacting biological and chemical species [34], high-dimensional chaotic Lorenz system [35], generalize Hirota-Satsuma coupled KdV and MKdV equations [36], time-fractional vibration model of large membranes [37, 38], and fractional Black–Scholes option pricing equations [39].

Abbreviations

KdV-BE:	The KdV-Burgers equation
PDEs:	Partial differential equations
CFD:	Caputo fractional derivative
ASSs:	Analytical approximate series solutions
L-RPSM:	Laplace residual power series method
LT:	Laplace transform
RPSM:	Residual power series method
F-PDEs:	Fractional partial differential equations.

Data Availability

There is no underlying data used to support the results and conclusion in the paper.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could appear to influence the work reported in this paper.

Acknowledgments

Aliaa Burqan was supported by the Department of Mathematics, Zarqa University, Zarqa, Jordan.

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Copyright © 2024 Aliaa Burqan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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