Wave solution for time fractional geophysical KdV equation in uncertain environment

1.   Introduction

Nonlinear evolution equations (NLEEs) have gained a significant importance in engineering and mathematical sciences, thereby influencing diverse fields such as nuclear physics, plasma physics, kinematics, chemistry, biology, bioengineering, mechanics, electromagnetism, nonlinear optics, mathematical biosciences, and so on. These equations are essential to describe complex scientific phenomena, enabling the modeling and analysis of dynamical systems with differential equations. In particular, mathematicians have made substantial progress in understanding the dynamics of nonlinear partial differential equations (NPDEs), particularly the soliton aspect, thus contributing valuable methods and insights. The exploration of physical behaviors within this framework not only enhances the understanding, but also opens doors to innovative applications, potentially leading to practical breakthroughs in various scientific domains.

A specific class of NLEEs includes the shallow water equations, a set of hyperbolic partial differential equations (PDEs), which have been widely used to model various real-world phenomena governed by PDEs of an arbitrary order. One prominent example of such an equation is the Korteweg-de Vries (KdV) equation, which can be expressed as follows:

This equation was introduced by Korteweg and de Vries in ^[1], which describes the evolution of small amplitude, long waves in shallow water, where factors such as depth and bottom topography influence the wave speed, thus leading to changes in wave profiles. The KdV equation, precisely solvable with suitable initial data, has captivated widespread interest among researchers for its significance in understanding phenomena like wave breaking and dispersion.

In contrast to the KdV equation, Geyer et al. ^[2] introduced a nonlinear equation to analyze single-layer oceanic fluids in equatorial regions by incorporating the effects of the rotatory motion of Earth on the fluid, commonly referred to as the Coriolis effect. This formulation, termed the geophysical KdV (GeoKdV) equation, contributes to a more comprehensive understanding of oceanic phenomena, and can be written as follows:

Here, $\eta$ is a function of $(\xi, \tau)$ and represents the free surface elevation, and $\omega$ is the Coriolis parameter.

In recent years, several authors have explored various aspects of the GeoKdV equation. For instance, Turgut et al. ^[3] investigated the Coriolis effect on oceanic flows in the equatorial region using Lie symmetries and implemented the finite element method for numerical solutions. Rizvi et al. ^[4] employed the Hirota bilinear method to obtain lump-kink and lump-periodic solutions. The solitons and complexions of the GeoKdV equation were derived using Kudryashov and Hirota methods, and the impact of the Coriolis parameter on nonlinear waves was studied by Hosseini et al. ^[5]. Additionally, Alharbi and Almatraf ^[6] investigated the GeoKdV equation by utilizing an improved exp-expansion method, and obtained various solitary solutions. Additionally, they applied shooting and adaptive moving methods to compute numerical solutions and used Fourier analysis to evaluate the accuracy and stability of these numerical schemes. In recent years, Sayed et al. ^[7] explored the modified and generalized forms of the GeoKdV equation. They utilized the extended Tanh method and the generalized Kudryashov method with the Riccati equation to derive sine and cosine hyperbolic solutions.

To address the limitations of integer order operators, which are localized and incapable of predicting complex behaviors, various mathematicians and scientists have introduced non-integer order operators. In recent years, scientific research has increasingly focused on exploring more complex changes through fractional calculus due to its non-local properties and memory-retaining characteristics. This shift has led to significant growth in its applications across various disciplines, including mathematics, physics, engineering, biotechnology, electrochemistry, and more; see ^{[8,9,10,11,12,13,14,15,16]}. However, despite considerable progress, obtaining precise or numerical solutions for models that involve fractional calculus remains a significant challenge. Addressing these complexities requires innovative approaches, as many existing methodologies in the literature employ nonlinear fractional differential equations (FDEs) to capture the behavior of physical phenomena within dynamic systems. In this regard, Karunakar and Chakraverty ^[17] utilized the homotopy perturbation method (HPM) to solve time-fractional third- and fifth-order KdV equations. Shah et al. ^[18] utilized the Mohand decomposition method with Caputo derivatives to tackle fractional and integer-order KdV equations, demonstrating that fractional-order solutions gradually converge to their integer-order counterparts. Similarly, Jena and Chakraverty ^[19] efficiently obtained solitary wave solutions for generalized Hirota-Satsuma coupled KdV and MKdV equations. Al-Sawalha et al. ^[20] made further advancements by applying a natural decomposition method alongside Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to obtain approximate solutions for the Kersten-Krasil'shchik coupled KdV and modified KdV systems. Cao et al. ^[21] explored variational principles for two types of nonlinear GeoKdV equations with fractal derivatives. In recent times, Shehzada et al. ^[22] delved into a modified form of the fractional GeoKdV equation with a power-law kernel. These studies collectively highlight the growing importance of fractional calculus to solve various types of KdV equations, thus demonstrating its potential to provide deeper insights into dynamic systems.

On the contrary, the information about the parameters involved in shallow water wave equations (SWWEs) is often incomplete. To address this, fuzzy parameters can be incorporated into the SWWEs. Zadeh ^[23] introduced the fuzzy set theory in 1965, offers a mathematical framework to handle uncertainty and imprecision. Unlike the classical set theory, where an element either fully belongs or does not belong to a set, fuzzy sets allow for partial membership, with elements belonging to a set to varying degrees, typically represented by values between 0 and 1. This approach is especially useful in cases where boundaries between the categories are not clearly defined, such as when dealing with linguistic variables or ambiguous data. In our study, we apply the fuzzy set theory to address the uncertainty present either in system parameters or initial conditions.

By representing uncertain variables as fuzzy numbers, we capture a range of possible values rather than relying on a single deterministic value. This approach increases the robustness of our model and provides a more comprehensive understanding of the system behavior, particularly in situations where precise information is unavailable. In this regard, several authors have also applied the fuzzy set theory to water wave equations, thus demonstrating its effectiveness in capturing uncertainties in such complex systems. As such, the behavior of two-dimensional SWWEs with fuzzy parameters was investigated by Karunakar and Chakraverty ^[24] using the HPM, treating water depth as uncertain and expressed in terms of fuzzy parameters. Sahoo and Chakraverty ^[25] introduced the Sawi transform-based HPM as a hybrid approach to approximate solutions to one-dimensional SWWEs in tsunami predictions. Additionally, they applied HPM to solve coupled SWWEs, considering the wave height as fuzzy ^[26]. Using this approach, the impact of the coastal slope and the sea depth on tsunami wave characteristics, particularly wave velocity and run-up height, was analyzed in uncertain environments. Extending the study of fuzzy uncertainty in these fields, they further derived an approximate solution for the GeoKdV equation by using the Adomian decomposition method within the fuzzy environment ^[27].

To the best of our knowledge, the proposed model has not been explored within the domain of fractional calculus and fuzzy environment. Drawing inspiration from valuable contributions in fractional calculus and fuzzy uncertainty, we aim to investigate the uncertain GeoKdV equation in the fractional order of Caputo's sense by using the fractional reduced differential transform method (FRDTM). Consequently, we consider the GeoKdV Eq (1.2) in the fuzzy fractional form as:

This study is motivated by the desire to understand the intricate behaviors exhibited by the nonlinear GeoKdV equation in the fuzzy environment and to investigate the effect of the fractional order on its dynamics. Unlike traditional methods, the FRDTM simplifies the complex dynamics of the GeoKdV equation under uncertain and fractional-order conditions, offering an improved accuracy and an ease of implementation. This study not only demonstrates the effectiveness of the FRDTM through multiple case analyses, but also establishes strong agreements between simulated results and provided data, highlighting the robustness of the method. These results confirm that the FRDTM technique is a reliable method for addressing fractional-order nonlinear problems.

The subsequent part of this article is structured as follows: Section 2 covers the fundamental concepts of fuzzy set theory and fractional calculus; in Section 3, the methodology is discussed, with a detailed analysis of the FRDTM; Section 4 aims to provide a comprehensive understanding of application of the FRDTM to solve the time fractional fuzzified GeoKdV equation; in Section 5, results and discussion are presented, where the research outcomes are thoroughly analyzed and interpreted; and finally, Section 6 offers a conclusive summary of the findings and insights for future research directions.

2.   Basic concepts of fuzzy sets and fractional calculus

In our subsequent discussions, we will look at some fundamental concepts related to fuzzy and fractional calculus.

Definition 2.1. ^[23,28,29] A fuzzy number denoted as $\tilde{S}$, is a type of fuzzy set defined on the real line $\mathbb{R}$. adhering to the conditions of convexity and normality. It is characterized by a piecewise continuous membership function $\mu_{\tilde{S}}(u)$ that satisfies the following:

Definition 2.2. ^[23,28,29] A triangular fuzzy number (TFN) $\tilde{R}$ can be represented as a triplet

It falls within the classification of triangular-shaped fuzzy numbers, as shown in Figure 1. The membership function $\mu_{\tilde{R}}(u)$ of a TFN $\tilde{R}$ may be stated as follows:

Definition 2.3. ^[23,28,29] Consider the following TFN

Utilizing the $p$-cut approach, it may be transformed into a fuzzy interval representation as follows:

where $p \in [0, 1].$

Definition 2.4. ^[23,28,29] By applying the $(p, r)$-cut technique, the above Eq (2.2) may be expressed in a double parametric form (DPF) as follows:

Definition 2.5. ^[23,28,29] For a function $\tilde{f}(t; p, r)$, with fuzzy values, the fuzzy fractional Riemann-Liouville operator of order $\alpha$ (i.e. ${J_t}^{\alpha}$) is defined as follows:

where $t > 0, \ \alpha > 0.$

Definition 2.6. ^[23,28,29] For a fuzzy valued function $\tilde{f}(t; p, r)$ based on its DPF, the fuzzy fractional differential operator ${D_t}^{\alpha}$ in the Riemann-Liouville sense is interpreted as follows:

In Caputo's sense, the fuzzy fractional differential operator ${D_t}^{\alpha}$ can be interpreted as follows:

3.   Method description

To enhance the understanding of the FRDTM, we will start by looking over the local fractional Taylor's theorem. Moreover, this theorem has been extended to incorporate fractional derivatives, thus establishing a thorough foundation for applying the FRDTM.

3.1.   Local fractional Taylor's expansion ^[30,31]

Assuming that

for $k = 0, 1, \ldots, n$ and $0 < \alpha \leq 1$, we may represent the relationship in the following manner:

where

and

Definition 3.1. ^[30,31] For an analytic function $\psi(\xi, \tau)$, the fractional reduced differential transform denoted as $\psi_k(\xi)$ can be expressed as follows:

where $k = 0, 1, \ldots n.$

Definition 3.2. ^[30,31] The fractional inverse differential transform of $\psi_k(\xi)$ can be expressed as follows:

The following results can be drawn from Definitions 3.2 and 3.3:

For a specific time $\tau_0 = 0$, we have

3.2.   Properties of FRDTM

Let $f(\xi, \tau)$, $g(\xi, \tau)$, and $h(\xi, \tau)$ be three analytical functions such that

Here, ${R_D}^{-1}$ is the inverse reduced differential transform operator. In this instance, we might have the following:

(1) If

then

(2) If

then

where $a$ is a constant.

(3) If

then

(4) If

then

(5) If

then

3.3.   Methodology of FRDTM ^[30,31]

Here, we have discussed the step-by-step explanation of the FRDTM in order to solve a non-linear fractional order PDE.

Consider the following fractional PDE in the operator form:

where $m-1 < m\alpha\leq m, m \in \mathbb{N},$ and

● $L$ represents the linear operator;

● $N$ represents the nonlinear operator;

● $z(\xi, \tau)$ is the inhomogenous source term.

Let us consider the initial conditions (ICs) associated with the above equation, i.e., Eq (3.7) is as follows:

Step 1. Implementation of properties of FRDTM

By employing the properties discussed in Section 3.4, Eq (3.7) simplifies to the following:

where

$g_k(\xi)$ and $z_k(\xi)$ are the differential transformed form of $g(\xi, \tau)$ and $z(\xi, \tau)$, respectively.

Now, applying the FRDTM to the given initial conditions in (3.8), we obtain

Step 2. Calculation of $g_k(\xi)$

Utilizing Eqs (3.9) and (3.10), the value of $g_k(\xi)$ for $k = 0, 1, 2, \ldots$ can be iteratively obtained.

Step 3. Perform the inverse transformation

After determining the values of $g_k(\xi)$, let us perform the inverse differential transformation to obtain the approximate solution for $g(\xi, \tau)$. The inverse transformation allows us to reconstruct the solution from the transformed terms.

The approximate solution for the $n$ term can be expressed as follows:

Step 4. Final solution

The required solution of Eq (3.7) is given by

4.   Application of FRDTM to the fuzzified GeoKdV equation

By incorporating fuzziness, we may express the time-fractional GeoKdV equation (Eq (1.3)) in an operator form as follows:

with ICs

Applying the FRDTM to Eqs (4.1) and (4.2), we arrive at the following recurrence relation:

Now, using the DPF of Eq (4.3), we obtain the following:

Solving the above for $k = 0, 1, 2, \ldots$, we obtain the following:

Continuing with the process, the remaining components, including $\tilde{\eta}_4(\xi; p, r), \tilde{\eta}_5(\xi; p, r), \tilde{\eta}_6(\xi; p, r)$ and so on, may be evaluated. By combining these results, we derive the series solution of Eq (4.1) as follows:

5.   Results and conversations

In this work, we obtained approximate solutions for $\tilde{\eta}$ using up to a four-term solution. This was done by considering the initial condition in the context of a TFN. Calculations up to the determination of $\tilde{\eta}_4(\xi, \tau; p, r)$ have been taken due to the complexity associated with presenting more terms. For an improved accuracy, one can calculate an increased number of terms as needed. It is important to note that the obtained solutions can be influenced by adjusting the parameters $p \text{ and } r$ along with the fractional order $\alpha$. To demonstrate convergence, we produced plots and compared with existing solutions we have assumed the fractional order index with $\alpha = 1$. Moreover, some graphs are generated with different values of $\alpha$. Additionally, exploring different values of the parameters linked to the governing equation enables the effective computation of the desired solutions.

To execute our approach, we considered the following initial condition ^[5] in the context of a TFN:

where

with $a, d$, and $C$ being arbitrary constants, whereas

is a TFN.

Additionally, the DPF can be applied to the provided initial condition to simplify the calculations. This leads to the following representation:

where $p, r \in [0, 1]$.

By applying the discussed FRDTM approach, the governing equation (Eq (4.1)) with the provided IC in Eq (5.2) can be solved. The solution is expressed in terms of $\tilde{\eta}(\xi, \tau; p, r)$, thereby considering up to four terms. To obtain the solution and analyze its physical behavior, the following sets of data are considered:

Now, when $\alpha = 1$, the series solution

efficiently converges to $\tilde{\eta}(\xi, \tau; p, r).$

From a graphical point of view, Figure 2 represents the convergence of the obtained fuzzy solutions which are in a triangular-shaped form. It can be observed that as the number of terms increases, the fuzzy approximate solutions demonstrate clear convergence. Some fuzzy solutions are obtained by taking different $\xi$ with a fixed time followed by the set of data 1, and this can be depicted in Figure 3. To derive the lower and upper bound solutions from the obtained results, it is essential to substitute, $r = 0$ and $1$, respectively, using $p = 0$. Furthermore, setting $p = 1$ allows determination of the central/crisp value of the fuzzy solution. By performing these operations, we plotted a 2D figure (Figure 4) to illustrate the lower, middle, and upper solutions of the obtained results. Similarly, Figure 5 represents a 3D plot of the obtained fuzzy solution for $\tau = 0.5$ and $1$.

The validation phase involves the verification of the accuracy and the reliability of the obtained fuzzy solution by comparing it with established results. A detailed description is given below.

The fuzzy solution is inherently represented as a range of possible values due to the uncertainty modeled by the fuzzy set theory. To facilitate a meaningful comparison, we extract the central values from the fuzzy solution. These central values correspond to specific cases where the uncertainty is minimized, representing a crisp solution derived from the fuzzy framework. This extraction is crucial as it allows us to compare these specific results with existing classical solutions. In this regard, the central values obtained from the fuzzy solution are then compared to an existing solution using dataset 2. The detailed comparison is presented in Table 1, which allows for a direct numerical comparison at different time levels. The visual representation of this comparison for different time levels $\tau = 0.25, 0.5, 0.75,$ and $1$ can be observed in Figure 6, which illustrates a good convergence with ^[5].

Furthermore, to enhance understanding, Figure 7 presents the absolute error plot using $\tau = 0.75$, $\omega = 1.5$, and other parameters from dataset 2. The central values of the obtained fuzzy solutions are used to calculate the absolute errors at each term's solution. As seen in the zoomed section of this figure, increasing the number of terms reduces the error. A similar pattern is observed in Table 2, where $\tau = 0.5$ and $\omega = 0.5$ are used for different $\xi$ values. For the purpose of this discussion, the fractional order $\alpha$ is taken to be $1$. The comparison helps to assess how closely the fuzzy-based results align with traditional methods, thereby testing the accuracy and reliability of the fuzzy solution.

In addition, to investigate the behavior of the solution, a 3D plot is presented in Figure 8 along with 2D and 3D filled contour plots. These visualizations provide a clearer understanding of the solution's evolution over time and space, thus illustrating the dynamic nature of the model. Another key part of the validation phase involves analyzing how external factors, such as the Coriolis parameter, influence the solution. The plot showcased in Figure 9 demonstrates the effect of increasing the Coriolis parameter on the dynamic behavior of $\eta$. By following the parameter regimes outlined in dataset 3, the observation indicates a decrease in the time required for the free surface elevation to approach zero as the Coriolis parameter increases ^[5].

Moreover, in the context of the fractional-order GeoKdV equation, Figure 10 displays a plot of the physical behavior and relationship between the exact solution of the integer-order GeoKdV equation ^[5] and the crisp case of the obtained fuzzy solution for different values of $\alpha$. It can be observed that as the fractional order increases towards 1, the approximated special case fuzzy solution converges towards the integer-order exact solution, thus highlighting the consistency between the two approaches. Additionally, Figure 11 presents a 3D representation of the obtained solution for various values of the fractional order derivative $\alpha$ under the crisp scenario.

6.   Conclusions

This article explored the application of the FRDTM to solve a time-fractional GeoKdV equation under fuzzy initial conditions. Specifically, triangular fuzzy initial conditions are considered, and the study observed that the approximate fuzzy series solution exhibited strong convergence as the number of terms in the series increased, demonstrating the method's robustness and reliability. Under crisp conditions, for the fractional order $\alpha = 1$, the results produced by the FRDTM closely align with existing precise solutions ^[5], further affirming the accuracy of this approach. This consistency emphasizes the FRDTM's effectiveness in handling fuzzy and crisp cases. The graphical representations provide a visual insight into the solution's behavior, thus showing the impact of different parameter values. A significant advantage observed is that FRDTM avoids the complications of linearization or complex transformations, thus making the method computationally efficient.

Building on the findings of this study, future research could extend the FRDTM to other nonlinear fractional differential equations, particularly those that model complex physical phenomena in higher-dimensional systems. Additionally, incorporating more generalized forms of fuzzy uncertainty, such as trapezoidal or Gaussian fuzzy numbers, could offer a broader perspective on handling imprecision in real-world applications. Furthermore, the implementation of higher dimensional fuzzy numbers, viz., interval type-2 fuzzy numbers and generalised type-2 fuzzy numbers, can be one interesting topic to be considered in the study of various shallow water wave equations.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The first author would like to express his gratitude to the University Grants Commission (UGC), New Delhi, for providing fellowship for conducting research work.

Conflict of interest

The authors declare that they have no conflicts of interest.