Generalized Riccati Equation Mapping Method Research Articles

On the study of solitary wave dynamics and interaction phenomena in the ultrasound imaging modelled by the fractional nonlinear system

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.

On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

Solitary wave type solutions of nonlinear improved mKdV equation by modified techniques

In this paper, the improved and modified version of the Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM) are manipulated effectively and generously to determine the exact solitary wave soliton solutions of the improved modified KdV (mKdV) equation. The purpose of this study is to provide novel exact solutions to the improved mKdV equation. Specifically, we utilized IMSSEM and IGREMM to study different solutions of the nonlinear improved mKdV equation, focusing on exponential, trigonometric, and trigonometric hyperbolic type solutions. Furthermore, the plotting of various solutions for direct viewing analysis is provided in two and three-dimensional graphs. The new strategies are straightforward, quick, and efficient and have many other advantages, whereas, they provide the most accurate and unique solution to many other types of nonlinear partial differential equations (NPDEs), which usually arise in engineering and applied sciences. It should be noted that these methodologies are novel mathematical instruments that have shown to be the most effective mathematical tools for solving higher-order nonlinear partial differential equations in mathematical physics. Symbolic computation was used to validate all of the solutions that were established. Thus, it is also hoped that these techniques will ultimately reduce the cumbersome workload involved during the process of solutions to complicated NLPDEs.

Unveiling new insights into soliton solutions and sensitivity analysis of the Shynaray-IIA equation through improved generalized Riccati equation mapping method

Unveiling new insights into soliton solutions and sensitivity analysis of the Shynaray-IIA equation through improved generalized Riccati equation mapping method

Optical soliton solutions in a distinctive class of nonlinear Schrödinger’s equation with cubic, quintic, septic, and nonic nonlinearities

The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

Investigation of Brownian motion in stochastic Schrödinger wave equation using the modified generalized Riccati equation mapping method

Investigation of Brownian motion in stochastic Schrödinger wave equation using the modified generalized Riccati equation mapping method

Comparisons of Numerical and Solitary Wave Solutions for the Stochastic Reaction–Diffusion Biofilm Model including Quorum Sensing

This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.

Investigation of soliton structures for dispersion, dissipation, and reaction time-fractional KdV–burgers–Fisher equation with the noise effect

ABSTRACT In this manuscript, the soliton structures for the time-fractional KdV–Burgers–Fisher equation with the effect of noise are investigated analytically. This is the dispersion–dissipation–reaction model. The third- and fifth-order time-fractional stochastic KdV–Burgers–Fisher equations are under consideration. These wave structures are constructed with the help of an extended generalized Riccati equation mapping method (EGREM). This method is a combined form of the G ′ / G expansion method with the generalized Riccati equation mapping method, and it will give the different forms of wave structures like shock, singular, combo, hyperbolic, trigonometric, mixed trigonometric, and rational solutions. These techniques are used symbolically with computational tools like Mathematica to demonstrate the efficiency and simplicity of the proposed strategy. Additionally, with the various relevant parameter values, the sketches of some solutions in the form of 3D and contour representations for the purpose of comprehending physical processes are drawn. These sketches clearly show the random behavior of these wave structures that are appearing in dispersion, dissipation, and reaction concentrations of these mathematical models.

Analytical study of reaction diffusion Lengyel-Epstein system by generalized Riccati equation mapping method

In this study, the Lengyel-Epstein system is under investigation analytically. This is the reaction–diffusion system leading to the concentration of the inhibitor chlorite and the activator iodide, respectively. These concentrations of the inhibitor chlorite and the activator iodide are shown in the form of wave solutions. This is a reaction†“diffusion model which considered for the first time analytically to explore the different abundant families of solitary wave structures. These exact solitary wave solutions are obtained by applying the generalized Riccati equation mapping method. The single and combined wave solutions are observed in shock, complex solitary-shock, shock singular, and periodic-singular forms. The rational solutions also emerged during the derivation. In the Lengyel-Epstein system, solitary waves can propagate at various rates. The harmony of the system’s diffusive and reactive effects frequently governs the speed of a single wave. Solitary waves can move at a variety of speeds depending on the factors and reaction kinetics. To show their physical behavior, the 3D and their corresponding contour plots are drawn for the different values of constants.

Stability analysis and consistent solitary wave solutions for the reaction–diffusion regularized nonlinear model

The reaction-nonlinear diffusion model is analyze analytically, under both logistic and bistable growth regularization. These regularization techniques have been widely apply and employ in models of chemical phase separation. In this study, we mainly focus on construct the different wave structures are with the help of the generalized Riccati equation mapping method. This method is provided us the different hyperbolic, trigonometric, and rational solutions. These solutions help for the dynamic study of the reaction-nonlinear diffusion model. Moreover, for the physical significance, we construct the 3D and their corresponding contour by choosing the different values of the parameters. Additionally for the computational interests of the readers stability and consistency analysis were proved.

Optical fiber application of the Improved Generalized Riccati Equation Mapping method to the perturbed nonlinear Chen-Lee-Liu dynamical equation

This work considers the improved generalized Riccati equation mapping (IGREM) technique with its Bäcklund transformation to obtain the wave solutions of the perturbed Chen-Lee-Liu (CLL) equation. The CLL equation actually depicts pulse propagation in optical fibers and plasma. We utilize proposed technique to obtain hyperbolic, periodic trigonometric, rational and exponential function solutions. To make this work worthy and for efficient illustration of this methodology, 3D, contours and 2D plots of some selected solutions are constructed by selecting the values of involved parameters. The results of this paper prove that the used method is effective to improve the nonlinear dynamical behavior of a system.

A study of propagation of the ultra-short femtosecond pulses in an optical fiber by using the extended generalized Riccati equation mapping method

A study of propagation of the ultra-short femtosecond pulses in an optical fiber by using the extended generalized Riccati equation mapping method

Abundant closed-form solutions of the (3+1)-dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering

This research article is a study of the (3+1)-dimensional Vakhnenko-Parkes equation that governs the propagation of high-frequency waves in a relaxing medium. To find a variety of new analytic exact solitary wave solutions, we apply three effective schemes, such as the generalized Kudryashov method (GKM), the generalized exponential rational function method (GERFM), and the generalized Riccati equation mapping method (GREMM). Obtained solutions are crucial to understand some existing, complicated physical phenomena. Many solutions are obtained, which are expressed by the exponential, trigonometric, and hyperbolic functions, including tanh, coth, sech, csch, tan, cot, sech, csch, and their combinations. These solutions include periodic-wave solutions, kink waves, combinations of kink and multi solitons, singular solitons, singular-kink structures, and periodic solitons. The 3D and 2D plots are shown by considering the possibility of arbitrary parameters occurring in the solutions better to understand the underlying mechanisms of the stated family. The computational results of this study assure us that the proposed approaches seem useful, efficient, and reliable tools for estimating the solutions attained in the study. Moreover, the method is expected to be frequently applied to examine various types of nonlinear PDEs, which emerge in mathematical physics, ocean engineering, and science.

New perturbed conformable Boussinesq-like equation: Soliton and other solutions

In recent years, finding exact solutions to nonlinear differential equations has become a fascinating study topic. In the present study, using the modified Kudryashov method and the improved generalized Riccati equation mapping method in the conformable sense, we acquired the exact soliton-type solutions for the new time-fractional perturbed Boussinesq-like equation. This equation has been successfully obtained by the use of asymptotic methods on the defocusing Camassa–Holm nonlinear Schrödinger equation. We obtain the exponential, trigonometric, hyperbolic, and rational type solutions comprising dark solitons, kink solitons, bisymmetry solitons, singular solitons, combined complex solitons, and periodic solutions. These solutions are of great importance in various fields of applied sciences, coastal, and ocean engineering. Furthermore, all acquired solutions have been validated by putting them back into the original equation with the help of the Mathematica package software.

Propagation of new dynamics of longitudinal bud equation among a magneto-electro-elastic round rod

In this survey, structure solutions for the longitudinal suspense equation within a magneto-electro-elastic (MEE) circular judgment are extracted via the implementation of two one-of-a-kind techniques that are viewed as the close generalized technique in its field. This paper describes the dynamics of the longitudinal suspense within a MEE round rod. Nilsson and Lindau provided the visual proof of the arrival concerning longitudinal waves within thin metal films. They recommended removal of anomalies between the ports concerning emaciated ([Formula: see text] Å) Ag layers preserved on amorphous silica because of being mildly [Formula: see text]-polarized at frequency tier according to the brawny plasma frequency. Anomalies have been due to confusion over the longitudinal waves mirrored utilizing the two borders. The wavelength is connected according to this wave’s property, which is an awful lot smaller than the wave over light. The wave is outstanding only for altogether superfine films. However, metallic movies defended amorphous substrates that are intermittent within the forward levels of their evolution. This made researchers aware of the possibility on getting ready altogether few layers with a desire to discussing the promulgation about longitudinal waves up to expectation colorful each among reverberation or transport on [Formula: see text]-polarized light. The mated options via the use of generalized Riccati equation mapping method or generalized Kudryashov technique show the rule and effectiveness regarding its methods then its ability because of applying one kind of forms over nonlinear incomplete differential equations.

Dynamics of optical solitons in higher-order Sasa–Satsuma equation

Higher order and multidimensional generalizations of the nonlinear Schrödinger equation are useful in a variety of applications. A generalized (3+1)-dimensional nonlinear Sasa–Satsuma equation is studied via the improved generalized Riccati equation mapping method and its Bäcklund transformation and the extended hyperbolic function method. This nonlinear model can be use to describe the physical processes of wave propagation in optical fibers. We explicitly retrieved varieties of optical solitons like bright, dark, singular, combined bright–dark and combined singular-bright to this multidimensional generalized nonlinear model. Other solutions, such as periodic, rational, and exponential solutions of different structures, are also produced as a result of the evolution of this improved technique. Furthermore, the graphs in two and three dimensions are provided to visualize the structure of the solutions acquired by choosing certain parameter values.

Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime

The generalized Riccati equation mapping method (GREMM) is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion. Later, we use the q-homotopy analysis method combined with the Laplace transform (q-HATM) to obtain approximate solutions of the bright and dark optical solitons. The q-HATM illustrates the solutions as a rapid convergent series. In addition, to show the physical behavior of the solutions obtained by the proposed techniques, the graphical representation has been provided with some parameter values. The findings demonstrate that the proposed techniques are useful, efficient and reliable mathematical method for the extraction of soliton solutions.

Construction of optical solitons for conformable generalized model in nonlinear media

In the current analysis, the conformable generalized Kudryashov equation of pulses propagation with power non-linearity is processed. The considered higher order equation represents a generalized mathematical model of many well-known ones in nonlinear media. A variety of multiple kinks, bi-symmetry, periodic, singular, bright and dark optical solitons are extracted via the generalized Riccati equation mapping method. Basing on the Riccati differential equation, the theoretical algorithm extracts a number of empirical solutions that do not exist in the literature. The obtained results showed that the present technique is an effective and strong tool for solving nonlinear fractional partial differential equations and produces a very large number of solutions.

Two improved techniques for the perturbed nonlinear Biswas–Milovic equation and its optical solitons

In this article, abundant optical soliton solutions of the Biswas–Milovic equation with Kudryashov’s law and nonlinear perturbation terms in polarization preserving fibers is constructed by employing two improved analytical schemes, namely, the improved Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM). As a result of these improved approaches, many constraint conditions emerge that are required for the existence of soliton solutions. These solutions are the rational, exponential, trigonometric hyperbolic, and trigonometric which can be classified into the bright soliton, W-shaped soliton, dark soliton, singular soliton, periodic, and the mixed complex soliton. All solutions obtained have been verified using symbolic computations. Moreover, the dynamics of some of the achieved solutions are presented by plotting two and three-dimensional graphs.

New travelling wave solutions to (2+1)-Heisenberg ferromagnetic spin chain equation using Atangana’s conformable derivative

In this study, some new optical soliton solutions are obtained with generalized Riccati equation mapping method for space time conformable (2+1) dimensional Heisenberg ferromagnetic spin chain equation by using Atangana’s conformable derivative. We construct bright and dark solitons, combined bright-dark solitons, periodic wave solutions for this equation. 3D and 2D graphs of few obtained solutions are plotted to show some interesting physical features as these new results might be helpful in the study of nonlinear ferromagnetic materials and optical fibers. The procured results show that this is very robust method to originate abundant solutions for various fractional differential equations. All the results are verified by Maple program.