Abstract
The fractional-stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) is considered here. To attain new hyperbolic, elliptic, rational, and trigonometric stochastic-fractional solutions, we use two various methods such as the sine-cosine and the Jacobi elliptic function methods. The solutions acquired are important in understanding some interesting physical phenomena due to the significance of the Radhakrishnan-Kundu-Lakshmanan equation in designing the propagation of solitons through an optical fiber. Furthermore, we graph some of the obtained solutions in 3D to display the influence of fractional derivative and multiplicative noise on these solutions. Finally, we show that when the order of fractional derivative decreases, the surface shrinks, while the multiplicative noise stabilizes the solutions of FSRKLE a round zero.
1. Introduction
Partial differential equations (PDEs) are found in several areas of applied science, including quantum mechanics, plasma physics, nonlinear optics, surface of water waves, hydrodynamics, molecular biology, fluid dynamics, elastic media, and biology. Obtaining solutions of PDEs is crucial for understanding physical phenomena. Therefore, many effective methods, including exp-function method [1], auxiliary equation [2], Darboux transformation [3], sine-cosine [4], Jacobi elliptic function [5], -expansion [6], sine-Gordon expansion [7], -expansion [8–10], generalized Kudryashov [11], perturbation [12–14], extended trial equation [15, 16], Jacobi elliptic function [17, 18], Riccati equation [19], tanh-coth [20], homotopy perturbation [21], modified decomposition [22], and F-expansion [23], have been constructed to attain exact solutions of PDEs.
Researchers and scientists have focused their attention over the last two decades on fractional differential equations (FDEs) that have been found to be more precise than classical differential equations in explaining complex physical phenomena in the real life. The idea of fractional derivative has been used to define various phenomena including fluid dynamics porous medium, signal processing, viscoelastic materials, ocean wave, electromagnetism, photonic, chaotic systems, wave propagation, optical fiber communication, plasma physics, and nuclear physics. Recently, Atangana and Goufo [24] have suggested the new conformable fractional derivative called beta-derivative. From this point, let us define the Atangana conformable derivative (ACD) for the function of order as follows:
The ACD satisfies the following properties for any constant and : (1) , (2) , (3) , (4) If then .
Stochastic partial differential equations (SPDEs), on the other hand, have been widely addressed as theoretical equations for spatial-temporal physical, chemical, and biological systems related to random perturbations. The significance of involving stochastic impacts in complex system modeling has been emphasized. For example, there is gaining awareness in using SPDEs to mathematically model complex phenomena in information systems, condensed matter physics, biology, climate systems, electrical and mechanical engineering, materials sciences, and finance.
It is worth noting that two forms widely utilized for stochastic integral are Itô and Stratonovich [25]. Modeling problems primarily determine which form is acceptable; however, once that form is selected, an equivalent equation of the other form can be produced using the same solutions. As a result, the following relationship can be utilized to switch between Itô (written as ) and Stratonovich (written as ): where is a Brownian motion (BM).
To satisfy a higher degree of quality agreement, the following stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) [26–28] is considered: where , for are constants and is the noise strength and is multiplicative Brownian motion in the Stratonovich sense. Recently, many investigators have created exact solutions of FSRKLE (3), with and , using different methods such as extended simple equation method [29], first integral method [30], sine-cosine method [31], Lie group analysis [32], and trial equation method [33].
The motivation of this article is to attain the exact solutions for FSRKLE (3). We use two separate approaches, the sine-cosine and the Jacobi elliptic function methods, to provide a wide range of solutions, including hyperbolic, trigonometric, rational, and elliptic functions. The acquired solutions are helpful for understanding several fascinating scientific events because of the significance of the RKL in describing the propagation of solitons through an optical fiber. Also, by creating 3D representations of the obtained FSRKLE (3) solutions, we examine the effect of BM on these solutions.
The article is in the following format: in Section 2, we determine the wave equation of the FSRKLE (3) by applying a suitable wave transformation. To develop the analytical solutions for the FSRKLE in Section 3, we use two different approaches (3). In Section 4, the impact of the BM on the solutions obtained is examined. The final section of the document is the conclusion.
2. Wave Equation for FSRKLE
To obtain the wave equation of the FSRKLE (3), the following transformation is utilized: where the function is deterministic and and are unknown constants. Putting Equation (4) into Equation (3) and utilizing where is the Itô correction term, and we get for imaginary part and for real part
Taking expectation on both sides for Equations (7) and (8) and using we have
Integrating Equation (10), we get
We obtain the next constraint conditions where the same function achieves both Equations (11) and (12): whenever
Plugging Equation (14) into Equation (11), we have the wave equation as follows: where
3. The Exact Solutions of the FSRKLE
We employ two various methods such as the Jacobi elliptic function [18] and sine-cosine [4], to determine the exact solutions to Equation (16). As a consequence, we can obtain the solutions of the FSRKLE (3).
3.1. Jacobi Elliptic Function Method
We suppose the solutions of Equation (16) has the type where for is Jacobi elliptic sine function and , and are undefined constants. Differentiate Equation (19) twice, we get
Putting Equations (19) and (20) into Equation (16), we obtain
Equating each coefficient of to zero, we have for ,
The outcomes of solving the previous equations are
As a result, using (19), the solution of Equation (16) is
Hence, the exact solution of the FSRKLE (3) is for and If , then solution (25) tends to
Analogously, we can replace in (19) with and to obtain the following solutions of Equation (16):
Thus, the exact solutions of the FSRKLE (3) are as follows: for , and for , respectively. If , then the Equations (28) and (29) tends to for ,
3.2. Sine-Cosine Method
Suppose the solution of Equation (16) takes the form where
Setting Equation (31) into Equation (16), we get rewriting the above equation
Balancing the term of in Equation (34), we obtain
Plugging Equation (35) into Equation (34),
We get by setting each coefficient of and equal to zero
By solving Equations (37) and (38), we get
Hence, the solution of Equation (16) is
Depending on the sign of and , there are numerous cases:
Case 1. If and then FSRKLE (3) has the solutions or
Case 2. If and then the analytical solutions of FSRKLE (3) have the form or
Case 3. If and then the solutions of FSRKLE (3) are or
Case 4. If and then FSRKLE (3) has the solutions or where are defined in (17).
Remark 1. Setting and in Equations (41), (42), and (45), we get the identical solutions as asserted in [31].
4. Effect of BM and Fractional Derivative on the Solutions
Here, the impact of BM and the fractional derivative on the exact solutions of the FSRKLE (3) is described. Fix the constants , and . Hence, , and . Now, we present some diagrams for various value of (intensity of noise) and for , . We apply the MATLAB to simulate the solutions of Equation (3).
Firstly, the impact of noise: in Figure 1, when we note that the surface fluctuates.
In Figure 2, if the noise appeared, then after small transit behaviors, the surface gets more planer when the intensity of noise increases as follows:
Secondly, the impact of fractional derivative: in Figures 3 and 4, if , we can observe that as increases, the surface extends:
5. Conclusions
In this paper, we obtained the exact solutions of the fractional-stochastic Radhakrishnan-Kundu-Lakshmanan Equation (3). To obtain rational, elliptic, trigonometric, and hyperbolic stochastic solutions, we used two different methods: the Jacobi elliptic function and the sine-cosine. Because of the priority of the FSRKLE in fluid dynamics and plasma physics, the results produced are useful for understanding some exciting physical phenomena. Finally, we plotted the obtained solutions using MATLAB tools to provide a number 3D diagram to demonstrate the impact of fractional derivative and multiplicative noise on these solutions. In future work, we can consider the FSRKLE (3) with additive noise.
Data Availability
All data are available in this paper.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.