In this research, the He-Laplace algorithm is extended to generalized third order, time-fractional, Korteweg-de Vries (KdV) models. In this algorithm, the Laplace transform is hybrid with homotopy perturbation and extended to highly nonlinear fractional KdVs, including potential and Burgers KdV models. Time-fractional derivatives are taken in Caputo sense throughout the manuscript. Convergence and error estimation are confirmed theoretically as well as numerically for the current model. Numerical convergence and error analysis is also performed by computing residual errors in the entire fractional domain. Graphical illustrations show the effect of fractional parameter on the solution as 2D and 3D plots. Analysis reveals that the He-Laplace algorithm is an efficient approach for time-fractional models and can be used for other families of equations.

1. Introduction

In the last few decades, fractional calculus has outperformed ordinary calculus because basic calculus has reached to its peak. Engineers and scientists are focusing on the fractional models and their solutions due to their ability to provide more meaningful insight of physical phenomena with memory effects such as fractional Casson fluid with ramped wall temperature [1], fractional SEIR model of Covid 19 [2], fractional dual-phase-lag thermoelastic model [3], novel fractional time-delayed grey Bernoulli forecasting model [4], fractal fractional model of drilling nono-liquids [5] and stability of fractional quasi-linear impulsive integro-differential systems [6]. This permits a more accurate description of real-world situations than the basic integral order. Well-known scientists including Joseph [7], Miller and Ross [8], Caputo [9], and Riemann [10] have made a significant contribution towards the foundation of fractional calculus. Fractional calculus provides a more accurate and realistic depiction of various phenomena in quantum physics [11], oceanography [12], fluid mechanics [13], and engineering [14]. In addition, fractional calculus is used to simulate the damping behavior of different materials and substrates, financial models, and many other scenarios.

Solitary wave equations like (1 + 1)-dimensional Mikhailov–Novikov–Wang equation (15), RLW equation (16), complex Ginzburg–Landau model [17], and Korteweg and de Vries equations [18] have assembled a lot of interest from researchers. Among them, the most relevant family is KdV equations which also provide a foundation for other models. During 1895, Korteweg and de Vries first modeled the classical KdV equation [18]. These equations are highly nonlinear and describe wave structures in crystal lattice, plasma, water, and density stratified ocean waves, etc., that are explored by many researchers. Heydari et al. observed fractional KdV-burger’s equation by discrete Chebyshev polynomials [19], second order difference schemes for time-fractional KdV-burger’s is solved by Cen et al. [20], fractional Kaup–Kupershmidt equation is analyzed by Shah et al. [21], Iqbal et al. [22] applied Atangana-Baleanu derivative on fractional Kersten–Krasil’shchik coupled KdV-mKdV system. KdV equations are also used in string theory with continuum limit. Similarly, the study of many physical aspects through KdV equations in quantum field theory, general relativity, and fluid mechanics are explored in recent studies [2326]. Crabb et al. observed the complex korteweg-de Vries equation [27], forced korteweg-de Vries equation on critical flow over a hole is solved by Veeresha et al. [28], Yavuz et al. utilized fractional order with Mittag–Leffler on Schrodinger-KdV equation [29], Wang and Mei analyzed [30] KdV and Boussinesq hierarchy with a lax triple.

In the present work, generalized third order time-fractional KdV models are investigated through an extended He-Laplace algorithm. The proposed approach is applied to three KdV models, namely, Korteweg-de Vries-Burgers (KdVB) [31], potential Korteweg-de Vries (p-KdV) [32], and time-fraction dispersive KdV equation [31]. The generalized KdVB equation was proposed by Su and Gardner in 1969 [33] by combining the classical KdV equation [18] with the Burgers equation [34]. General form of KdVB model iswhere and are spatial and temporal variables while is the wave profile, and are nonzero real constants, is viscous loss, and are dispersion and convective nonlinearity, respectively. KdVB equation depicts several physical phenomena like the flow of liquids containing gas bubbles, propagation of waves in an elastic tube filled with a viscous fluid, plasma waves, and propagation of bores in shallow water etc.

On the other hand, the p-KdV equation [32] replicates waves on much greater frequency such as tsunami waves. The standard form of P-KdV is

Here, is the evolution term, is the nonlinear term and is the dispersion term. Moreover, , indicate spatial and temporal variables while and is the wave profile, respectively. and are nonzero real constants.

Computing the exact or approximate solutions of fractional differential equations (FDEs) is very important in all the mentioned fields, but due to the complex nature of FDEs, the exact solution is not possible in most of the cases. As a result, it is essential to compute approximate solutions through analytical or numerical methods like the fractional natural decomposition method [35], Fourier spectral method [36], Lie symmetry analysis [37], auxiliary function method [38], consistent Riccati expansion method [39], and homotopy perturbation method [40]. For better accuracy while dealing with nonlinear problems, various modifications of HPM are also employed on different equations. A few of these modifications are; HPM coupled with the PSEM method [41], Li-He’s modified homotopy perturbation method [42], modified HPM for the solution of parametric cubic-quintic nonconservative duffing oscillator [43], novel homotopy perturbation method with exponential-decay kernel [44], He-Laplace method [45]. The he-Laplace technique combines Laplace transform with classic HPM. This modified algorithm can easily solve nonlinear problems with reasonable accuracy. In the current manuscript, He-Laplace is extended to nonlinear generalized third order time-fractional KdV models. The algorithm is tested against the potential KdV model (high frequency model mostly used for tsunami waves), KdV Burgers model, and dispersive KdV model. In the rest of the manuscript, Sections 2 and 3 present the definitions and general methodology of the He-Laplace algorithm for generalized third-order time-fractional KdV models. Section 4 is showing the convergence and error estimation. The application of the He-Laplace approach to KdV models is in Section 5. Discussion results are presented in Section 6 while the conclusion is in Section 7.

2. Definitions

Definition 1. The Laplace transform coupled with the Riemann–Liouville time-fractional integral [46] is described as follows [47]:

Definition 2. The Laplace transform coupled with Caputo’s time-fractional derivative [46] is described as follows [47]:

3. Fundamental Concept of He-Laplace Algorithm for Third-Order Time-Fractional KdV Models

Consider a general third order, time-fractional KdV equation as follows:that depend on initial conditionswhere is an unknown function that has time-fractional derivative , is a known function with and as space and time variables respectively. is the domain of and and are symbols of linear and nonlinear operators, respectively.

Start the procedure by applying Laplace transform on (5), which gives

By using Def. (4), we have

Homotopy of the above-given equation is:where represent the initial guess.

By expanding using Taylor series with regard to , we get

Substituting the (10) in (9), and after that equating the coefficients of with identical powers, we acquire the following equation.

The equation at first order:

Implementing inverse Laplace transform leads to

In general, equation at order is

Inverse Laplace transform of the above-given equation is

The approximate series solution of the general third order, time-fractional KdV equation is

Substituting (15) in (5) gives residual function as follows:

4. Convergence Analysis and Error Estimation

4.1. Convergence

Theorem 1. Let a Banach space (, ) has functions and defined in it. Then, the series solution given in equation (16) converges towards the solution of (5) with constant (0, 1).

Proof. For the sequence of partial sums of (16), we have to verify that is a Cauchy sequence in (, ). ConsiderBy the successive use of triangle inequality on the partial sums and with and , we haveUsing (17), we getGiven that 0 1, hence, 1. Thus, is bounded so, it givesHence, we have proved that is a Cauchy sequence in Banach space. Therefore, the series solution given in (15) converges towards the solution of (5).

4.2. Error Estimation

Theorem 2. For a third order, time-fractional KdV equation (5), the maximum absolute truncation error of its solution (16) is

Proof. Equation (20) givesSince , therefore , which givesHence proved.

5. Solutions of Time-Fractional KdV Models Using He-Laplace Algorithm

Example 1. Consider the time-fractional potential KdV equationwith initial conditionThe exact solution of p-KdV (25) is

Solution 1. Initiating Laplace transform of given p-KdV (25) and then adapting definition (4) givesHomotopy of (28) iswhere, the initial guess isUsing (10) and then equalizing the same coefficients of gives.
Equation at first order:Implementing inverse Laplace transform gives:Equation at second order:Solution at second order:Equation at third order:Solution at third order:Obtained approximate series solution of (25) isResidual function ‘’ of (25) is

Example 2. Consider the time-fractional dispersive KdV equationassociated with initial conditionThe exact solution is

Solution 2. Procedure given in Section 3 leads to:
The initial guess that isEquation at first order:Solution at first orderEquation at the second order:Solution at the second order:Equation at third order:Solution at third order:Approximate series solution of (39) by the He-Laplace algorithm can be observed byResidual function ‘’ of (39) is

Example 3. Consider the time-fractional KdV Burgers equationthat has initial conditionThe exact solution of KdV Burgers equation is

Solution 3. Procedure in Section 3 gives:
The initial guess that isEquation at first order:Equation at second order:Equation at third order:After taking inverse Laplace transform of these problems the approximate series isResidual errors of (51) can be examined by the following equation:

6. Results and Discussion

In this paper, several third order KdV models, KdV-Burger’s, time-fractional dispersive KdV, and potential-KdV models are examined at both fractional and integral orders. The solution and analysis of these models are depicted in graphical and tabular form. For this purpose, an efficient semianalytical technique, He-Laplace method is utilized. For Example 1, the convergence of the p-KdV equation for fractional parameter , can be examined from Table 1. Solutions at third, sixth, and ninth iterations are compared. Increasing iterations by using the He-Laplace algorithm decrease in errors is observed. Table 2 expressed the residual errors calculated by residual function (39) at  = 0.18, 0.44, 0.76, and 0.92. This table indicates that as the value of approaches 1, the errors decline. 3-D Figure 1 demonstrates the solutions and errors in graphical form at  = 1. To illustrate the nature of on surface waves throughout the domain, 2-D diagrams (Figure 2) and (Figure 3) are plotted. Figure 2 shows that at a fixed time, increasing the value of the fractional parameter reduces water level for whereas it increases the level at . On the other hand, Figure 3 reveals that increase in values of at fixed increases the water level. Figure 4 is the effect of the constant parameter on the velocity profile which shows that increasing value of increases the velocity profile.

Table 3 of Example 2 displays the comparison between errors obtained by the modified generalized Mittag–Leffler function method (MGMLFM) and the He-Laplace method. It is concluded that the He-Laplace method shows better results in terms of accuracy as compared with other methods. Table 4 indicates that as the quantity of fractional parameter rises, error reduces. Figure 5 presents the solution and error plots in 3-D format whereas, Figure 6 is the 2-D representation of effect of on water waves profile at a certain time. It can be seen from Figure 7 that as increases throughout the domain, wave profile decreases. Moreover, Figure 8 conveys that the increment in the value of constant parameter decreases the velocity profile.

In Example 3, the efficiency of the He-Laplace algorithm over other methods is depicted in Table 5. Moreover, Table 6 displays residual errors of the KdVB equation at various . Figure 9 depicts the approximate solution and absolute error. Figure 10 shows the behaviour of waves at different values of the fractional parameter for a fixed time. At a certain value of a spatial variable, by boosting the water waves drop (see Figure 11). Also, from Figure 12 we can observe that with higher values of the constant parameter , velocity profile elevates.

7. Conclusion

The focus of this study is to solve and analyze third-order time-fractional KdV models of three different kinds. This investigation can assist researchers to illustrate complex, fractional, and nonlinear real-world problems effectively. In this regard, p-KdV, KdVB, and KdV equations are considered. A highly efficient technique, the He-Laplace method, which is the combination of the homotopy perturbation method (HPM) and Laplace transform, is adapted for solution purpose. Approximate solutions and residual errors are depicted in form of 3-D graphs. The effect of fractional parameter on wave profile is shown in graphical and tabular form. The convergence of the model is verified through tabular results. With an increase in the number of iterations, series form solutions converge to the analytical solution. Moreover, the reliability of the method is confirmed through comparison with existing results in the literature. The he-Laplace algorithm proved to be an effective technique in solving specified KdV models with least errors and enhanced series solutions.

Data Availability

All the data is within the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.