Browsing by Author "Kaplan M."

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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 new exploration of some explicit soliton solutions of q-deformed Sinh-Gordon equation utilizing two novel techniques
(2023-03-01) Raza N.; Butt A.R.; Arshed S.; Kaplan M.
In the present work, the q-deformed Sinh-Gordon equation considered by performing new extended generalized Kudryashov method and improved tan(Φ2)-expansion method. Thanks to these methods, we can effectively obtain rational, hyperbolic and trigonometric solutions by specially choosing the parameters for the existence of solutions. The proposed equation expands the possibilities for modeling physical systems in which symmetry is broken. The obtained solutions are graphically illustrated.
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.
A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water
(2020-06-16) Kumar D.; Kaplan M.; Haque M.R.; Osman M.S.; Baleanu D.
For different nonlinear time-conformable derivative models, a versatile built-in gadget, namely the generalized exp(−φ(ξ))-expansion (GEE) method, is devoted to retrieving different categories of new explicit solutions. These models include the time-fractional approximate long-wave equations, the time-fractional variant-Boussinesq equations, and the time-fractional Wu-Zhang system of equations. The GEE technique is investigated with the help of fractional complex transform and conformable derivative. As a result, we found four types of exact solutions involving hyperbolic function, periodic function, rational functional, and exponential function solutions. The physical significance of the explored solutions depends on the choice of arbitrary parameter values. Finally, we conclude that the GEE method is more effective in establishing the explicit new exact solutions than the exp(−φ(ξ))-expansion method.
Abundant soliton-type solutions to the new generalized KdV equation via auto-Bäcklund transformations and extended transformed rational function technique
(2022-08-01) Jannat N.; Kaplan M.; Raza N.
This study investigates new soliton-type solutions to the new generalized KdV (ngKdv) equation. For this purpose, the homogeneous balance method is used to create Auto-Bäcklund transformations of the regarded equation and with the help of the transformations, abundant exact and explicit solutions have been found. We found complexiton solutions to the dealt equation by using the extended transformed rational function technique. We have also given the 3D graphics of the obtained solutions.
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.
An exploration of novel soliton solutions for propagation of pulses in an optical fiber
(2022-07-01) Raza N.; Arshed S.; Kaplan M.; Butt A.R.
In this article, the propagation of pulses in optical fiber has been studied by considering the nonlinear partial differential equation (NPDE). The proposed model is investigated using two analytical techniques namely the Sine-Gordon expansion (SGE) procedure and the modified auxiliary equation (MAE) method. The trigonometric function, hyperbolic function, and rational function solutions have been extracted from the proposed methods. The employed procedures are compatible in obtaining traveling wave solutions. Moreover, the obtained results are assisted with 3D graphs to demonstrate the physical significance and dynamical behaviors by using different parameter values.
Application of the modified Kudryashov method to the generalized Schrödinger–Boussinesq equations
(2018-09-01) Kumar D.; Kaplan M.
In the paper, the modified Kudryashov method is applied to find new exact solutions for the generalized Schrödinger–Boussinesq equation with the help of symbolic computation package Maple through the complex transform. The obtained solutions have been checked by substituting back into its corresponding equation with the aid of Maple package program.
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.
Applications of two reliable methods for solving a nonlinear conformable time-fractional equation
(2017-09-01) Kaplan M.
The current work presents analytical solutions of a nonlinear conformable time-fractional equation by using two different techniques. These are the modified simple equation method and the exponential rational function method. Based on the conformable fractional derivative and traveling wave transformation, the fractional partial differential equation is turned into the nonlinear non-fractional ordinary differential equation. Therefore, we implement the algorithms to this nonlinear non-fractional ordinary differential equation. To the best of our knowledge, the exact solutions obtained in this paper might be very useful in various areas of applied mathematics in interpreting some physical phenomena.
Applications of two reliable methods for solving a nonlinear conformable time-fractional equation
(2017-09-01) Kaplan M.; Kaplan, M
The current work presents analytical solutions of a nonlinear conformable time-fractional equation by using two different techniques. These are the modified simple equation method and the exponential rational function method. Based on the conformable fractional derivative and traveling wave transformation, the fractional partial differential equation is turned into the nonlinear non-fractional ordinary differential equation. Therefore, we implement the algorithms to this nonlinear non-fractional ordinary differential equation. To the best of our knowledge, the exact solutions obtained in this paper might be very useful in various areas of applied mathematics in interpreting some physical phenomena.
Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation
(2018-01-01) Kaplan M.; Ozer M.
In the present work, according to the concept of extended homogeneous balance method and with help of Maple, we get auto-Bäcklund transformations for a (2 + 1)-dimensional nonlinear evolution equation. Subsequently, by using these auto-Bäcklund transformation, exact explicit solutions of this equation are obtained.
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.
Complexiton and resonant multi-solitons of a (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation
(2022-02-01) Raza N.; Kaplan M.; Javid A.; Inc M.
This work is concerned with the extraction of auto-Bäcklund transformations of (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation in accordance with the extended homogeneous balance (HB) method incorporating Maple. Subsequently, these transformations are used to study analytic explicit solutions of this equation. Also, based on the Hirota bilinear form and the extended transformed rational function method, complexiton solutions have been found of the (4 + 1)-dimensional BLMP equation through a direct symbolic computation with Maple. This method is the improved form of the transformed rational function method. The obtained complexiton solutions, including trigonometric and hyperbolic trigonometric solutions, have been verified utilizing Hirota bilinear forms. Some of the solutions are depicted in 3D graphs to understand physical properties. The reported results are new and have applications in several scientific fields such as incompressible fluids.
Construction of complexiton-type solutions using bilinear form of Hirota-type
(2023-02-01) Kaplan M.; Raza N.
In this paper, based on the Hirota bilinear form and the extended transformed rational function method, complexiton solutions have been found of the Hirota-Satsuma-Ito (HSI) equation and generalized Calogero-Bogoyavlenskii-Schiff equation through a direct symbolic computation with Maple. This method is the improved form of the transformed rational function method. The obtained complexiton solutions, includes trigonometric and hyperbolic trigonometric solutions, have verified utilizing Hirota bilinear forms. Also, a graphical representation of the obtained solutions is given.
Different Types of Progressive Wave Solutions via the 2D-Chiral Nonlinear Schrödinger Equation
(2020-07-07) Osman M.S.; Baleanu D.; Tariq K.U.H.; Kaplan M.; Younis M.; Rizvi S.T.R.
A versatile integration tool, namely the protracted (or extended) Fan sub-equation (PFS-E) technique, is devoted to retrieving a variety of solutions for different models in many fields of the sciences. This essay presents the dynamics of progressive wave solutions via the 2D-chiral nonlinear Schrödinger (2D-CNLS) equation. The solutions acquired comprise dark optical solitons, periodic solitons, singular dark (bright) solitons, and singular periodic solutions. By comparing the results gained in this work with other literature, it can be noticed that the PFS-E method is an useful technique for finding solutions to other similar problems. Furthermore, some new types of solutions are revealed that will help us better understand the dynamic behaviors of the 2D-CNLS model.