Abstract
The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.
1. Introduction
In recent years, fractional calculus has gained significant attention from scientists and engineers due to its broad applicability and its ability to capture the complexities of real-world problems in various fields such as plasma physics, fluid dynamics, quantum mechanics, optics, and signal processing. Its use has allowed for a more accurate representation of these phenomena and has provided insights that traditional calculus cannot capture [1, 2]. Fractional partial differential equations (FPDEs) are widely used in various scientific and technological fields, and many researchers have been studying them lately. These fractional equations can describe many interesting phenomena in the areas of fluid and quantum mechanics, waves and optical fiber, electrodynamics, material science, plasma physics, and more [3, 4].
Numerous techniques have been suggested in previous studies to solve these equations. Some of these are FADM [5], RDTM [6], ILTM [7], ETDM [8], and more [9, 10]. Our goal in this paper is to use LRPSM to solve the following two equations.
1.1. Fractional Generalized Kuramoto–Sivashinsky Equation
The fractional generalized Kuramoto–Sivashinsky equation is a nonlinear partial differential equation that can be used to describe traveling waves in dispersive media, such as plasma and porous media [11], and also the dynamics of flame propagation in turbulent combustion. It is a generalization of the Kuramoto–Sivashinsky equation, which exhibits chaotic behavior and arises in a wide range of physical systems [12]. The fractional generalized Kuramoto–Sivashinsky equation involves fractional derivatives with nonsingular kernels, which can capture the memory and nonlocal effects of complex phenomena.
The fractional generalized Kuramoto–Sivashinsky equation is given by the following equation:
An electrostatic variable is denoted by , and the parameters , , and are coefficients that determine the strength of the second-, third-, and fourth-order derivatives, respectively. They can affect the stability and dynamics of the traveling waves. The parameter is the fractional order of the time derivative, which can capture the memory and nonlocal effects of complex phenomena.
The GKSE models various phenomena in science and engineering, such as chemical reactions, flows, flames, and reaction-diffusion systems. It also occurs in ocean engineering problems, such as viscous flow, magneto hydrodynamics, weather and climate, microtide turbulence, and off-shore industry. The Kuramoto–Sivashinky equation (KSE) [13] is a special case of the generalized Kuramoto–Sivashinky (1) when = and . The Laplace transformation and the variational iteration method were used by Shah et al. to solve (1) [14]. The modified Kudrayshov method was used to solve (1) with = = in [15]. The approximated solution of the KSE was analyzed by Veeresha and Prakasha using novel computational technique [16].
1.2. The Fractional Generalized Regularized Long Wave (FGRLW) Equation
A mathematical formula called the FGRLW equation is used to describe how small-amplitude extended waves travel through a fluid’s surface, and it also describes the behavior of long water waves in the ocean, including their propagation and interaction with different coastal structures. So, it is suitable to investigate this equation in fractional derivative to predict any unusual formulation of waves. It can be stated as follows:
The FGRLW equation involves several parameters, including constant , a positive integer , and the order of the fractional derivative . These parameters determine the behavior of the system and its solutions. The term represents the time derivative of the dependent variable , while represents its spatial derivative. The term is a nonlinear term that describes the effects of wave-wave interactions. Numerous disciplines, including fluid dynamics, plasma physics, and quantum mechanics, use the FGRLW equation extensively. Its solutions exhibit interesting phenomena such as solitary waves, which are waves that maintain their shape and speed over long distances. The study of the FGRLW equation and its solutions can provide insights into the behavior of complex systems and phenomena in nature.
When , the equation becomes the regularized long wave equation, which is a significant equation in physics media. It models phenomena that involve weak nonlinearity and dispersive waves equation [17]. When , the FGRLW becomes a special case that is called the modified regularized long wave (FMRLW) equation. Different approaches for solving the GRLW equation have been proposed in the literature. Nuruddeen et al. [18] introduced the METEM method for solving the FRLW equation. Nikan et al. used a finite difference method for solving the FRLW equation [19]. The optimal homotopy asymptotic method was presented by Nawaz et al. [20] for solution for the DGRLW equation. Also, there are many methods that have been used to solve this equation.
This paper intends to clarify the LRPS, a straightforward and successful technique for resolving differential equations with variable coefficients. The LRPS method was suggested in [21, 22] and provides a more straightforward and precise way to compute solutions for the equations mentioned earlier. We use the Laplace transform (LT) and power series method to deal with nonlinear differential equations. This involves changing the equations to Laplace space and using a suitable expansion to solve the equation that results from the power series method. To do this, we have created a new expansion that represents the solution of the equation in Laplace space. The coefficients of the series are then determined by the LRPS method. The LRPS method is simpler and more efficient than the conventional residual power series method, as it determines the coefficients based on the concept of the limit instead of on fractional derivation. This reduces the calculations, avoiding the need to repeatedly calculate fractional derivatives as required in the RPS method. Our suggested approach makes it possible to obtain precise and accurate approximations by adding a fast convergent series.
This paper consists of the following sections. In Section 1, we explain some important terms and ideas related to fractional calculus, and in Section 2, we present the specific type of fractional series that we will use in our study. Next, in Section 3, we describe the LRPS method, which is a useful technique for finding and predicting unique solutions to nonlinear fractional differential equations. Then, in Section 4, we demonstrate how the LRPS method works on two different differential equations. In addition, Section 5 shows graphs of the solutions that we obtained in Section 4. In Section 6, we discuss the results and their significance. Finally, we conclude by summarizing our main points and implications.
2. Preliminaries
Here, we introduce fundamental definitions and concepts of fractional calculus [23], alongside theorems related to Laplace transform [21].
Definition 1. If a real function , where , satisfies the condition that there exists a real number such that , where , then it is said to belong to the space , where . Similarly, if it satisfies the same condition but with being a natural number, then it is said to belong to the space , where .
Definition 2. The fractional integral of R–L of order , of a function is defined as follows:
Some properties of are, for , given by the following equation:
Definition 3. Caputo fractional derivative is defined as follows:for , , and .
Lemma 4. If ,, and ,
For further details, see [23]. Because it permits the formulation of our work by incorporating traditional initial and boundary conditions, we adopt the Caputo fractional derivative.
Definition 5. Suppose we have a function that is piecewise continuous on and has an exponential order of . In this case, we can define the Laplace transform of , denoted by , as follows [24]:using inverse Laplace transform.The Laplace transform’s integral converges absolutely only in the right half-plane, where is located.
Lemma 6. Suppose we have a piecewise continuous function on , with its Laplace transform denoted by . The function has the following characteristics:(1), for (2), for (3), for Here, represents the Caputo derivative.
The demonstration for this lemma can be found in [25].
Theorem 7. Consider a function that is sequentially consistent on the interval and for which the exponential order exists on the time interval . Let be the Laplace transform of . Suppose that the fractional expansion of is given by the following equation:then the coefficients are equal to the derivative of with respect to time evaluated at , denoted by .
This theorem provides a useful tool for solving fractional differential equations and other mathematical problems involving the Laplace transform, particularly those with a fractional expansion. The evidence of theorem (1) can be found in [25].
Remark 8. It is stated that the inverse Laplace transform of the equation presented in (3) can be expressed in the following manner:The fractional Taylor’s formula, as described in [26], is associated with the expression given for the inverse Laplace transform.
3. An Overview of the LRPSM Methodology
Using LRPSM, which is a method for solving complex equations involving fractional derivatives, we will explore the basic concepts and techniques for dealing with nonlinear fractional PDEs, which are equations that describe phenomena with fractional order of change.subject to
and the path of motion of a solitary wave is given by an unknown function that depends on and . The Caputo derivative is presented by , while and are constant values. The functions and can be either linear or nonlinear.
First, applying LT to equation (11), we have the following equation:
After applying the fact that and using the initial condition (12), we can express equation (11) as follows:where .
Second, we express the function that has been modified by a fractional Laplace transform as an infinite sum of terms.
Then, we write the function that is the truncated of the series of (15) as follows:
We define the LRF of (14) which is used to determine the unknown coefficients of the series in (16) by applying the LRPS method.and the LRF is as follows:
Several properties that are present in the standard residual power series method [26] can also be extended to the LRPSM. These properties include the following properties:(1) and for (2)(3)
We can recursively derive a system for obtaining the coefficient functions by satisfying the following condition:
After obtaining the coefficient functions through a recursive system, we use them to compute for a given Laplace variable and then apply the inverse Laplace transform to obtain the approximate solution as a function of the time variable . This procedure allows us to solve the original problem using an iterative approach that yields increasingly accurate solutions with each iteration.
3.1. Convergence Analysis
Since the proposed technique lead to the truncated power series of the following form:with exact solution , we can prove the convergence using the same manner as follows.
The investigated equation is written in the following form:
Theorem 9. Let be an operator from ⟶ (where is the Hilbert space) and let be the exact solution of equation (20), then the approximated solution (13) is convergent to if there exist a constant in which , for all .
Proof. We aim to prove that is a convergent Cauchy sequence as follows:For , we obtain the following equation:Hence, is a convergent Cauchy sequence in .
4. Applications
We show how the LRPS method can find solitary solutions for some FPDEs that are common in many fields, such as the fractional generalized Kuramoto–Sivashinsky and fractional generalized regularized long wave equations. We give two examples to illustrate the advantages and performance of the LRPS method for these problems. We used MATHEMATICA 11 for all the symbolic and numerical computations in this paper.
4.1. Application 1
Given the following, the fractional generalized Kuramoto–Sivashinsky equation is as follows:with initial condition
4.1.1. Case I
Let us consider FKS (23) for [16]:with initial conditionwhere , and are constants. The exact solution at is as follows:
We apply the LT on equation (25) and the initial condition from equation (26) to obtain the following equation:
We presume that the following form is the series solution of equation (25):
The truncate series of equation (25) can be obtained by performing the following steps:
Furthermore, we provide the following description for the LRF of equation (25):
To find for , we begin by substituting the truncate series (31) into the LRF equation (31). We then multiply both sides of (31) by . Next, we solve the resulting relation recursively to obtain . We can now obtain the first few elements of the sequence .
We can write the solution of (25) using the LRPS method as an infinite series.
The -approximate solution of our problem is obtained by taking the Laplace inverse of (33):
We have the option to compute additional coefficients; however, we will limit our calculation to . We will then compare the errors between the exact solution and the approximate solution obtained from this series.
Table 1 presents the results obtained by applying the LRPS method to solve the FKS equation (25) (Case I) for different combinations of time , spatial variable , and fractional derivative order . It demonstrates the convergence of solutions for different values of .
4.1.2. Case II
Let us consider the FKS (25) for [16]:with initial statewhere , and are constants.The exact solution at is
By performing a LT on both sides of equation (35) and utilizing the initial condition provided in equation (36), we can derive the following expression:
We make the assumption that the series solution of equation (35) takes the following form:
The truncated series of equation (35) can be obtained by the following equation:
We can define the LRF of equation (35) in the following manner:
To find for , we begin by substituting the truncate series (40) into the LRF (41). We then multiply both sides of equation (41) by . Next, we solve the resulting relation recursively to obtain . We can now obtain the first few elements of the sequence .
The solution to (35) via the LRPS method can be expressed as an infinite series.
By applying Laplace inverse to (43), we can obtain the -approximate solution to our problem.
We could calculate more coefficients, but for now, we will only calculate . Afterward, we will compare the errors between the precise solution and the estimated solution obtained from this series.
Table 2 presents the results obtained by applying the LRPS method to solve the FKS equation (35) (Case II) for different combinations of time , spatial variable , and fractional derivative order . It demonstrates the convergence of solutions for different values of .
4.1.3. Case III
Let us consider FKS (16) for [16]:with initial conditionwhere , and are constants. The exact solution at is
Following the initial condition specified in equation (46) while performing the Laplace transform of each side of equation (45), we obtain the following equation:
We presume that the following form can be used to express the series answer of equation (45):
The truncated series of equation (45) can be obtained by performing the following steps:
Furthermore, we provide the following definition for the LRF of equation (45):
In order to compute for values of from 1 to infinity, we first insert the truncated series expression (50) for the term into the corresponding LRF (51). We then multiply equation (51) by and proceed to solve the resulting equation:recursively to determine .
By following this method, we are able to calculate the initial terms of the sequence as follows:
We can write the solution of (45) using the LRPS method as an infinite series.
We can obtain the approximate solution to our problem by utilizing LI on equation (54).
For now, our main goal is to find the value of . We might compute more coefficients later. Then, we will measure the differences between the exact solution and the approximation from this series. This allows us to assess the quality of the estimated solution.
Table 3 presents the results obtained by applying the LRPS method to solve the FKS equation (45) (Case III) for different combinations of time , spatial variable , and fractional derivative order . It demonstrates the convergence of solutions for different values of .
4.2. Application 2
Given the following time-fractional GRLWE,where , and are real parameters.
With initial condition,where , and are constants. The exact solution is as follows:
The parameters and are usually related to the amplitude and the phase of the wave packet, respectively. The parameter represents the speed of the wave packet, and is the time variable. The parameter is related to the dispersion of the medium, and is the position of the wave packet’s peak at time .
4.2.1. Case I
When , the equation becomes the regularized long wave equation, which is a significant equation in physics media.
Given the following time-fractional RLWE,
The initial condition according to [27] is as follows:
For , the exact solution is as follows:
After performing the LT on each sides of equation (59) and applying the initial condition described in equation (60), we can derive the following expression:
We assume that the structure is taken by the series solution of equation (59):and the truncated series of equation (59) is obtained by the following equation:
To further clarify, we can define the LRF of equation (59) with the following formulation:
To compute the values of for ranging from 1 to infinity, we employ a recursive method that involves inserting the truncated series expression (64) into the corresponding Laplace residual function (65). We then multiply the resulting equation by and solve for in a recursive manner to determine . By using this approach, we can calculate the initial terms of the sequence without having to compute the entire infinite series.
We can write the solution of (59) using the LRPS method as an infinite series.
By utilizing the Laplace inverse on equation (67), we can obtain an approximate solution to our problem for the iteration.
Our primary purpose is to compute the numerical value of the third coefficient, . Additional coefficients may need to be computed at a later time. Following that, we will carefully examine any inconsistencies between the exact solution and the estimated solution produced from this series. This will allow us to assess the precision and reliability of the calculated solution.
Table 4 displays the results of solving FRLW equation (59) using the LRPS method for different values of and , as well as various values of (namely, and ). We notice from the results of the table a convergence of solutions at different values of .
4.2.2. Case II
When , the FGRLW becomes a special case that is called the modified regularized long wave (FMRLW) equation. Given the following time-fractional RLWE,
The initial condition according to [28] is
For , the exact solution is as follows:and following the same steps mentioned earlier, we will obtain the following equation:
We can write the solution of (69) using the LRPS method as an infinite series.
Applying the Laplace inverse to equation (73) can yield the approximate solution to our problem.
Our primary objective is to determine the numerical value of the third coefficient, . It may be necessary to compute additional coefficients at a later time. Subsequently, we will compare the exact solution with the estimated solution derived from this series and closely scrutinize any discrepancies. This will enable us to evaluate the accuracy and dependability of the computed solution.
The outcomes obtained from applying the LRPS method to solve the FRLW equation (69) for various and values, in addition to different values (specifically, and ), are presented in Table 5. We notice from the results of the tables a convergence of solutions at different values of .
5. Graphical Illustrations
To illustrate the relationship between the various parameters of a solution, graphs are a powerful method. Hence, this section uses 2D and 3D graphs to show the solution with different values of and . The approximate solution of (25) is shown in Figures 1(a) and 1(b) for various values of and . We notice convergence of results despite the difference in values between and . Figures 1(c) and 1(d) show that for , the approximate solution converges to the exact solution as increases. Similarly, the solutions obtained through approximation for (35) are presented in Figures 2(a) and 2(b), depicting different values of and . Figures 2(c) and 2(d) demonstrate that as increases, the accuracy of the approximate solution improves, approaching the exact solution for the case where equals 1. Also, the approximate solutions of (45) are shown in Figures 3(a) and 3(b) for various values of and . We notice convergence of results despite the difference in values between and . Figures 3(c) and 3(d) show that for , the approximate solution converges to the exact solution as increases. Additionally, in order to provide a more accurate representation of the method’s efficiency, we calculated the residual errors for the results of equations (25), (35), and (45) at different values of . This is depicted in Figures 4(a)–4(c). Finally, Figures 5(a), 5(b), 6(a), and 6(b) show the approximate solution to the RLWE (52) and the MRLWE (69) for different values of and , illustrating a two-dimensional graph that displays the soliton wave solution. The graph plots against and traces the wave trajectory for different values. The results show that the outcomes obtained through fractional-order analysis converge to those obtained via integer-order analysis. Additionally, the second 2D graph presents the wave solutions for at different values of , revealing that the soliton’s amplitude remains constant while it moves to the right. Furthermore, in Figures 5(c), 5(d), 6(c), and 6(d), the shape of both the approximate and the exact solutions is represented, highlighting the convergence of the two solutions. Additionally, in order to provide a more accurate representation of the method’s efficiency, we calculated the residual errors for the results of (59) and (69) at different values of . This is depicted in Figures 7(a) and 7(b).
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6. Discussion
In this section, we will review the results we have obtained in solving a set of nonlinear fractional partial differential equations using the LRPS method and compare these results with other methods used to solve the same applications. In Tables 6–8, we compare the solutions obtained using the LRPS method with those obtained using q-HATM [16] method for the first application, which represents the FKSE in its three cases. The results showed a convergence between the two methods for different values of , as well as a convergence to the exact solution. Furthermore, in Table 9, which represents the absolute error results for RLWE equation (59), a comparison was made with the q-HATM method [28]. The results showed that the method used in this study provided much better solutions than the other method. Finally, a comparison was made between the absolute error results obtained using the proposed method and the HPSTM [27] method in Table 10, which represents the results of solving equation MRLWE (69). The results showed that the proposed method outperformed the HPSTM [27] method in terms of accuracy. In general, this approach is highly effective and can be easily utilized for various nonlinear fractional partial differential equations along with their initial conditions. Furthermore, it offers a comprehensive framework that can be employed for different physical systems. Nevertheless, one drawback of this method is that it may not be suitable for all types of NFPDEs and the accuracy of the outcomes might rely on the specific system that is being modeled. Additionally, the method’s constraints in terms of the fractional derivative order and time range should be taken into account. Despite these limitations, this proposed method is a significant contribution to the field of NFPDEs and opens up promising avenues for future research.
7. Conclusion
In this paper, we have presented a new analytical method for solving nonlinear fractional partial differential equations, namely, the LRPS method. The LRPS method is based on the Laplace transform and the residual power series technique. We have applied the LRPS method to two nonlinear fractional partial differential equations: the fractional GRLWE and FKSE. We have shown that the LRPS method can provide accurate and efficient approximate solutions for these equations. We have also compared the LRPS method with some existing methods and found that the LRPS method has some advantages over them. The LRPS method is a simple and powerful mathematical tool that can be used to solve various nonlinear fractional partial differential equations arising in different fields of science and engineering.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Disclosure
The authors confirm that all the results they obtained are new.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The first and second authors contributed to the planning and design of the study. The third author conducted data collection and analysis. The fourth and fifth authors made figure preparation, manuscript preparation, and editing. All authors reviewed the final version of the paper and agreed to publish.