Browsing by Author "Akbulut A."

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A Mathematical Analysis of a Model Involving an Integrable Equation for Wave Packet Envelope
(2022-01-01) Kaplan M.; Akbulut A.
This study investigates new optical solutions to a model with an integrable equation for wave packet envelopes. For this purpose, we have employed two reliable techniques involving the modified extended tanh function and the exponential rational function procedures. We have also given the 3D graphics of the obtained solutions.
A new computational approach to the fractional-order Liouville equation arising from mechanics of water waves and meteorological forecasts
(2022-01-01) Yue X.G.; Zhang Z.; Akbulut A.; Kaabar M.K.A.; Kaplan M.
The current analysis employs the improved F-expansion, modified extended tanh, exponential rational function, and (g′)−expansion procedures to find a divergent collection of the fractional Liouville equation's exact solutions in the context of beta-derivative. Also, we have given the graphical representations of the obtained results. These plots are useful in describing the dynamic characteristics of the solutions. The investigated equation is very essential in studying the mechanics of water waves and atmospheric predictability. Therefore, our techniques provide a great help in investigating various nonlinear models formulated in the contexts of fractional derivatives arising from oceanography and mathematical physics.
A new computational investigation to the new exact solutions of (3 + 1)-dimensional WKdV equations via two novel procedures arising in shallow water magnetohydrodynamics
(2022-01-01) Zhou M.; Akbulut A.; Kaplan M.; Kaabar M.K.A.; Yue X.G.
Various new exact solutions to (3 + 1)-dimensional Wazwaz-KdV equations are obtained in this work via two techniques: the modified Kudryashov procedure and modified simple equation method. The 3D plots, contour plots, and 2D plots of some obtained solutions are provided to describe the dynamic characteristics of the obtained solutions. Our employed techniques are very helpful in constructing new exact solutions to several nonlinear models encountered in ocean scientific phenomena arising in stratified flows, shallow water, plasma physics, and internal waves.
A novel exploration for traveling wave solutions to the integrable equation of wave packet envelope
(2022-06-01) Kaplan M.; Akbulut A.
In this paper, with the aid of symbolic computation, different types of traveling wave solutions to a model involving an integrable equation for wave packet envelope have been presented. The exp(−Φ(ξ))-expansion and modified Kudryashov procedures have been adopted and finally 3D and 2D graphics of the obtained solutions have been plotted. The obtained results are including dark, breather, bright, and periodic soliton solutions.
A novel investigation of exact solutions of the coupled nonlinear Schrodinger equations arising in ocean engineering, plasma waves, and nonlinear optics
(2022-01-01) Gu J.; Akbulut A.; Kaplan M.; Kaabar M.K.A.; Yue X.G.
We have analyzed the two coupled nonlinear Schrödinger equations (CNLSE) in the current work. This model has applications in high birefringence fibers. To generate different types of analytical solutions, including exponential, periodic, and soliton-type, we operated generalized Kudryashov, modified Kudryashov and exponential rational function procedures. Thanks to this implementation, we contributed to ocean engineering, plasma waves, nonlinear birefringence phenomena, and pulse compression features. In addition, in our article, we have included 2D and 3D graphics of solutions to help understand the physical meaning of solutions. The article highlights the method's ability to provide various solutions to various physical problems.
A unique computational investigation of the exact traveling wave solutions for the fractional-order Kaup-Boussinesq and generalized Hirota Satsuma coupled KdV systems arising from water waves and interaction of long waves
(2022-01-01) Wang X.; Yue X.G.; Kaabar M.K.A.; Akbulut A.; Kaplan M.
A novel technique, named auxiliary equation method, is applied in this research work for obtaining new traveling wave solutions for two interesting proposed systems: the Kaup-Boussinesq system and generalized Hirota-Satsuma coupled KdV system with beta time fractional derivative. Our solutions were obtained using MAPLE software. This technique shows a great potential to be applied in solving various nonlinear fractional differential equations arising from mathematical physics and ocean engineering. Since a standard equation has not been used as an auxiliary equation for this technique, different and novel solutions are obtained via this technique.
An effective computational approach and sensitivity analysis to pseudo-parabolic-type equations
(2021-01-01) Kaplan M.; Butt A.R.; Thabet H.; Akbulut A.; Raza N.; Kumar D.
The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly.
Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative
(2018-05-04) Kaplan M.; Akbulut A.
The current paper devoted on two different methods to find the exact solutions with various forms including hyperbolic, trigonometric, rational and exponential functions of fractional differential equations systems with conformable farctional derivative. We have employed the modified simple equation and exp(Φ(ξ)) method here for the approximate long water wave equation. We have adopted here the fractional complex transform accompanied by properties of conformable fractional calculus for reduction of fractional partial differential equation systems to ordinary differential equation systems.
Auxiliary equation method for time-fractional differential equations with conformable derivative
(2018-02-01) Akbulut A.; Kaplan M.
In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.
Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation
(Springer, 2023) Jannat N.; Raza N.; Kaplan M.; Akbulut A.
In this investigation, we address a particular variant of the Korteweg–de Vries (KdV) equation, specifically focusing on the (2+1)-dimensional KdV equation. The equation can model various physical phenomena in different fields, including fluid dynamics, plasma physics, nonlinear optics, and other areas where coupled wave interactions are important. To commence, we establish the Auto-Bäcklund and Cole–Hopf transformations for the given model, resulting in the derivation of numerous soliton-like solutions characterized by hyperbolic, trigonometric, and exponential function waves. Furthermore, we effectively elucidate the behavior of lump, lump–kink, breather, two-wave, and three-wave solutions using the Hirota bilinear technique. Extensive numerical simulations employing 3-D profiles are conducted with meticulous consideration of pertinent parameter values, providing additional insights into the distinctive traits of the obtained solutions. Moreover, employing the extended transformed rational function method grounded in the bilinear form of the underlying equation, we uncover complexiton solutions. These solutions are depicted using 3-D and 2-D visualizations to portray their dynamics. Our findings reveal that the approach adopted to derive analytical solutions for nonlinear partial differential equations proves to be both efficient and potent. The combination of numerical simulations and visual representations enhances our understanding of these solutions, ultimately affirming the effectiveness and robustness of the employed methodology in tackling nonlinear partial differential equations.
Extraction of Exact Solutions of Higher Order Sasa-Satsuma Equation in the Sense of Beta Derivative
(2022-11-01) Fadhal E.; Akbulut A.; Kaplan M.; Awadalla M.; Abuasbeh K.
Nearly every area of mathematics, natural, social, and engineering now includes research into finding exact answers to nonlinear fractional differential equations (NFDES). In order to discover the exact solutions to the higher order Sasa-Satsuma equation in the sense of the beta derivative, the paper will discuss the modified simple equation (MSE) and exponential rational function (ERF) approaches. In general, symmetry and travelling wave solutions of the Sasa-Satsuma equation have a common correlation with each other, thus we reduce equations from wave transformations to ordinary differential equations with the help of Lie symmetries. Actually, we can say that wave moves are symmetrical. The considered procedures are effective, accurate, simple, and straightforward to compute. In order to highlight the physical characteristics of the solutions, we also provide 2D and 3D plots of the results.
New conservation laws and exact solutions of the special case of the fifth-order KdV equation
(2022-08-01) Akbulut A.; Kaplan M.; Kaabar M.K.A.
The current study deals with the Kaup-Kupershmidt (KK) equation to construct formal Lagrangian, conservation laws, and exact solutions. KK is basically a special case of the 5th-order KdV equation. The conservation laws obtained by using the conservation theorem are trivial conservation laws. In addition, exact solutions are found via the modified simple equation (MSE) method. For a suitable value of solutions, the 3D surfaces have been plotted using MAPLE. These plots giving novel exact solutions are made to reveal important wave characteristics. Our obtained results in this work concerning our investigated equation are essential to explain many physical and oceanographic applications involving ocean gravity waves and many other related phenomena.
New exact solutions of the Mikhailov-Novikov-Wang equation via three novel techniques
(2023-01-01) Akbulut A.; Kaplan M.; Kaabar M.K.A.
The current work aims to present abundant families of the exact solutions of Mikhailov-Novikov-Wang equation via three different techniques. The adopted methods are generalized Kudryashov method (GKM), exponential rational function method (ERFM), and modified extended tanh-function method (METFM). Some plots of some presented new solutions are represented to exhibit wave characteristics. All results in this work are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in ocean engineering and physics. This equation provides new insights to understand the relationship between the integrability and water waves’ phenomena.
New Solitary Wave Patterns of the Fokas System in Fiber Optics
(2023-04-01) Kaplan M.; Akbulut A.; Alqahtani R.T.
The Fokas system, which models wave dynamics using a single model of fiber optics, is the design under discussion in this study. Different types of solitary wave solutions are obtained by utilizing generalized Kudryashov (GKP) and modified Kudryashov procedures (MKP). These novel concepts make use of symbolic computations to come up with a dynamic and powerful mathematical approach for dealing with a variety of nonlinear wave situations. The results obtained in this paper are original and have the potential to be useful in mathematical physics.
On the conservation laws and exact solutions to the (3+1)-dimensional modified kdv-zakharov-kuznetsov equation
(2021-05-01) Akbulut A.; Almusawa H.; Kaplan M.; Osman M.S.
In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′ /G, 1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.
Research on sensitivity analysis and traveling wave solutions of the (4 + 1)-dimensional nonlinear Fokas equation via three different techniques
(2022-01-01) Kaplan M.; Akbulut A.; Raza N.
In the current manuscript, (4 + 1) dimensional Fokas nonlinear equation is considered to obtain traveling wave solutions. Three renowned analytical techniques, namely the generalized Kudryashov method (GKM), the modified extended tanh technique, exponential rational function method (ERFM) are applied to analyze the considered model. Distinct structures of solutions are successfully obtained. The graphical representation of the acquired results is displayed to demonstrate the behavior of dynamics of the nonlinear Fokas equation. Finally, the proposed equation is subjected to a sensitivity analysis.
Solitary wave solutions of coupled nerve fibers model based on two analytical techniques
(2023-07-01) Razzaq W.; Akbulut A.; Zafar A.; Kaplan M.; Raheel M.
This paper focuses on a few innovative solutions to the coupled nerve fibers model. The constructed solutions can be used to expose this model in a noticeable way. The verified solutions are including the trigonometric, exponential, and hyperbolic functions. Utilizing the Mathematica tool, the results are verified. We employed two approaches, named as modified extended tanh expansion and modified (G′G2) -expansion methods, to obtain the results. We gave the 2-D and 3-D plots of the obtained results. The obtained results are dissimilar from previous results in the literature. The used methods are powerful and effective. The obtained results have potential to be conducive for the model’s future development.
The analysis of conservation laws, symmetries and solitary wave solutions of Burgers-Fisher equation
(2021-09-10) Akbulut A.; Kaplan M.; Kumar D.; Taşcan F.
In this paper, the conservation laws, significant symmetries' application, and traveling wave solutions are obtained for Burger-Fisher equation (BFE). Conservation laws have a great importance for partial and fractional differential equations and their solutions, especially in physics implementations. The conservation theorem and partial Noether approach are implemented for conservation laws for this equation, and the extended sinh-Gordon expansion method (esGEM) is presented for new solitary wave solutions. All obtained conservation laws are trivial conservation laws. The new and comprehensive solitary wave solutions of the equation by the esGEM are also obtained.
The analysis of the soliton-type solutions of conformable equations by using generalized Kudryashov method
(2021-09-01) Kaplan M.; Akbulut A.
In this study, we applied the generalized Kudryashov method to two different conformable fractional differential equations and one system namely Burgers’ equation with conformable derivative, Wu-Zhang system with conformable derivative, and the conformable sinh-Gordon equation. Obtained solutions are soliton-type solutions. All obtained solutions have been verified and considered correct by placing them in the original equation with the help of the Maple software program. Also, we have plotted three-dimensional (3D) graphs of some of these solutions with the help of Maple.