Hyperbolic Function Solutions Research Articles

New Exact Solutions of Some Important Nonlinear Fractional Partial Differential Equations with Beta Derivative

In this work, the F-expansion method is used to find exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrödinger equation with beta derivative. One of the most efficient and significant methods for obtaining new exact solutions to nonlinear equations is this method. With the aid of Maple, more exact solutions defined by the Jacobi elliptic function are obtained. Hyperbolic function solutions and some exact solutions expressed by trigonometric functions are gained in the case of m modulus 1 and 0 limits of the Jacobi elliptic function.

A variety of closed-form solutions, Painlevé analysis, and solitary wave profiles for modified KdV–Zakharov–Kuznetsov equation in (3+1)-dimensions

The major goal of the article is to use the generalized exponential rational function method to seek abundant closed-form solutions and dynamics of solitary wave profiles to the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov (MKdV–ZK) equation. We also apply the Painlevé analysis with Kruskal simplification to investigate the integrability of the said equation. The (3+1)-dimensional MKdV–ZK model is a significant model with great advantages in various disciplines of physics, such as nonlinear optics, fluid dynamics, plasma physics, mathematical physics, and quantum mechanics. By taking advantage of the generalized exponential rational function technique, we extracted various soliton solutions, including exponential form, trigonometric form, and hyperbolic form function solutions. From both mathematical and physical points of view, it is important to obtain bright–dark soliton forms and other solitonic wave profiles. Furthermore, the results have been graphically revealed using 3D, 2D, and contour plots to demonstrate the underlying dynamics of the interactive waves. The obtained findings indicate the method’s efficacy and dependability, allowing it to be widely applied in a variety of advanced nonlinear evolution equations.

Modified Exp-Function Method to Find Exact Solutions of Ionic Currents along Microtubules

A number of solitary wave solutions for microtubules (MTs) are observed in this article by using the modified exp-function approach. We tackle the problem by treating the results as nonlinear RLC transmission lines, and then finding exact solutions to Nonlinear Evolution Equation (NLEE) containing parameters of particular importance in biophysics and nanobiosciences. For this equation, we find trigonometric, hyperbolic, rational, and exponential function solutions, as well as soliton-like pulse solutions. A comparison with other approach indicates the legitimacy of the approach we devised as well as the fact that our method offers extra solutions. Finally, we plot 2D, 3D and contour visualizations of the exact results that we observed using our approach using appropriate parameter values with the help of software Mathematica 10.

New Classifications of All Single Traveling Wave Solutions of the Fractional Approximate Long Water Wave Equations by Using the Complete Discrimination System

First, our work is to transform the space-time fractional approximate long water wave equations into nonlinear ordinary differential equations via the traveling wave transformation in the sense of conformable fractional derivative. Second, we simplify the nonlinear ordinary differential equations into an ordinary differential equation with only one variable by integration and some transformations. Finally, we can further get all single traveling wave solutions of the space-time fractional approximate long water wave equations by the complete discrimination system for the four-order polynomial method; these solutions include the hyperbolic function solutions, rational function solutions, and implicit solutions.

Exact solutions of the different dimensional CBS equations in mathematical physics

The focus of this article is to find the exact traveling wave solutions to Calogero–Bogoyavlenskii–Schiff (CBS) equation by employing the exp−Φζ expansion method. The traveling wave solutions are expressed by the hyperbolic function solutions, trigonometric function solutions and rational function solutions. 3D and 2D charts are drawn from the obtained solutions by Mathematica software and discussed the graphical behavior of solutions. Finally, an effective mathematical tools for solving nonlinear evolution equations in mathematical physics are provided by the proposed method. This method is more common and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations.

Abundant closed-form wave solutions to the simplified modified Camassa-Holm equation

The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been applied to the SMCH equation. Base on the method, we have obtained some novel analytical solutions such as hyperbolic, trigonometric, exponential, and rational function solutions of the SMCH equation. For appropriate values of parameters, three dimensional (3D) and two dimensional (2D) graphs are designed by Mathematica. The stability of the model is also discussed in this manuscript. The dynamic and physical behaviors of the solutions derived from the SMCH equation have been extensively discussed by these plots. All our solutions are indispensable for understanding the nonlinear phenomena of dispersive waves that are important in ocean engineering and science. In addition, our results are essential to clarify the various oceanographic applications containing ocean gravity waves, offshore rig in water, energy associated with a moving ocean wave and numerous other related phenomena. Finally, the obtained solutions are helpful for studying wave interactions in many new structures and high-dimensional models.

On the structure of unsteady korteweg-de vries model arising in shallow water

In this article, the modified Kudryashov approach and (1G′)-expansion approach are utilized to extract some new analytical solutions to the unsteady Korteweg-de Vries equation. In nonlinear sciences, this equation is very important. As a result, a variety of new exact solutions are acquired for the aforementioned nonlinear model. Moreover, the two dimensional, three dimensional, and contour shapes are visualized with the aid of latest scientific tools. We found four forms of explicit solutions such as the hyperbolic, trigonometric, exponential, and rational function solutions. It has been demonstrated that the proposed techniques are highly efficient and practical for the aforementioned issues, as well as additional NLEEs that appear in engineering domains and mathematical physics.

New solitary wave solutions for fractional Jaulent–Miodek hierarchy equation

In this paper, the generalized rational function procedure was used to produce new solitary solutions to the fractional Jaulent–Miodek hierarchy equation. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials. The equation is converted into a fourth-order partial differential equation using a transformation. In order, we find some solitary solutions such as soliton, rational and hyperbolic function solutions of Jaulent–Miodek hierarchy equation by the help of generalized exponential rational function procedure. Then, for some parameters, we draw three-dimensional graphics of real values of some exact solutions that we acquired by using this procedure.

Traveling wave solutions of conformable time-fractional Klien-Fock-Gordon equation by the improved [formula omitted]-expansion method

This paper considers the improved tan(Ψ(ζ)/2)-expansion method to retrieve the traveling wave solutions of the time-fractional Klien-Fock-Gordon equation. The quadratic case has been discussed for the proposed model. The conformable derivative has been applied on the proposed model for investigating the fractional effects. As a result, abundant traveling wave solutions are found. The method contributes four kinds of solutions which are exponential, hyperbolic, trigonometric and rational function solutions for the aforementioned model. The obtained results are also explained graphically for some particular choices of fractional order α,0<α<1. The fractional effects on solutions have been illustrated through 3D plots and line graphs.

The classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation

The main purpose of this paper is to study the single traveling wave solutions of the fractional coupled nonlinear Schrödinger equation. By using the complete discriminant system method and computer algebra with symbolic computation, a series of new single traveling wave solutions are obtained, which include trigonometric function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, solitary wave solutions and rational function solutions. As you can see, we give all the classification of single traveling wave solutions for the fractional coupled nonlinear Schrödinger equation. The obtained results substantially improve or complement the corresponding conditions in the literature (Esen and Sulaiman in Optik 167:150-156, 2018), (Eslami in Appl. Math. Comput. 258:141-148, 2016), (Han et al. in Phys. Lett. 395:127217, 2021). Finally, in order to further explain the propagation of the fractional coupled nonlinear Schrödinger equation in nonlinear optics, two-dimensional and three-dimensional graphs are drawn.

Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters

The exact solutions of the space-time fractional Fokas-Lenells equation with parameters in nonlinear optics are obtained by using the complete discrimination system for polynomial method, including rational function solutions, periodic solutions, solitary wave solutions, Jacobi elliptic function solutions and hyperbolic function solutions. The relevant images of the exact solution are drawn, and the influence of parameters on the structure of the solution is analyzed.

Optical solitons and single traveling wave solutions of Biswas-Arshed equation in birefringent fibers with the beta-time derivative

&lt;abstract&gt;&lt;p&gt;This article describes the construction of optical solitons and single traveling wave solutions of Biswas-Arshed equation with the beta time derivative. By using the polynomial complete discriminant system method, a series of traveling wave solutions are constructed, including the rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and inverse trigonometric function solutions. The conclusions of this paper comprise some new and different solutions that cannot be found in existing literature. Using the mathematic software Maple, the 3D and 2D graphs of the obtained traveling wave solutions were also developed. It is worth noting that these traveling wave solutions may motivate us to explore new phenomena which may be appear in optical fiber propagation theory.&lt;/p&gt;&lt;/abstract&gt;

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable-coefficients

&lt;abstract&gt;&lt;p&gt;Based on the variable separation method, the Kadomtsev-Petviashvili equation is transformed into a system of equations, in which one is a fractional ordinary differential equation with respect to time variable $ t $, and the other is an integer order variable coefficients partial differential equation with respect to spatial variables $ x, y $. Assuming that the coefficients of the obtained partial differential equation satisfy certain conditions, the equation is further reduced. The extended homogeneous balance method is used to find the exact solutions of the reduced equation. According to the solutions of some special fractional ordinary differential equations, we obtain some hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable coefficients.&lt;/p&gt;&lt;/abstract&gt;

New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.

Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation

&lt;abstract&gt;&lt;p&gt;In this paper, our main purpose is to study the soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. New soliton solutions have been obtained by the extended $ (G'/G) $-expansion method, first integral method and complete discrimination system for the polynomial method, respectively. The solutions we obtained mainly include hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. Moreover, we draw its three-dimensional graph.&lt;/p&gt;&lt;/abstract&gt;

A novel integration approach to study the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index

Modelling the movement of waves and the propagation of waves is extremely important in the field of fluid dynamics. All types of waves phenomena are very important in real-life implementations. We use an efficient solver and the improved F-expansion approach to obtain precise traveling wave solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index. All acquired solutions are in the form of trigonometric, hyperbolic, and rational function solutions, and the resulting solutions are applicable to real-world problems in fluid dynamics, optical fibers, plasma physics, and so on. Furthermore, we depicted the three-dimensional and contour graphs to illustrate the behavior of some obtained solutions.

Dynamics of optical pulses in fiber optics

In this paper, we pay attention to the nonlinear dynamical behavior of ultra-short pulses in optical fiber. The new Hamiltonian amplitude equation is used as a governing model to analyze the pulses. We secure the ultra-short pulses in the forms of bright, dark, singular, combo and complex soliton solutions by the utilization of three of sound computational integration techniques that are protracted (or extended) Fan-sub equation method (PFSEM), the generalized exponential rational function method (GERFM), extended Sinh-Gordon equation expansion method (ShGEEM). Moreover, Jacobi’s elliptic, trigonometric, and hyperbolic functions solutions are also discussed as well as the constraint conditions of the achieved solutions are also presented. In addition, we discuss the different wave structures by the assistance of logarithmic transformation. The findings demonstrate that the examined equation theoretically contains a large number of soliton solution structures. By selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in 3D and 2D profiles. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex phenomena. The resulting solutions are novel, intriguing, and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including nonlinear optics. Our new results have been compared to these in the literature.

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

Deeper properties of the nonlinear Phi-four and Gross-Pitaevskii equations arising mathematical physics

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program.