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In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional (2 + 1) -dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported. [ABSTRACT FROM AUTHOR]

This research paper's primary goal is to find fresh approaches to the Hirota–Maccari system. This system explains the dynamical features of the femto-second soliton pulse in single-mode fibers. The bright soliton, dark soliton, dark-bright soliton, dark singular, bright singular, periodic soliton, and singular solutions are developed utilizing the modified auxiliary equation technique. To make the physical significance of each unique solution clearer, it is mapped in both 2D and 3D. The primary Hirota–Maccari system is being verified by all new solutions, and the constraint condition is also provided. The obtained optical solitons may be important for the analysis of nonlinear processes in optic fiber communication and signal processing. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

This document discusses recent trends in mathematical models and solitons used to analyze real-world problems. The focus is on the application of fractional calculus in modeling phenomena such as anomalous diffusion, non-Markovian processes, and random walk. The document provides summaries of 15 papers accepted for publication in a special issue, covering topics such as digestive system modeling, COVID-19 pandemic modeling, control systems for motors, symmetry in pine wilt disease, soliton solutions for various equations, and epidemic modeling. The papers employ various mathematical techniques and provide insights into different aspects of real-world problems. [Extracted from the article]

The modified exponential function method is used to obtain some new analytical solutions of the nonlinear Gerdjikov–Ivanov equation with the M-fractional operator. The novel obtained solutions are expressed in hyperbolic, trigonometric, and exponential function forms. Moreover, the instability modulation and gain spectra of the Gerdjikov–Ivanov equation are also analyzed. Constraints conditions are utilized to verify the existence of the solutions. Presented solutions are novel, satisfy and verify the M-fractional Gerdjikov–Ivanov equation. [ABSTRACT FROM AUTHOR]

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J -transform method ( O HA J TM) and J -variational iteration transform method (J -VITM) have been adopted. The OHA J TM is the hybrid method, where optimal-homotopy analysis method ( O HAM) is utilized after implementing the properties of J -transform (J T), and in J -VITM is the J -transform-based variational iteration method. Banach's fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that J -VITM is T -stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via O HA J TM performs better in producing more accurate behavior in comparison to J -VITM and the methods introduced recently. [ABSTRACT FROM AUTHOR]

The time-fractional generalized biological population model and the (2, 2, 2) Zakharov–Kuznetsov (ZK) equation are significant modeling equations to analyse biological population, ion-acoustic waves in plasma, electromagnetic waves, viscoelasticity waves, material science, probability and statistics, signal processing, etc. The new generalized G ′ / G -expansion method is consistent, computer algebra friendly, worthwhile through yielding closed-form general soliton solutions in terms of trigonometric, rational and hyperbolic functions associated to subjective parameters. For the definite values of the parameters, some well-established and advanced solutions are accessible from the general solution. The solutions have been analysed by means of diagrams to understand the intricate internal structures. It can be asserted that the method can be used to compute solitary wave solutions to other fractional nonlinear differential equations by means of fractional complex transformation. [ABSTRACT FROM AUTHOR]

This article deals with the applications of (1 / G ′) -expansion method which has recently been presented to the literature. Applying this tool on the generalized (2 + 1)-dimensional Hirota–Maccari system, perturbed nonlinear Schrödinger's equation and dispersive cubic-quintic nonlinear Schrödinger equation, new complex traveling wave solutions are founded. The solution of partial differential equations representing the phenomenon of fiber optic communication and the interrelationship of the internal dynamics of traveling wave solutions, which play an important role in the transport of energy, have been discussed. Furthermore, for a better physical understanding of the results, various graphs with the appropriate values of the parameters have been presented and the simulation has been used to support the discussion. [ABSTRACT FROM AUTHOR]

Bernoulli sub‐equation function method is applied to obtain exact solutions of Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation. As a result of this, exact traveling‐wave and some new oscillating solutions to CDGSK are obtained. It may be observed that Bernoulli sub‐equation function method employed here is very effective and reliable to get explicit solutions for this nonlinear partial differential equation. Profiles of all constructed solutions are graphically illustrated entirely as well. [ABSTRACT FROM AUTHOR]

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

In this paper, some new exact traveling and oscillatory wave solutions to the Kudryashov-Sinelshchikov non-linear PDE are investigated by using Bernoulli sub-equation function method. Profiles of obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In this paper, perturbed Chen–Lee–Liu equation is considered to describe pulse propagation in optical fibers. Chen–Lee–Liu equation is a derivative form of the nonlinear Schrödinger equation. Two renowned analytical techniques namely, exp (- φ (η)) -expansion method and the generalized Kudryashov method are applied to analyze this model. The algorithm of these methods is also presented. Different structures of soliton solutions are successfully investigated for perturbed this model. Additionally, constraint conditions on coefficients of perturbation terms are also defined to assure the existence of such solitons. The graphical representation of these solutions is shown to demonstrate the dynamics of pulse propagation governed by Chen–Lee–Liu equation in optical fibers. [ABSTRACT FROM AUTHOR]

This paper focuses on the generalized Calogero–Bogoyavlenskii–Schiff equation to extract new complex solutions by using two analytical methods, namely, Bernoulli sub-equation function method, Modified exponential function method. For better understanding of physical meanings of solutions, simulations are reported by using a computational package program. Moreover, strain conditions for validity of complex solutions are also archived. Finally, a conclusion part completes the paper by mentioning the novelties of paper. [ABSTRACT FROM AUTHOR]

This paper focuses on the instability modulation and new travelling wave solutions of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar equation via the tanh function method. Dark, mixed dark–bright, complex solitons and periodic wave solutions are archived. Strain conditions for the validity of results are also reported. Instability modulation properties of the governing model are also extracted. Various wave simulations in 2D, 3D and contour graphs under the strain conditions are presented. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the higher-order dispersive extended nonlinear Schrödinger equation, which demonstrates the propagation of ultrashort pulses in optical communication networks. In this study, both the sinh-Gordon expansion method and the generalized exponential rational function method are used to offer some novel optical solutions. These optical soliton solutions are dark soliton, bright soliton, singular, periodic, and dark-bright soliton solutions. The obtained optical soliton solutions are presented graphically in 2D and 3D to clarify the behavior of solutions more effectively. The constraint conditions are also used to verify the exitances of the new analytical solutions. Moreover, all solutions compared to solutions obtained previously are new, and all the new wave solutions have verified Eq. (1) after we substituted them into the studied equation. In the future, these novel soliton solutions will be very helpful in developing fluid dynamics, biomedical issues, dynamics of adiabatic parameters, industrial research, and many other areas of science. To our acknowledgment, the presented optical solutions are novel, and also beforehand these methods have not been applied to this studied equation. [ABSTRACT FROM AUTHOR]

In this manuscript, new dark and trigonometric function traveling wave soli- ton solutions to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation by using the modified exponential function method are successfully obtained. Along with novel dark structures, trigonometric solutions are also extracted. For deeper investigating of waves propagation on the surface, 2D and 3D graphs along with contour simulations via computational programs such as Wolfram Mathematica, Matlap softwares and so on are presented. [ABSTRACT FROM AUTHOR]

In this work, we investigate the perturbed optical solitons to the Gerdjikov-Ivanov equation consisting of group velocity dispersion and quintic nonlinearity coefficients, which communicate the propagation of pulses in nonlinear optics. The various kinds of solitons in single and combined forms like dark, singular, dark-singular, bright-dark are derived by Fan-extended sub equation method. Moreover, the singular periodic, triangular type solutions are also obtained. And, we also discuss the stability analysis of the studied nonlinear model with the concept of linear stability, we prove that the governing model is stable. Parametric conditions on physical parameters to ensure the existence criteria of optical solitons are also listed. We also plot 3D profiles for the physical behavior of the obtained solutions by selecting the suitable values of the parameters. [ABSTRACT FROM AUTHOR]

According to the report presented by the World Health Organization, a new member of viruses, namely, coronavirus, shortly 2019-nCoV, which arised in Wuhan, China, on January 7, 2020, has been introduced to the literature. The main aim of this paper is investigating and finding the optimal values for better understanding the mathematical model of the transfer of 2019-nCoV from the reservoir to people. This model, named Bats-Hosts-Reservoir-People coronavirus (BHRPC) model, is based on bats as essential animal beings. By using a powerful numerical method we obtain simulations of its spreading under suitably chosen parameters. Whereas the obtained results show the effectiveness of the theoretical method considered for the governing system, the results also present much light on the dynamic behavior of the Bats-Hosts-Reservoir-People transmission network coronavirus model. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the various dimensional nonlinear Fokas and Breaking soliton equations via a powerful analytical method, namely, sine-Gordon expansion method. Many new solutions such as complex combined dark-bright soliton solutions, singular and hyperbolic functions are derived. Choosing the suitable values of these parameters, various novel simulations are also plotted. Such results explain the wave behavior of the governing models, physically. [ABSTRACT FROM AUTHOR]

An editorial is presented on mathematical models in the form of numerical solutions, analytical solutions, approximate solutions, and periodic properties. Topics include Sumudu decomposition method (SDM) finding approximate solutions to two-dimensional fractional partial differential equations; and applying an integral transform-based technique for solving a fractional order Volterra-type integro-differential equation (FVIDE).

The motivation of this work is to analyse the nonlinear models and their complex nature with generalized tools associated with material and history-based properties. With the help of well-known and widely used numerical scheme, we study the stimulating behaviours of the financial system in this work. The impact of parameters on price index, rate of interest, investment demand, influence changes and investment cost with respect to saving amount, and the elasticity of commercial markets demand are discussed. The consequences of generalizing the model within the arbitrary order are derived. The existence of the solution for the considered system is presented. This study helps beginner researchers to investigate complex real-world problems and predict the corresponding consequences. [ABSTRACT FROM AUTHOR]

In this study, using the extended sinh-Gordon equation expansion method, we construct the dark, bright, combined dark-bright optical, singular, combined singular solitons and singular periodic waves solutions to the complex cubic nonlinear Schrödinger equation with δ -potential. The conditions for the existence of the obtained solutions are given. To present the physical feature of the acquired result, the 2D and 3D graphs are plotted under the choice of suitable values of the parameters. [ABSTRACT FROM AUTHOR]

This study applies the powerful sine-Gordon expansion method in acquiring some new solutions to the strain wave equation in micro-structured solids arising in mathematical physics. The strain wave equation in micro-structured solids is used in modelling wave propagation in micro-structured materials. We successfully acquire some new solitary wave solutions with hyperbolic function structure. By taking some suitable values of parameters involved in the solutions, we plot the 2D, 3D and contour graphs of all the obtained solutions in this study. [ABSTRACT FROM AUTHOR]

In this study, with the aid of Wolfram Mathematica 11, the modified exp [ABSTRACT FROM AUTHOR]

In this paper, we use the homotopy perturbation transform method (HPTM) to offer an efficient semi-analytical technique for solving fractional Emden–Fowler equations. A mixture of Laplace transform, Caputo–Fabrizio derivative, and homotopy perturbation transformation process has the projected technique. To assess the efficacy of the suggested technique, test examples have been provided. The series have been used to represent semi-analytical solutions. Also, covered have the convergence position, estimation, and semi-analytical simulation results. The HPTM efficiently managed and controlled a series solution that quickly converges to a precise result in a narrow admissible region. The new findings essentially improve and simplify some of the previously published findings (see Malagia in Math. Comput. Simul. 190:362, 2021). By assigning appropriate values to free parameters, dynamical wave structures of some semi-analytical solutions are graphically demonstrated using 2-dimensional and 3-dimensional figures. Furthermore, various simulations are used to demonstrate the physical behaviors of the acquired solution with respect to fractional integer order. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and use the new modified exp ( − Ω ( ξ ) ) -expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease. [ABSTRACT FROM AUTHOR]

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions. [ABSTRACT FROM AUTHOR]

ABSTRACT This manuscript focuses on the new wave behaviors of the system of equations for the ion sound wave under the action of the ponderomotive force which results from a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field due to high-frequency field. The sine-Gordon expansion method which is one of the powerful methods has been considered for finding traveling wave solutions of the system of equations for the ion sound wave under the pressure of the Langmuir wave. This algorithm yields new complex hyperbolic function solutions to the system considered in this paper. Wolfram Mathematica 9 has been successfully used throughout the paper for mathematical calculations. [ABSTRACT FROM PUBLISHER]

Obtaining new and important travelling wave solutions of wave propagation modelling of waves such as nonlinear optics models, propagation and transmission models of waves by using different methods plays an important role in maritime engineering, ocean, beach science and floating structures, and also for understanding new physical meanings of coastal structural properties. New complex travelling wave solutions obtained by using different newly improved methods explain new and general properties of nonlinear optic structures, wave propagations and motions, major structures of the impact of environmental factors on the beach like tsunami, the impact of the waves on the vessel, the power of the effects of the waves on wave distribution panels. These solutions structures may be trigonometric, complex function, hyperbolic function, exponential and rational function. In this study, we apply two effective methods to the nonlinear evolution equation used to describe the new versions of different mathematical models for wave motion and propagations. The first is improved Bernoulli sub-equation function method (IBSEFM), the latter is modified exp ( - Ω ( ξ ) ) -expansion function method (MEFM). We obtain some new travelling wave structures such as complex function, hyperbolic function and rational function, exponential function solutions. We observe that all travelling wave solutions have been verified the nonlinear partial differential equation by using Wolfram Mathematica 9. Then, we plot the two and three dimensional surfaces for all travelling wave structures obtained in this paper by the same computer program. [ABSTRACT FROM AUTHOR]

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained. [ABSTRACT FROM AUTHOR]

In this study, a new method called improved Bernoulli sub-equation function method has been proposed. This method is based on the Bernoulli sub-ODE method. After we mention the general properties of proposed method, we apply this algorithm to the (2 + 1)-dimensional Boiti–Leon–Pempinelli equation system. This gives us some new prototype solutions such as exponential and rational function solutions. Then, we have plotted two- and three-dimensional surfaces of analytical solutions. Finally, we have submitted a comprehensive conclusion. [ABSTRACT FROM AUTHOR]

In this study, we have applied the modified exp(-Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two- and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program. [ABSTRACT FROM AUTHOR]

In this study, we obtain some new complex analytical solutions to the Kundu–Eckhaus equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics using improved Bernoulli sub-equation function method. After we have mentioned the general structure of improved Bernoulli sub-equation function method, we have successfully applied this method and then obtained some new complex hyperbolic and complex trigonometric function solutions. Two- and three-dimensional surfaces of analytical solutions have been plotted via wolfram Mathematica 9 version. At the end of this article, a conclusion has been submitted by mentioning important points founded in this study. [ABSTRACT FROM AUTHOR]

In this study, we have studied to obtain some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin–Bona–Mahony equation by using modified exp-function method. We have submitted the general structure of modified exp-function method. We have founded some new analytical solutions such as hyperbolic and rational function solutions. Afterward, we have plotted 2D and 3D surfaces of analytical solutions obtained in this study by using computer programming wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In this paper, the Natural transform decomposition method (NTDM) which has been constructed by combining Natural transform method (NTM) and Adomian decomposition method (ADM) has been proposed. This algorithm has been applied to obtain the approximate solutions for the modified Camassa-Holm equation (mCHE) and modified Degasperis-Procesi equation (mDPE). Then, two and three dimensional surfaces have been drawn by using Wolfram Mathematica 9. A table including numerical results between approximate solutions which are obtained by using NTDM in this paper and approximate solutions which have already founded by the help of using VIM by Jafari H. et al. with the same values of parameters has been formed for mCHE and mDPE. [ABSTRACT FROM AUTHOR]

In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams-Bashforth algorithm, show the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

This paper focuses on the application of an efficient technique, namely, the fractional natural decomposition method (FNDM). The numerical solutions of the model containing the water transport in unsaturated porous media, called Richards equation, are extracted. This model is used to describe the non-locality behaviors which cannot be modeled under the framework of classical calculus. To demonstrate the effectiveness and efficiency of the scheme used, two cases with time-fractional problems are considered in detail. The numerical stimulation is presented with results accessible in the literature, and corresponding consequences are captured with different values of parameters of fractional order. The attained consequences confirm that the projected algorithm is easy to implement and very effective to examine the behavior of nonlinear models. The reliable algorithm applied in this paper can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical and analytical study of the HIV‐1 infection of CD4+ T‐cells conformable fractional mathematical model. This model is studied analytically by Kudryashov and modified Kudryashov methods and numerically by finite difference method. A comparison between the results of the analytical and numerical methods is investigated. We also give some figures that show how accurate the solutions will be obtained from analytical and numerical methods. [ABSTRACT FROM AUTHOR]

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 + 1)-dimensional cubic Klein-Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]