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In this paper, the time-fractional Jaulent–Miodek system associated with energy-dependent Schrödinger potential is solved by the modified Laplace decomposition method. The Caputo fractional derivative is considered throughout the paper. The attained solutions using the method are analyzed and compared with the solutions of the existing studies in the literature to demonstrate the efficacy and applicability of the technique. The results are summarized in the tables and figures. We use Mathematica for all computations and figures in the paper. The method is competitive, easily computable, and adaptable to solving various nonlinear problems.

This paper deals with a new variant of the Biswas-Milovic equation, referred to as the perturbed Biswas-Milovic equation with parabolic-law nonlinearity in spatio-temporal dispersion. To our best knowledge, the considered equation which models the pulse propagation in optical fiber is studied for the first time, and the abundant optical solitons are successfully obtained utilizing the auxiliary equation method. Furthermore, we also investigate the impact of the parameters such as the spatio-temporal dispersion and the parabolic nonlinearity on the behavior of the soliton. The auxiliary equation method was employed due to its effectiveness in extracting abundant and diverse kinds of soliton solutions, including bright, kink, and singular. It has been tested and verified using Mathematica that all solutions obtained satisfy the main equation. 3D, 2D, and contour graphs of the solution functions are plotted and interpreted to understand the physical behavior of the model. The new model and findings may contribute to the understanding and characterization of the nonlinear behavior of pulse propagation in optical fibers, which is crucial for the development of optical communication systems.

In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n, n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebraic equations by utilizing the wavelets method including an integral operational matrix and then discretizing the equation at the collocation points. The system can be simply solved by several traditional methods. Finally, the algorithm is implemented for some numerical examples and the numerical solutions are compared with the exact solutions using MAPLE. The obtained results are demonstrated using figures and tables. When the results are compared, it is evinced that the algorithm is quite effective and advantageous due to its easily computable algorithm, high accuracy, and less process time. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

In this paper, some analytical solutions for a model of dual-core optical fbers governed by a system of coupled non-linear Schrödinger equations (NLSEs) and the efect of the coefcient of the group velocity dispersion term on the considered model are investigated. The group velocity dispersion (GVD) has a important role in the optical wave propagation. The enhanced modifed extended tanh expansion method (eMETEM) is successfully implemented to the governing model. The NLSE system is turned into a nonlinear ordinary diferential equation (NLODE) via appropriate wave transformations. Supposing that the NLODE has solutions in the form suggested by the method and utilizing the enhanced solutions of the Riccati equation, we gain a nonlinear system of algebraic equations. The solutions of the governing model are obtained after solving the system of algebraic equations. 2D, 3D and contour illustrative fgures for the physical interpretation of the attained solutions are presented. Besides, the result of the investigation, which is related to the efect of the coefcient of the group velocity dispersion term, is presented by supporting the various graphical scheme.

In this paper, we aim to derive new soliton solutions of (1+1)- and (2+1)-dimensional generalized Sasa–Satsuma equations via the new Kudryashov method. In optical fiber transmission systems, the Sasa–Satsuma equation describes the effects of third-order dispersion, self-steepening and stimulated Raman scattering in the propagation of ultrafast pulses. The considered equations are encountered in various physical applications such as ultra-short and femto-second pulse propagation in optical fibers and dynamics of deep water waves. So, investigation of the novel solutions of the equations is one of the important topics. We have successfully extracted some soliton solutions for the considered equation. The various graphs of the obtained solutions have been depicted in the figures by selecting appropriate parameters. The singular and bright soliton solutions have been revealed in the figures. All acquired solutions have been confirmed to satisfy the considered equations. The results show that the approach may be used to find exact solutions to various nonlinear evolution equations. The new solutions and the paper results may enrich the understanding of the wave propagation in the optical fibers and may shed light on new studies.

In this work, we aim to derive various soliton solutions to the Wazwaz–Benjamin–Bona–Mahony equation with conformable and M-truncated derivatives. The considered equation models long waves in the ocean engineering field. Unified Riccati equation expansion and Kudryashov auxiliary equation methods are used to the model, and so, kink, singular, and periodic-singular soliton solutions are successfully obtained. It is also reported on the constraint conditions that assure the validity of novel wave forms. By choosing appropriate parameters, numerical simulations of the obtained results are depicted by using two- and threedimensional plots and the comparative results between the solutions for the conformable and M-truncated derivative are shown in two-dimensional graphs for various orders α, β. Moreover, the effects of the parameters in the obtained solutions are shown. The methods might be useful for obtaining the analytical solutions of many physical phenomena in nature since they are effective, robust, and easily applicable. Finally, this study contributes to extract both various solutions to the literature and to investigate wave behavior while the parameters change.

In this paper, we have successfully extracted many analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger equation (NLSE) by the enhanced modifed extended tanh expansion method (eMETEM). The considered method is a recently enhanced version of the classical modifed extended tanh expansion method. So, we have successfully extracted the abundant solutions of the (1+2)-dimensional Chiral NLSE. Using a computer algebra system program, we have verifed that all derived solutions satisfy the Chiral NLSE. The plots of some solutions are demonstrated to explain the dynamics of the solutions. It is expected that the results of the paper might be helpful for future works in traveling wave theory.

This paper presents investigation of soliton solutions for the perturbed Fokas-Lenells (pFL) equation, has a vital role in optics, using Sardar sub-equation method. The equation models the propagation in ultrashort light pulses in optical fibers. Using appropriate wave transformation, the pFL equation is reduced to a nonlinear ordinary differential equation (NLODE). The solutions of this NLODE equation are assumed to be in the suggested form by the Sardar sub-equation method. Hence, an algebraic equation system is obtained by substituting the trial solutions and their necessary derivatives into the NLODE. After finding the unknowns in the system, the soliton solutions of the perturbed Fokas-Lenells equation are extracted. The method produces various kinds of solitons such as the dark, singular and periodic. To show physical representations of the solitons, 2D, 3D and contour plots of the solutions are demonstrated via computer algebraic systems. It is expected that derived solutions may be useful for future works in various fields of science, especially optics and so, it may contribute to optic fiber industry.

Objective – The main objective of this paper is to investigate analytic soliton solutions of a nonlinear Schrödinger equation (NLSE), including Kudryashov’s sextic power-law nonlinearity by introducing different approaches of two efficient analytical methods. The considered equation has been recently introduced by N. A. Kudryashov to describe pulse propagation in optical fibers. The refractive index in the equation comprises six terms, each of the terms containing a power-law component. Methods – Applying a wave transformation to the considered NLSE and splitting up the real and imaginary parts, the NLSE is converted to the nonlinear ordinary differential equations (NLODEs). Then, the solutions of NLODEs are considered as suggested in the proposed method and suggested solutions that include some unknown parameters are substituted into the NLODE. An algebraic equations system is acquired by collecting the same power of the unknown function and equating all coefficients to zero. The unknown parameters in the system, and so the solutions of the NLSE, are found by solving the system. In the proposed first method, the modified extended tanh method is enhanced by proposing more solutions. The proposed second method, a different version of auxiliary methods, can remarkably reduce calculations to easily get solutions for the NLSEs with higher-order or power-law nonlinearity. Results – The two proposed methods are successfully applied to the considered NLSE and the abundant solutions of the NLSE are attained. Besides, 2D, 3D and contour graphs are demonstrated in figures for the physical illustrations of the gained solutions. Conclusion – Obtaining the solutions of NLSEs with higher-order or power-law nonlinearity has crucial importance but still challenging work. So, we propose different approaches of two efficient analytical methods, namely, enhanced modified extended tanh expansion method and an auxiliary function method. The derived results imply that the used methods are very efficient, reliable and powerful such that they can be easily implemented to many nonlinear NLSEs with higher-orders or higher power-law nonlinearities that describe real-life phenomena.

Purpose: In this paper, optical solitons of higher-order nonlinear Schrödinger equation with Kudryashov's sextic power-law of nonlinear refractive index are investigated via the direct mapping method. The considered model identifies the optical soliton pulse propagation in the optical fibers. Deriving the optical solutions of investigated model such as sextic power is critical but difficult work. The primary aim of this paper is to graphically examine the impact of power-law nonlinearity (pLawNL) and chromatic dispersion (CD) parameters clarifying self-phase modulation (SPM) in the equation on soliton behavior as well as obtaining optical soliton solutions. Methodology: To according to the used technique, we first used the complex wave transform to generate the nonlinear ordinary differential equation (NLODE) form of the nonlinear Schrödinger equation (NLSE) with Kudryashov's sextic power-law the nonlinear refractive index (SPLawNRI). Then, we were able to produce a system of linear equations in polynomial form by using the approach. Different solution sets including the values of the parameters of the studied equation and the suggested approach were produced by solving the linear system of equations. Findings: We acquired the optical soliton solutions of the main equation after inserting the sets and wave transformation into the solution functions suggested by the approach. The constraint conditions for the related solutions were suggested. We proved that the gained solutions satisfied the NLSE with Kudryashov's SPLawNRI under the suggested constraint conditions. Originality: We present contour, 3D and 2D depictions in various simulations in figures to comment the obtained solution functions. Besides, we investigate the effects of the power-law nonlinearity parameter that expresses SPM in the main equation and the parameter that are the group velocity dispersion (GVD) or chromatic dispersion (CD) on soliton behavior. The results suggest that the utilized approach is efficient, reliable, and powerful to be readily applied to various NLSEs with higher-order or higher pLawNLs that characterize real-life problems.

In this paper, we have successfully extracted novel analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger (NLS) equation by modified extended tanh expansion method combined with new Riccati solutions (METEM-cNRCS) as far as we know. When a wave transformation is applied to the considered Chiral NLS equation, a nonlinear ODE is obtained. Assuming the solutions of ODE have a form as the method suggests, and substituting the trial solutions to the ODE, we get a polynomial. Gathering the coefficients with the same power in the polynomial, we acquire an algebraic equation system. So, we may obtain the abundant solutions of the (1+2)-dimensional Chiral NLS equation by solving the system via Maple. The plots of some solutions are demonstrated to explain the dynamics of the solutions. It is expected that the results of the paper are a guide for future works in traveling wave theory.

In this paper, we have investigated the perturbed Chen–Lee–Liu equation which describes the pulse propagation in the optical fbers, under the impact of the inter-modal dispersion, self-steepening and nonlinear dispersion terms. By using the enhanced modifed extended tanh expansion method, bright, singular, periodic singular and periodic bright solitons have been obtained and the efects of the coefcients of the inter-modal dispersion, self-steepening and nonlinear dispersion terms on the soliton’s dynamics have been examined in each case. In this respect, the review in the article has not been studied and reported before. The computations throughout this paper have been fulflled by Maple and also the graphical simulations are demonstrated via Matlab.

Purpose: When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology: By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings: Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality: Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.

In this study, the Jacobi wavelet collocation method is studied to derive a solution of the time-fractional Fisher?s equation in Caputo sense. Jacobi wavelets can be considered as a generalization of the wavelets since the Gegenbauer, and thus also Chebyshev and Legendre polynomials are a special type of the Jacobi polynomials. So, more accurate and fast convergence solutions can be possible for some kind of problems thanks to Jacobi wavelets. After applying the proposed method to the considered equation and discretizing the equation at the collocation points, an algebraic equation system is derived and solving the equation system is quite sim?ple rather than solving a non-linear PDE. The obtained values of our method are checked against the other numerical and analytic solution of considered equation in the literature and the results are visualized by using graphics and tables so as to reveal whether the method is effectiveness or not. The obtained results evince that the wavelet method is quite proper because of its simple algorithm, high accuracy and less CPU time for solving the considered equation.

In this research, our main motivation is to find the novel analytical solutions of $$(2+1)$$ dimensional Heisenberg ferromagnetic spin equation, which describes the nonlinear dynamics of the ferromagnetic materials by using the extended rational $$sine-cosine$$ and $$sinh-cosh$$ methods. The considered PDE is converted to an ODE by applying a wave transformation, and then the solutions of the ODE are supposed to be in the rational forms of trigonometric functions. After substituting the solutions to the ODE and doing some basic calculations, a system of algebraic equations is derived. So, finding the solutions of the PDE turns into a problem of solving an algebraic system of equations. The unknown coefficients in the solutions that are in the rational form are found by solving the obtained system. The methods are powerful and can be applied to find exact solutions to lots of PDEs in mathematical physics.

This study investigates various analytic soliton solutions of the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation in fluid dynamics and plasma physics using a recently introduced technique which is the New Kudryashov method. Moreover, it is examined how the wave propagation in both directions represented by the CBS equation occurs. The considered equation describes the interaction of the long propagating wave in the x axis with the Riemann propagating wave along the y axis. To get traveling wave solutions of the CBS equation, it is transformed into a nonlinear ordinary differential equation (NLODE) using a proper wave transformation. Supposing that the NLODE has some solutions in the form provided by the method, one can obtain a nonlinear system of algebraic equations. The unknowns in the system can be found by solving the system via computer algebraic systems such as Mathematica and Maple, etc. Substituting the unknowns into the trial solutions provided by the method, we get the solutions of the NLODE. Then, putting wave transformations back into the solutions of NLODE, we get the solutions of the considered CBS equation. We present the 2D, 3D and contour plots to illustrate the physical behavior of the obtained solutions using the appropriate parameters. Besides, the schematic representation of wave motion of the soliton along both spatial axes and its interpretation are given. The used novel technique can be used for a wide range of partial differential equations (PDEs) in the real world. It is expected that the derived soliton solutions might be helpful for better understanding the wave behavior and so, it might contribute to future studies in various disciplines.

In this paper, we study the novel exact solutions of the ( 2 + 1 ) -dimensional Biswas–Milovic equation for the description of pulse propagation in optical fiber. We successfully constructed some important solutions, such as dark, singular, combined dark-singular soliton, singular periodic wave and rational function solutions, have been analytically obtained using the extended tanh method. These solutions might play an important role in the engineering and physics fields. It is shown that the considered method provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic and other physical phenomena.

In this chapter, we are planning to make a comparison between conventional Time Series Models and Fuzzy Time Series Models by an application in an e-commerce company. Future sales of furniture will be predicted. The performance of different models and forecasting periods are going to be analyzed to discuss advantages and disadvantages of each method. MAE is chosen as performance indicators of each model and forecasting period combination. As a conclusion to this chapter, generic strategies for prediction in an e-commerce company will be formulated in consideration of these indicators.