Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method

. Te Laplace residual power series (LRPS) method uses the Caputo fractional derivative defnition to solve nonlinear fractional partial diferential equations. Tis technique has been applied successfully to solve equations such as the fractional Kur-amoto–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. Te resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. Te LRPS approach ofers both computational efciency and solution accuracy, making it an efective technique for solving nonlinear fractional partial diferential equations (NFPDEs). Te 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 efort.


Introduction
In recent years, fractional calculus has gained signifcant attention from scientists and engineers due to its broad applicability and its ability to capture the complexities of real-world problems in various felds such as plasma physics, fuid 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 diferential equations (FPDEs) are widely used in various scientifc and technological felds, and many researchers have been studying them lately.Tese fractional equations can describe many interesting phenomena in the areas of fuid and quantum mechanics, waves and optical fber, 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.

Fractional Generalized Kuramoto-Sivashinsky Equation.
Te fractional generalized Kuramoto-Sivashinsky equation is a nonlinear partial diferential equation that can be used to describe traveling waves in dispersive media, such as plasma and porous media [11], and also the dynamics of fame 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].Te fractional generalized Kuramoto-Sivashinsky equation involves fractional derivatives with nonsingular kernels, which can capture the memory and nonlocal efects of complex phenomena.
Te fractional generalized Kuramoto-Sivashinsky equation is given by the following equation: An electrostatic variable is denoted by φ(ξ, t ¸), and the parameters μ, ϑ, and δ are coefcients that determine the strength of the second-, third-, and fourth-order derivatives, respectively.Tey can afect the stability and dynamics of the traveling waves.Te parameter α is the fractional order of the time derivative, which can capture the memory and nonlocal efects of complex phenomena.

Te 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 fuid's surface, and it also describes the behavior of long water waves in the ocean, including their propagation and interaction with diferent 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: Te FGRLW equation involves several parameters, including constant μ, a positive integer a, and the order α of the fractional derivative D α t ¸.Tese parameters determine the behavior of the system and its solutions.Te term D α t ¸φ represents the time derivative of the dependent variable φ(ξ, t ¸), while φ ξ represents its spatial derivative.Te term φ ξ φ p is a nonlinear term that describes the efects of wavewave interactions.Numerous disciplines, including fuid 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.Te study of the FGRLW equation and its solutions can provide insights into the behavior of complex systems and phenomena in nature.
When p � 1, the equation becomes the regularized long wave equation, which is a signifcant equation in physics media.It models phenomena that involve weak nonlinearity and dispersive waves equation [17].When p � 2, the FGRLW becomes a special case that is called the modifed regularized long wave (FMRLW) equation.Diferent 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 fnite diference method for solving the FRLW equation [19].Te 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.
Tis paper intends to clarify the LRPS, a straightforward and successful technique for resolving diferential equations with variable coefcients.Te 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 diferential equations.Tis 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.Te coefcients of the series are then determined by the LRPS method.Te LRPS method is simpler and more efcient than the conventional residual power series method, as it determines the coefcients based on the concept of the limit instead of on fractional derivation.Tis 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.
Tis 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 specifc 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 fnding and predicting unique solutions to nonlinear fractional diferential equations.Ten, in Section 4, we demonstrate how the LRPS method works on two diferent diferential 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 signifcance.Finally, we conclude by summarizing our main points and implications.

Preliminaries
Here, we introduce fundamental defnitions and concepts of fractional calculus [23], alongside theorems related to Laplace transform [21].
Defnition 1.If a real function φ(t ¸), where t ¸> 0, satisfes the condition that there exists a real number ρ > υ such that φ(t ¸) � t ¸ρφ 1 (t ¸), where φ 1 (t ¸) ∈ C(0, ∞), then it is said to belong to the space Cυ, where υ ∈ R. Similarly, if it satisfes the same condition but with ρ > υ being a natural number, then it is said to belong to the space C ℵ υ, where ℵ ∈ N.

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International Journal of Diferential Equations Defnition 2. Te fractional integral of R-L of order α, α > 0 of a function φ(t ¸) is defned as follows: Some properties of J α are, for c, α ⩾ 0, υ ⩾ − 1, given by the following equation: Defnition 3. Caputo fractional derivative is defned as follows: 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.
Defnition 5. Suppose we have a function φ(ξ, t ¸) that is piecewise continuous on I × [0, ∞) and has an exponential order of δ.In this case, we can defne the Laplace transform of φ(ξ, t ¸), denoted by Φ(x, S), as follows [24]: using inverse Laplace transform.
Te Laplace transform's integral converges absolutely only in the right half-plane, where ϵ 0 is located.
Here, D nα t ¸� z n /zt ¸n(D α t ¸) represents the Caputo derivative.
Te demonstration for this lemma can be found in [25].
Theorem 7. Consider a function φ(ξ, t ¸) that is sequentially consistent on the interval I and for which the exponential order δ exists on the time interval [0, ∞).Let Φ(ξ, S) be the Laplace transform of φ(ξ, t ¸).Suppose that the fractional expansion of Φ(ξ, S) is given by the following equation: then the coefcients ρ n (x) are equal to the n th derivative of φ(ξ, t ¸) with respect to time evaluated at t ¸� 0, denoted by Tis theorem provides a useful tool for solving fractional diferential equations and other mathematical problems involving the Laplace transform, particularly those with a fractional expansion.Te 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: Te fractional Taylor's formula, as described in [26], is associated with the expression given for the inverse Laplace transform.

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.
Ten, we write the function Φ k (ξ, S) that is the k th truncated of the series of (15) as follows: We defne the LRF of ( 14) which is used to determine the unknown coefcients of the series in ( 16) by applying the LRPS method.
and the k th LRF is as follows: Several properties that are present in the standard residual power series method [26] can also be extended to the LRPSM.Tese properties include the following properties: . .We can recursively derive a system for obtaining the coefcient functions ρ n (ξ) by satisfying the following condition: After obtaining the coefcient functions ρ n (ξ) through a recursive system, we use them to compute Φ k (ξ, S) for a given Laplace variable S and then apply the inverse Laplace transform to obtain the k th approximate solution φ k (ξ, t ¸) as a function of the time variable t ¸.Tis procedure allows us to solve the original problem using an iterative approach that yields increasingly accurate solutions with each iteration.

Convergence Analysis.
Since the proposed technique lead to the truncated power series of the following form: with exact solution φ κ (ξ, t ¸), we can prove the convergence using the same manner as follows.
Te investigated equation is written in the following form: Theorem 9. Let F be an operator from H ⟶ H (where H is the Hilbert space) and let φ be the exact solution of equation (20), then the approximated solution ( 13) is convergent to φ if there exist a constant W, (0 Proof.We aim to prove that f ι || ∞ ι�0 is a convergent Cauchy sequence as follows:

International Journal of Diferential Equations
For ι, m ∈ N, ι > m, we obtain the following equation: Hence, f ι || ∞ ι�0 is a convergent Cauchy sequence in H.

Applications
We show how the LRPS method can fnd solitary solutions for some FPDEs that are common in many felds, 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 MATHEMA-TICA 11 for all the symbolic and numerical computations in this paper.

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 μ � − 1, ϑ � 0, δ � 1 [16]: with initial condition where β, v, and κ are constants.Te exact solution at α � 1 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): Te k th truncate series of equation ( 25) can be obtained by performing the following steps: Furthermore, we provide the following description for the kth LRF of equation ( 25):
We can write the solution of (25) using the LRPS method as an infnite series.
Te k th -approximate solution of our problem is obtained by taking the Laplace inverse of (33): We have the option to compute additional coefcients; however, we will limit our calculation to ρ 2 .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 diferent combinations of time (t ¸), spatial variable (ξ), and fractional derivative order (α).It demonstrates the convergence of solutions for diferent values of α.

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International Journal of Diferential Equations

Case II.
Let us consider the FKS (25) for μ � 1, ϑ � 0, δ � 1 [16]: with initial state where β, v, and κ are constants.Te exact solution at α � 1 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: Te k th truncated series of equation ( 35) can be obtained by the following equation: We can defne the k th LRF of equation ( 35) in the following manner: To fnd ρ k (ξ) for k � 1, 2, 3, . .., we begin by substituting the k th truncate series (40) into the k th LRF (41).We then multiply both sides of equation ( 41) by S kα+1 .Next, we solve the resulting relation lim S ⟶ ∞ S kα+1 L ¸Res k (ξ, S) � 0, k � 1, 2, 3, 4, . . .recursively to obtain ρ k (ξ).We can now obtain the frst few elements of the sequence ρ k (ξ).
Te solution to (35) via the LRPS method can be expressed as an infnite series.
By applying Laplace inverse to (43), we can obtain the k th -approximate solution to our problem.
We could calculate more coefcients, but for now, we will only calculate ρ 2 .Afterward, we will compare the errors between the precise solution and the estimated solution obtained from this series.
with initial condition where β, v, and κ are constants.Te exact solution at α � 1 is International Journal of Diferential Equations Following the initial condition specifed 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): Te k th truncated series of equation ( 45) can be obtained by performing the following steps: Furthermore, we provide the following defnition for the k th LRF of equation ( 45): In order to compute ρ k (ξ) for values of k from 1 to infnity, we frst insert the truncated series expression (50) for the k th term into the corresponding LRF (51).We then multiply equation (51) by S kα+1 and proceed to solve the resulting equation: By following this method, we are able to calculate the initial terms of the sequence ρ k (ξ) as follows: International Journal of Diferential Equations ) 32768 We can write the solution of (45) using the LRPS method as an infnite series.
We can obtain the k th approximate solution to our problem by utilizing LI on equation (54).
For now, our main goal is to fnd the value of ρ 2 .We might compute more coefcients later.Ten, we will measure the diferences between the exact solution and the approximation from this series.Tis 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 diferent combinations of time (t ¸), spatial variable (ξ), and fractional derivative order (α).It demonstrates the convergence of solutions for diferent values of α.

Application 2. Given the following time-fractional GRLWE,
) where p, a, and μ are real parameters.

With initial condition,
where c, A, and ξ 0 are constants.Te exact solution is as follows: Te parameters A and p are usually related to the amplitude and the phase of the wave packet, respectively.Te parameter c represents the speed of the wave packet, and t ¸is the time variable.Te parameter μ is related to the dispersion of the medium, and ξ 0 is the position of the wave packet's peak at time t ¸� 0.

Case I.
When p � 1, the equation becomes the regularized long wave equation, which is a signifcant equation in physics media.Given the following time-fractional RLWE, Te initial condition according to [27] is as follows: 10 International Journal of Diferential Equations For α � 1, 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 k th truncated series of equation ( 59) is obtained by the following equation: To further clarify, we can defne the k th LRF of equation (59) with the following formulation: To compute the values of ρ k (ξ) for k ranging from 1 to infnity, 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 S kα+1 and solve for lim S ⟶ ∞ S kα+1 L ¸Res k (ξ, S) � 0, k � 1, 2, 3, 4, . . . in a recursive manner to determine ρ k (ξ).By using this approach, we can calculate the initial terms of the sequence ρ k (ξ) without having to compute the entire infnite series.International Journal of Diferential Equations We can write the solution of (59) using the LRPS method as an infnite series.
By utilizing the Laplace inverse on equation (67), we can obtain an approximate solution to our problem for the k th iteration.
Our primary purpose is to compute the numerical value of the third coefcient, ρ 2 .Additional coefcients 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.Tis 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 diferent values of t ¸and ξ, as well as various values of α (namely, α � 0.6, 075 and 1).We notice from the results of the table a convergence of solutions at diferent values of α.

Case II.
When p � 2, the FGRLW becomes a special case that is called the modifed regularized long wave (FMRLW) equation.Given the following time-fractional RLWE, Te initial condition according to [28] is For α � 1, 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 infnite series.
Applying the Laplace inverse to equation (73) can yield the k th approximate solution to our problem.
12 International Journal of Diferential Equations Our primary objective is to determine the numerical value of the third coefcient, ρ 2 .It may be necessary to compute additional coefcients at a later time.Subsequently, we will compare the exact solution with the estimated solution derived from this series and closely scrutinize any discrepancies.Tis will enable us to evaluate the accuracy and dependability of the computed solution.
Te outcomes obtained from applying the LRPS method to solve the FRLW equation (69) for various t ¸and ξ values, in addition to diferent α values (specifcally, α � 0.6, 0.75 and 1), are presented in Table 5.We notice from the results of the tables a convergence of solutions at diferent values of α.

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 φ(ξ, t ¸) with diferent values of α and t ¸.Te approximate solution of ( 25) is shown in Figures 1(a) and 1(b) for various values of α and t ¸.We notice convergence of results despite the diference in values between α and t ¸.Figures 1(c) and 1(d) show that for α � 1, the approximate solution converges to the exact solution as α increases.Similarly, the solutions obtained through approximation for (35) are presented in Figures 2(a) and 2(b), depicting diferent values of α and t ¸.Figures 2(c) and 2(d) demonstrate that as α increases, the accuracy of the approximate solution improves, approaching the exact solution for the case where α equals 1.Also, the approximate solutions of (45) are shown in Figures 3(a) and 3(b) for various values of α and t ¸.We notice convergence of results despite the diference in values between α and t ¸.Figures 3(c) and 3(d) show that for α � 1, the approximate solution converges to the exact solution as α increases.Additionally, in order to provide a more accurate representation of the method's efciency, we calculated the residual errors for the results of equations ( 25  ¸, illustrating a two-dimensional graph that displays the soliton wave solution.Te graph plots φ(ξ, t ¸) against ξ and traces the wave trajectory for diferent α values.Te 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 α � 1 at diferent values of t ¸, 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 efciency, we calculated the residual errors for the results of (59) and (69) at diferent values of α.Tis is depicted in Figures 7(a

Discussion
In this section, we will review the results we have obtained in solving a set of nonlinear fractional partial diferential 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 frst application, which represents the FKSE in its three cases.Te results showed a convergence between the two methods for diferent values of α, 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].Te 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).Te results showed         International Journal of Diferential Equations that the proposed method outperformed the HPSTM [27] method in terms of accuracy.In general, this approach is highly efective and can be easily utilized for various nonlinear fractional partial diferential equations along with their initial conditions.Furthermore, it ofers a comprehensive framework that can be employed for diferent 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 specifc 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 signifcant contribution to the feld of NFPDEs and opens up promising avenues for future research.

Conclusion
In this paper, we have presented a new analytical method for solving nonlinear fractional partial diferential equations, namely, the LRPS method.Te 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 diferential equations: the fractional GRLWE and FKSE.We have shown that the LRPS method can provide accurate and efcient 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.Te LRPS method is a simple and powerful mathematical tool that can be used to solve various nonlinear fractional partial diferential equations arising in diferent felds 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
Te authors confrm that all the results they obtained are new.