Trigonometric Function Solutions Research Articles

Regarding New Traveling Wave Solutions for the Mathematical Model Arising in Telecommunications

This research paper focuses on the application of the tanh function method to find the soliton solutions of the (2+1)-dimensional nonlinear electrical transmission line model. Materials used to form a transmitting line are very important to transmit electric charge. In this sense, we find some new voltage behaviors such as dark, trigonometric, and complex function solutions. Choosing some suitable values of parameters, we present some various surfaces of results obtained in this paper. These results play an important role in telecommunications lines used to stand for wave propagations.

Various Exact Solutions for the Conformable Time-Fractional Generalized Fitzhugh–Nagumo Equation with Time-Dependent Coefficients

In this paper, the subequation method and the sine-cosine method are improved to give a set of traveling wave solutions for the time-fractional generalized Fitzhugh–Nagumo equation with time-dependent coefficients involving the conformable fractional derivative. Various structures of solutions such as the hyperbolic function solutions, the trigonometric function solutions, and the rational solutions are constructed. These solutions may be useful to describe several physical applications. The results show that these methods are shown to be affective and easy to apply for this type of nonlinear fractional partial differential equations (NFPDEs) with time-dependent coefficients.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

A variety of exact solutions for fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain in the semi classical limit

This paper investigates exact voyaging (2 + 1) dimensional Heisenberg ferromagnetic spin chain solutions with conformable fractional derivatives, an important family of nonlinear equations from Schrödinger (NLSE) for the construction of hyperbolic, trigonometric and complex function solutions. The detailed rational sine-cosine system and rational sinh-cosh system were used to locate dim, special and periodic wave solutions successfully. These findings suggest that the proposed approaches may be useful to investigate a range of solutions inside a repository of applied sciences and engineering, with success, quality, and trust. In addition, graphical representations and physical expresses of such solutions are represented by a set of the required values of the parameters involved. The methods are essentially adequate and can be extended to different dynamic models that create the nonlinear processes in today’s research.

Investigation of shallow water waves and solitary waves to the conformable 3D-WBBM model by an analytical method

In this article, we elucidate the dynamical behavior of exact solitary waves to the conformable 3D Wazwaz-Benjamin- Bona-Mahony (3D-WBBM) equation emerging in shallow water waves. A variety of nonlinear dynamical solitary wave structures are extracted in different shapes like hyperbolic, trigonometric, and exponential function solutions including some special known solitary wave solution like bell shaped, shock, singular and multiple soliton by an analytical mathematical tool namely the generalized exponential rational function method (GERFM). Besides, we also secure singular periodic wave solutions with unknown parameters. All the secured solutions are verified by substituting back to the original equation through soft computation Mathematica. The outcomes show that the governing model theoretically possesses extremely rich structures of exact solitary wave solutions. The physical characterization of some reported results are figured out graphically in 3D, 2D and their corresponding contour profiles by selecting appropriate values of parameters.

Thirty traveling wave solutions to the systems of ion sound and Langmuir waves

Nine kinds of general solutions of certain popular first-order ordinary differential equation are obtained by direct calculations. Compared with complete discrimination system for polynomials, our classifications of solutions straightly depend on the coefficients of the ordinary differential equation and hence are easier to be employed. According to the nine kinds of solutions, we establish thirty traveling wave solutions to the coupled systems of ion sound and Langmuir waves, including rational function solution, exponential function solution, trigonometric function solutions, hyperbolic function solutions and Jacobi elliptic function solutions. To the best of our knowledge, many of them are new. Solutions discussed in two recent articles are showed to be equivalent to some special cases of our thirty traveling wave solutions. Our solutions include the kink and anti-kink soliton solutions, dark and bright soliton solutions as well as periodic soliton solutions and solitary wave solutions. We believe that these solutions will be of great use to researchers concerning with nonlinear physical phenomena. In addition, the famous (1+1)-dimensional dispersive long wave equations is taken to illustrate the applicability of these thirty solutions.

Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation

This study’s prime goal is to explore new and general analytic solutions to the conformable time-fractional Phi-four equation by utilizing the advanced exp ( − φ ( ξ ) ) -expansion approach with the conformable derivative principle’s help. This approach is essentially a modified particular case of the generalized form of exp ( − φ ( ξ ) ) -expansion method. Concerning hyperbolic and trigonometric function solutions, an enormous class of modern precise moving wave solutions is revealed. Some interaction solutions obtained, such as lump-periodic and lump-periodic-kink structure and periodic wave, are seen by showing their three-dimensional (3D) and two-dimensional (2D) combination plots with the aid of computational software Maple. We have portrayed the figures of the estimated solutions to understand the physical phenomena.

Exact and explicit traveling wave solution to the time-fractional phi-four and (2+1) dimensional CBS equations using the modified extended tanh-function method in mathematical physics

This current study’s primary aim is to discover new and exact traveling wave solutions to the time-fractional phi-four equation and the (2+1) dimensional Calogero–Bogoyavlanskil schilf (CBS) equation in the perspective of nonlinear traveling wave phenomena. The modified extended tanh-function method is imposed on the phi-four and the (2+1) dimensional CBS equations in this case. Consequently, lump, mixed lump, lump-periodic, lump-periodic-kink, kink, singular kink, kink soliton, periodic, and singular solutions are exhibited in trigonometric, hyperbolic, and rational function solutions. To enucleate, the underlying traveling​ structures, achieved solutions are established by making their dynamic comportment of the exact solutions presented in three-dimensional (3D), contour, and two-dimensional (2D) chart with computational software MATLAB. In terms of conformable derivative, fractional traveling wave transformation, and the applied procedure, all the exact solutions obtained are considered to be novel. To comprehend the physical processes, we have portrayed the figures of the estimated solutions.

Closed-form solutions to the new coupled Konno–Oono equation and the Kaup–Newell model equation in magnetic field with novel statistic application

This paper employs the new robust solver to retrieve exact solutions to a Kaup–Newell model equation and the nonlinear coupled Konno–Oono equation. This solver yields the closed formula for the solutions. Different types of travelling wave solutions, i.e., rational function, hyperbolic function and trigonometric function solutions with many capricious parameters are revealed. Subsequently, by utilizing Matlab 18 we plot 2D and 3D surfaces of obtained analytical solutions for suitable values of the free parameters. The depiction of the solver is direct, vital, sturdy and can be applied to other nonlinear partial differential equations. We also show that some proposed rational solutions can be rapprochement with some known probability distributions.

Construction of traveling wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

The present paper applies the exp (−φ(ξ))-expansion and the extended tanh-function methods to the (2+1)-dimensional Heisenberg Ferromagnetic Spin Chain (HFSC) equation. The applied methods acquire some new exact traveling wave solutions to the HFSC equation, which are representing the hyperbolic, trigonometric, exponential and rational function solutions. All solutions exhibit distinct physical configurations, such as the periodic, dark and singular soliton solutions. Three dimensional (3D) and two dimensional (2D) cross sectional graphics of some obtained solutions are confirmed the periodic, dark and singular behaviors. Furthermore, the conformable derivative is also considered and discussed for aforesaid methods to the HFSC equation. As outcomes, some new optical solutions are also attained in terms of fractionality. Obtained new solutions ensured that aforesaid methods are the reliable treatment for seeking nonlinear phenomena to HFSC equation as well as any NLEEs arising in mathematical physics and fiber optics.

Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion

A new strategy is proposed to analytically construct stochastic fractional solutions of a wick-type SFNLSE driven from the Brownian motion by altogether using the Hermite transformation, modified Riemann–Liouville fractional derivative rule and fractional mapping method with white noise theory. These stochastic fractional solutions include fractional bright, dark solitons and trigonometric function solutions with or without chirped phase. By the dynamical analysis of stochastic solutions, it is found that the evolution of soliton shows randomness and random wave packets appear. Moreover, the randomness of integer order soliton evolution becomes more obvious compared with fractional solitons. They have discontinuous random phenomena with the evolution of time t, and some random independent wave packets appear. These results will play an important role in the related research on the physical application of the random model in the white noise environment.

Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrodinger equation by beta-fractional derivative

In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractional-order Schr¨odinger equation in fractional time derivative form of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr¨odinger equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.

Optical soliton to multi-core (coupling with all the neighbors) directional couplers and modulation instability

n this research paper under consideration is the nonlinear multi-core coupler (coupling with all neighbors) with the parabolic law nonlinearity and optical metamaterials parameters. The integration scheme, namely the New sub-ODE method, is used to extract bright, dark, trigonometric function solutions, Jacobian elliptic function solutions and Weierstrass elliptic function solutions. To better understand the physical phenomena of the gained solutions, some physical explanation of the presented solutions under suitable values for the physical parameters via the 3D, 2D and contour simulations is given. The confrontation between the nonlinear and dispersion terms has permitted to study the behavior of the modulation instability (MI) gain spectra. The results obtained in this article are more general than those contained in the following papers [41–43] and will help to explain the virtue of the nonlinear optical birefringent.

Analytical survey of the predator–prey model with fractional derivative order

This work addresses the analytical investigation of the prey–predator behavior modeled by nonlinear evolution equation systems with fractional derivative order. Through the New Extended Algebraic Method (NEAM), we unearthed diverse types of soliton solutions including bright, dark solitons, combined trigonometric function solutions, and singular solutions. Besides the results obtained in the work of Khater, some new complex soliton solutions are also unearthed. The NEAM can also be used like the synthesis of the two mathematical tools.

The [formula omitted]-model expansion method for solving the Radhakrishnan-Kundu-Lakshmanan equation with Kerr law nonlinearity

In this paper, the Φ6-model expansion method is applied to obtain the new exact solutions of the perturbed Radhakrishnan-Kundu-Lakshmanan equation that appears with Kerr law nonlinearity. The obtained exact solutions are based on the Jacobi elliptic function, hyperbolic function and trigonometric function solutions. Specific values of the parameters consistent with the constraint conditions are also assigned to reveal the dynamic behavior of the solutions.

MEFM For Exact Solutions Of The (3+1) Dimensional KZK Equation and (3+1) Dimensional JM Equation

In this study, we have applied the modified exp -expansion function method (MEFM) to obtain the exact travelling wave solutions for the (3+1) dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and (3+1)-dimensional Jimbo-Miwa (JM) equation. Dark soliton solution, dark-bright soliton solution, hyperbolic function solution and trigonometric function solution of the (3+1) dimensional KZK equation and (3+1)-dimensional JM equation have been found by using this method. After that, we have scratched the 2D and 3D graphs for all exact solutions obtained in this study by using Wolfram Mathematica 9. Thus, the graphical simulations openly show force of this method.

Optical solitons and other solutions to the (2+1)-dimensional coupled system of NLSE by two integration approaches

Based on the new mapping approach and the new extended auxiliary equation approach, we obtain solitons and other solutions of the (2+1)-dimensional coupled nonlinear Schrödinger equations (NLSE). Bright, dark and singular soliton solutions as well as Jacobi elliptic function solutions, solitary wave solutions and trigonometric function solutions are found. This equation describes the transverse effects in an optical fiber, time-independent corporation and field of optical soliton.

Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

In this article, the exact wave structures are discussed to the Caudrey-Dodd-Gibbon equation with the assistance of Maple based on the Hirota bilinear form. It is investigated that the equation exhibits the trigonometric, hyperbolic, and exponential function solutions. We first construct a combination of the general exponential function, periodic function, and hyperbolic function in order to derive the general periodic-kink solution for this equation. Then, the more periodic wave solutions are presented with more arbitrary autocephalous parameters, in which the periodic-kink solution localized in all directions in space. Furthermore, the modulation instability is employed to discuss the stability of the available solutions, and the special theorem is also introduced. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 2-dimensional graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters for stability analysis. The concluded results are helpful for the understanding of the investigation of nonlinear waves and are also vital for numerical and experimental verification in engineering sciences and nonlinear physics.

New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extendedG′/G-expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

Dynamics of solitons to the coupled sine-Gordon equation in nonlinear optics

This paper employs the principle of undetermined coefficients to establish the hyperbolic and trigonometric function solutions of the coupled sine-Gordon equation (CSGE) which describes the propagation of an optical pulse in fiber waveguide. Lie point symmetry of the CSGE is derived. Previously, it was noticed that the concept of nonlinear self-adjointness (NSA) was not applied on the equation under consideration. Here, we apply the concept of NSA to find an explicit form of the differential substitution. By means of the obtained substitution, we establish a new variant of conserved vectors by a new conservation theorem.