Exact soliton solutions and the significance of time-dependent coefficients in the Boussinesq equation: theory and application in mathematical physics.

This article effectively establishes the exact soliton solutions for the Boussinesq model, characterized by time-dependent coefficients, employing the advanced modified simple equation, generalized Kudryashov and modified sine-Gordon expansion methods. The adaptive applicability of the Boussinesq system  to coastal dynamics, fluid behavior, and wave propagation enriches interdisciplinary research across hydrodynamics and oceanography. The solutions of the system obtained through these significant techniques make a path to understanding nonlinear phenomena in various fields, surpassing traditional barriers and further motivating research and application. Significant impacts of the coefficients of the equation, wave velocity, and related parameters are evident in the profiles of soliton-shaped waves in both 3D and 2D configurations when all these factors are treated as variables, which are not seen in the case for constant coefficients. This study enhances the understanding of the significant role played by nonlinear evolution equations with time-dependent coefficients through careful dynamic explanations and detailed analyses. This revelation opens up an interesting and challenging field of study, with promising insights that resonate across diverse scientific disciplines.

A study of the wave dynamics of the space-time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach.

The space-time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions.

Compatible extension of the (G'/G)-expansion approach for equations with conformable derivative.

In this study, the compatible extensions of the (G'/G)-expansion approach and the generalized (G'/G)-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique.

Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq-Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

Numerical and analytical solutions of new Blasius equation for turbulent flow.

The Blasius equation for laminar flow comes from the Prandtl boundary layer equations. In this article, we establish a new and generic Blasius equation for turbulent flow derived from the turbulent boundary layer equation that can be used for turbulent as well as laminar flow. The analytical and numerical solutions have been investigated under specific conditions to the developed new Blasius equation. The analytical and numerical results have been compared through tables and graphs to validate the established model. In fluid dynamics, analytical solutions to complicated systems are tedious and time-consuming. Changing one or more constraints can introduce new challenges. In this case, symbolic computation software provides an easier and more flexible solution for fluid dynamical systems, even if boundary conditions are adjusted to explain reality. Therefore, the MATLAB code is used to investigate the new third-order Blasius equation. The comparison and graphical representations demonstrate that the achieved results are encouraging.

Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation.

The low-pass nonlinear electrical transmission lines and the Cahn-Allen equation are important nonlinear model equations to figure out different tangible systems, namely, electrical engineering, fluid dynamics etc. The contrivance of this study is to introduce advanced Bernoulli sub-equation function method to search for stable and effective solitary solutions of the described wave equations. Stable solitary solutions are reported as an integration of exponential functions, hyperbolic functions, etc., and the graphical implications for specific values of the corresponding parameters are explained in the solutions obtained in order to uncover the inmost structure of the tangible phenomena. It is establish that the IBSEF method is reliable, contented and might be used in further works to found ample novel soliton solutions for other types of NLEEs arising in physical science and engineering.

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis.

The Estevez-Mansfield-Clarkson (EMC) equation and the (2+1)-dimensional Riemann wave (RW) equation are important mathematical models in nonlinear science, engineering and mathematical physics which have remarkable applications in the field of plasma physics, fluid dynamics, optics, image processing etc. Generally, through the sine-Gordon expansion (SGE) method only the lower-dimensional nonlinear evolution equations (NLEEs) are examined. However, the method has not yet been extended of finding solutions to the higher-dimensional NLEEs. In this article, the SGE method has been developed to rummage the higher-dimensional NLEEs and established steady soliton solutions to the earlier stated NLEEs by putting in use the extended higher-dimensional sine-Gordon expansion method. Scores of soliton solutions are figure out which confirms the compatibility of the extended SGE method. The solutions are analyzed for both lower and higher-dimensional nonlinear evolution equations through sketching graphs for alternative values of the associated parameters. From the figures it is notable to perceive that the characteristic of the solutions depend upon the choice of the parameters. This study might play an impactful role in analyzing higher-dimensional NLEEs through the extended SGE approach.

Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics.

The objective of this article is to construct new and further general analytical wave solutions to some nonlinear evolution equations of fractional order in the sense of the modified Riemann-Liouville derivative relating to mathematical physics, namely, the space-time fractional Fokas equation, the time fractional nonlinear model equation and the space-time fractional (2 + 1)-dimensional breaking soliton equation by exerting a rather new mechanism ( G ' / G , 1 / G ) -expansion method. We use the fractional complex transformation and associate the fractional differential equations to the solvable integer order differential equations. A comprehensive class of new and broad-ranging exact traveling and solitary wave solutions are revealed in terms of trigonometric, rational and hyperbolic functions. The attained wave solutions are sketched graphically by using Mathematica and make a comparison to the results attained by the presented technique with other techniques in a comprehensive manner. It is notable that the method can be considered as a reduction of the reputed ( G ' / G ) -expansion method commenced by Wang et al. It is noticeable that, the two variable ( G ' / G , 1 / G ) -expansion method appears to be more reliable, straightforward, computerized and user-friendly.

What Has Been Learned from Magnetic Resonance Imaging Examination of the Injured Human Spinal Cord: A Canadian Perspective.

Magnetic resonance imaging (MRI) has transformed the way surgeons and researchers study and treat spinal cord injury. In this narrative review, we explore the historical context of imaging the human spinal cord and describe how MRI has evolved from providing the first visualization of the human spinal cord in the 1980s to a remarkable set of imaging tools today. The article focuses in particular on the role of Canadian researchers to this field. We begin by outlining the clinical context of traumatic injury to the human spinal cord and describe why current MRI standards fall short when it comes to treating this disabling condition. Parts 2 and 3 of this work explore an exciting and dramatic shift in the use of MRI technology to aid in our understanding and treatment of traumatic injury to the spinal cord. We explore the use of functional imaging (part 2) and structural imaging (part 3) and explore how these techniques have evolved, how they are used, and the challenges that we face for continued refinement and application to patients who live with the neurological and functional deficits caused by injury to the delicate spinal cord.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.

In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

Solitary wave solutions of the fourth order Boussinesq equation through the exp(-Ф(η))-expansion method.

ABSTRACT: The exp(-Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(-Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(-Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering. MATHEMATICS SUBJECT CLASSIFICATIONS: 35C07; 35C08; 35P99.

Study of analytical method to seek for exact solutions of variant Boussinesq equations.

ABSTRACT: In this paper, we have been acquired the soliton solutions of the Variant Boussinesq equations. Primarily, we have used the enhanced (G'/G)-expansion method to find exact solutions of Variant Boussinesq equations. Then, we attain some exact solutions including soliton solutions, hyperbolic and trigonometric function solutions of this equation. MATHEMATICS SUBJECT CLASSIFICATION: 35 K99; 35P05; 35P99.

Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation.

ABSTRACT: Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. MATHEMATICS SUBJECT CLASSIFICATION: 35C07; 35C08; 35P99.

Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G'/G)-expansion method.

ABSTRACT: Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G'/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G'/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering. PACS: 05.45.Yv, 02.30.Jr, 02.30.Ik.

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations.

ABSTRACT: In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions-hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters. PACS NUMBERS: 02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg.

A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations.

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.