Form Of Rational Function Research Articles

Exploring Kink Solitons in the Context of Klein–Gordon Equations via the Extended Direct Algebraic Method

This work employs the Extended Direct Algebraic Method (EDAM) to solve quadratic and cubic nonlinear Klein–Gordon Equations (KGEs), which are standard models in particle and quantum physics that describe the dynamics of scaler particles with spin zero in the framework of Einstein’s theory of relativity. By applying variables-based wave transformations, the targeted KGEs are converted into Nonlinear Ordinary Differential Equations (NODEs). The resultant NODEs are subsequently reduced to a set of nonlinear algebraic equations through the assumption of series-based solutions for them. New families of soliton solutions are obtained in the form of hyperbolic, trigonometric, exponential and rational functions when these systems are solved using Maple. A few soliton solutions are considered for certain values of the given parameters with the help of contour and 3D plots, which indicate that the solitons exist in the form of dark kink, hump kink, lump-like kink, bright kink and cuspon kink solitons. These soliton solutions are relevant to actual physics, for instance, in the context of particle physics and theories of quantum fields. These solutions are useful also for the enhancement of our understanding of the basic particle interactions and wave dynamics at all levels of physics, including but not limited to cosmology, compact matter physics and nonlinear optics.

A quantum-corrected Peng‒Robinson equation of state for helium-4 from 3 K to 50 K considering quantum swelling effects through the Feynman‒Hibbs correction of the EXP-6 potential

Accurate and efficient prediction of the properties of helium-4 (He-4) via an equation of state (EOS) is a prerequisite for evaluating liquid helium-4 (LHe-4) storage technology. However, the performance of the Peng–Robinson (PR) EOS in predicting the density of LHe-4 and vapour helium-4 (VHe-4) deteriorates within the thermodynamic ranges of the LHe-4 tank: 3–50 K and 60–600 kPa. To modify the PR EOS, we establish first-order and second-order Feynman–Hibbs (FH)-corrected EXP-6 potentials and propose a reduced effective particle diameter (REPD) correlation with four parameters to consider the quantum swelling effects of LHe-4. On the basis of the rational function form of the REPD correlation, we introduce a quantum-corrected covolume term to develop a regression model for the FH-corrected Peng–Robinson (FH-PR) EOS. Moreover, to improve the effectiveness of regression near the saturation curve, we propose a hypothetical boundary consisting of the saturation curve from 3 K to the critical temperature and a virtual saturation curve from the critical pressure to 600 kPa. The results indicate that the FH − PR EOS shows satisfactory engineering application performance in predicting the density of He-4 within the studied range. Under verification conditions, the average absolute relative deviation (AARD) of the density determined via the FH − PR EOS ranges from 0.72 % to 1.77 %, and the maximum relative deviation (MRD) ranges from 2.17 % to 5.62 %. Under test conditions, the AARD of the density ranged from 1.06 % to 1.71 %, and the MRD ranged from 3.77 % to 7.38 %.

Diverse variety of exact solutions for some nonlinear models via the [formula omitted]-expansion method

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the (G′G)-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

Analytic wave solutions to the beta-time fractional modified equal width equation based on two efficient approaches

By using the two distinct methods known as the exp(-ϕ(η))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\phi (\\eta ))$$\\end{document} and the Expa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Exp_a$$\\end{document}-function methods, various forms of soliton solutions of the modified equal width wave equation (MEWE) with beta time derivative (BTD) are produced in this study. This model is used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The obtained solutions are in the form of rational, trigonometry and hyperbolic trigonometry functions. Using Mathematica software, the resulting solitons are validated. Graphs are also used at the conclusion to explain the findings. These soliton solutions imply that these two methods are more dependable, simple and efficient than other methods. The findings can be used to explain how studious structures and other comparable non-linear physical structures are substantially understood. The obtained results are very helpful in the fields of optics, kinetics solid-state physics and hydro-magnetic waves in cold plasma.

Exploring solitary wave solutions to the simplified modified camassa-holm equation through a couple sophisticated analytical approaches

The simplified modified Camassa-Holm equation, a developed Korteweg-de Vries equation, is an integrable nonlinear dispersive water wave equation with potential applications in diverse fields such as shallow water waves, liquid drop patterning, water surface waves in channels, tsunamis in coastal regions, ocean engineering, etc. In this study, we derived solitary wave solutions to the described equation, employing two reliable approaches: the enhanced modified simple equation and the extended Kudryashov methods. The enhanced modified simple equation technique yields solutions with variable coefficients through a generalized wave transformation. On the other hand, the extended Kudryashov method exploits the solutions of the Riccati and Bernoulli differential equations. The determined results include diverse soliton solutions expressed in the form of trigonometric, hyperbolic, and rational functions. Notably, for specific parameter values, these solutions exhibit wave of distinct shapes, including kink, smooth kink, flat kink, periodic, singular periodic, one-sided kink, etc. To apparently explain the physical behavior and the impact of the fractional parameter in the solutions, we plotted three- and two-dimensional graphs. The obtained solutions have potential applications in oceanography, spanning offshore rigs, ocean gravity waves, and wave energy.

Exact wave solutions of new generalized Bogoyavlensky–Konopelchenko model in fluid mechanics

In this paper, we examine a novel generalized [Formula: see text]-dimensional Bogoyavlensky–Konopelchenko (gBK) model analytically via application of five mathematical methods, namely called Extended Simple Equation (ESE) method, Modified Extended Auxiliary equation mapping (MEAEM) method, [Formula: see text]-expansion method, [Formula: see text]-expansion method and modified F-expansion method, respectively, to obtain different types solitary wave solutions in the form of trigonometric, hyperbolic, rational and exponential functions as compared to solutions exist in [S. T. Chen and W. X. Ma, Math. China 13 (2018) 525; A. Sonia, A. Jamshad, S.-U. Rehman and A. Ali, Int. J. Appl. Comput. Math. 9 (2023)]. The research focuses on physical systems, physiology, mathematical applications, engineering fields, and the chemical processes of species in the porous nanoparticles, examining the impact of various parameters on the nonlinear model. The graphs of some solutions are plotted in the form of two-dimensional and three-dimensional by assigning the particular values to the paraments with the assistance of mathematica software which illustrated the physical behavior of the concern model. The derived solution has numerous applications in fluid dynamics and the interaction of Riemann waves and long waves.

Soliton solutions to generalized (3 + 1)-dimensional shallow water-like equation using the (ϕ'/ϕ,1/ϕ)-expansion method

The (3 + 1)-dimensional generalized shallow water equation is a significant mathematical framework for analyzing the dynamic behavior of waves in ocean physics. The purpose of this article is to investigate some more generic soliton solutions of the generalized shallow water-like model in three dimensions. The investigation is conducted utilizing the sophisticated mathematical methodology known as the double variables ( ϕ ′ / ϕ , 1 / ϕ ) expansion technique. With this approach, we produce new propagating wave solutions for this model in the form of hyperbolic, trigonometric, and rational functions. In addition, we offer two- and three-dimensional graphical representations to help visualize the intricate physical phenomena of the system. We have constructed many soliton solutions, such as kink shape soliton solutions, anti-bell shape solutions, single periodic solutions, singular soliton solutions, and anti-kink shape solutions for different values of the free parameters involved in the obtained solutions. These graphical representations are predicated on certain parameter selections, which facilitate the understanding of the complicated general behavior for this model. Through the presentation of new findings in the field of soliton solutions for the aforementioned equation, this paper offers fresh perspectives and highlights hitherto overlooked aspects of this fascinating mathematical challenge. The paper illuminates new results on soliton solutions with different geometrical structures for the given equation, revealing hitherto overlooked facets of this intriguing mathematical challenge.

Simplified implicit finite-difference method of spatial derivative using explicit schemes with optimized constant coefficients based on L1 norm

The explicit finite-difference (EFD) method is widely used in numerical simulation of seismic wave propagation to approximate spatial derivatives. However, the traditional and optimized high-order EFD methods suffer from the saturation effect, which seriously restricts the improvement of numerical accuracy. In contrast, the implicit FD (IFD) method approximates the spatial derivatives in the form of rational functions and thus can obtain much higher numerical accuracy with relatively low orders; however, its computational cost is expensive due to the need to invert a multidiagonal matrix. We derive an explicit strategy for the IFD method to reduce the computational cost by constructing the IFD method with the discrete Fourier matrix; then, we transform the inversion of the multidiagonal matrix into an explicit matrix multiplication; next, we construct an objective function based on the [Formula: see text] norm to reduce approximation error of the IFD method. This explicit strategy of the IFD method can avoid inverting the multidiagonal matrix, thus improving the computational efficiency. This constant coefficient optimization method reduces the approximation error in the medium-wavenumber range at the cost of tolerable deviation (smaller than 0.0001) in the low-wavenumber range. For the 2D Marmousi model, the root-mean-square error of the numerical results obtained by this method is one-fifth that of the traditional IFD method with the same order (i.e., 5/3) and one-third that of the traditional EFD method with much higher orders (i.e., 72). The significant reduction of numerical error makes the developed method promising for numerical simulation in large-scale models, especially for long-time simulations.

Exact solutions of the (3+1)-generalized fractional nonlinear wave equation with gas bubbles

In this manuscript, we implement the travelling wave solutions of the fractional (3+1) generalized computational nonlinear wave equation with gas bubbles via application of five mathematical methods. Liquids with gas bubbles primarily arise in various applications like science, engineering, and mathematical physics. The obtained solitary waves solutions have fruitful applications in engineering, science, life, nature and physics. Several novel soliton solutions of concerned model are established in the form of hyperbolic, trigonometric, exponential and rational functions. To handle all calculations and verification of obtained results, computational software Mathematica 12.1 is used. For the demonstration of the physical behaviour of concern model, some solutions are plotted graphical in 2-dimensional and 3-dimensional by imparting specific values to the parameters under constrain conditions. Finally, we intrigue both two and three dimensional to explain the physical behavior of the model.

Solitary waves of the generalized Zakharov equations via integration algorithms

<abstract><p>In many applications, the investigation of traveling wave solutions is essential in obtaining an accurate description of the dynamical behavior of most physical phenomena. The exact solutions to nonlinear equations can provide more physical descriptions and insightful details for many problems of practical interest. This paper focuses on investigating the solitary wave solutions of the generalized Zakharov equations (GZEs) by using four integration algorithms, namely, the modified $ (g'/g^{2}) $-expansion method, the modified $ (g') $-expansion method, the generalized simple ($ w/g $)-expansion method, and the addendum to Kudryashov's method. The GZEs have been widely used to describe the propagation of Langmuir waves in the field of plasma physics. The efficiency and simplicity of these methods are evaluated based on their application to GZEs, which have yielded multiple new optical solitary wave solutions in the form of rational, trigonometric, and hyperbolic functions. By using a suitable wave transformation, the coupled nonlinear partial differential equations are converted into ordinary differential equations. The derived optical solutions are graphically depicted in $ 2 $D and $ 3 $D plots for some specific parameter values. The traveling wave solutions discovered in the current study constitute just one example of the desired solutions that may enable the exploration of the physical properties of many complex systems and could also contribute greatly to improving our understanding of many interesting natural phenomena that arise in different applications, including plasma physics, fluid mechanics, protein chemistry, wave propagation, and optical fibers.</p></abstract>

Probing the diversity of soliton phenomena within conformable Estevez-Mansfield-Clarkson equation in shallow water

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water.

Modeling of dynamic characteristics for hydraulic drive under nonstationary load

The methodology for estimating the transient response and frequency characteristics for volumetric hydraulic drive under nonstationary load is proposed. The methodology is based on obtaining the transfer function for the speed of movement for the drive output link according to the load. To obtain the transfer function, the nonlinear mathematical model of translational hydraulic drive has been compiled, including the motion equations for the hydraulic cylinder piston, the equations of continuity in the pressure and drain cavities, taking into account compressibility of the working fluid. Hydraulic losses are described by quadratic dependence. In addition to the generally arbitrary load acting on the hydraulic cylinder rod, the resistance force proportional to the speed of movement of the hydraulic cylinder piston is also considered. By linearization of the mathematical model, the transfer function for the speed of movement on load is established in the form of fractional rational function of the third order. The dependences for calculating the coefficients of the transfer function is determined. The example of calculating the transient response, amplitude and phase frequency characteristics is given. The presented results can be used in the design and modernization of technological equipment with hydraulic drives.

Kink soliton solution of integrable Kairat-X equation via two integration algorithms

<p>In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.</p>

Exploration of nonlinear traveling wave phenomena in quintic conformable Benney-Lin equation within a liquid film

<abstract><p>In this article, we use the modified extended direct algebraic method (mEDAM) to explore and analyze the traveling wave phenomena embedded in the quintic conformable Benney-Lin equation (CBLE) that regulates liquid film dynamics. The proposed transformation-based approach developed for nonlinear partial differential equations (PDEs) and fractional PDEs (FPDEs), efficiently produces a plethora of traveling wave solutions for the targeted CBLE, capturing the system's nuanced dynamics. The methodically determined traveling wave solutions are in the form of rational, exponential, hyperbolic and trigonometric functions which include periodic waves, bell-shaped kink waves and signal and double shock waves. To accurately depict the wave phenomena linked to these solutions, we generate 2D, 3D, and contour graphs. These visualizations not only improve understanding of the CBLE model's dynamics, but also provide a detailed way to examine its behavior. Moreover, the use of the proposed techniques contributes to a better understanding of the other FPDEs' distinct characteristics, enhancing our comprehension of their underpinning dynamics.</p></abstract>

Gene Transfer-Based Phylogenetics: Analytical Expressions and Additivity via Birth-Death Theory.

The genomic era has opened up vast opportunities in molecular systematics, one of which is deciphering the evolutionary history in fine detail. Under this mass of data, analyzing the point mutations of standard markers is often too crude and slow for fine-scale phylogenetics. Nevertheless, genome dynamics (GD) events provide alternative, often richer information. The synteny index (SI) between a pair of genomes combines gene order and gene content information, allowing the comparison of genomes of unequal gene content, together with order considerations of their common genes. Recently, genome dynamics has been modeled as a continuous-time Markov process, and gene distance in the genome as a birth-death-immigration process. Nevertheless, due to complexities arising in this setting, no precise and provably consistent estimators could be derived, resulting in heuristic solutions. Here, we extend this modeling approach by using techniques from birth-death theory to derive explicit expressions of the system's probabilistic dynamics in the form of rational functions of the model parameters. This, in turn, allows us to infer analytically accurate distances between organisms based on their SI. Subsequently, we establish additivity of this estimated evolutionary distance (a desirable property yielding phylogenetic consistency). Applying the new measure in simulation studies shows that it provides accurate results in realistic settings and even under model extensions such as gene gain/loss or over a tree structure. In the real-data realm, we applied the new formulation to unique data structure that we constructed-the ordered orthology DB-based on a new version of the EggNOG database, to construct a tree with more than 4.5K taxa. To the best of our knowledge, this is the largest gene-order-based tree constructed and it overcomes shortcomings found in previous approaches. Constructing a GD-based tree allows to confirm and contrast findings based on other phylogenetic approaches, as we show.

New traveling wave solutions, phase portrait and chaotic pattern for the stochastic modified Korteweg–de Vries equation

This article mainly studies the new traveling wave solutions of the stochastic modified Korteweg–de Vries equation with multiplicative noise. The traveling wave solutions in the form of hyperbolic function, trigonometric function, rational function and Jacobi elliptic function are obtained by transforming the modified Korteweg–de Vries equation into an ordinary differential equation through traveling wave transformation, and combining it with the complete discriminant system of polynomials. This article fully applies visual analysis technology and provides graphical representations of the obtained solutions in multiple dimensions such as phase maps, sensitivity maps, 3D maps and 2D maps, providing an intuitive and convenient channel for people to further understand the physical characteristics of the solutions for the stochastic modified Korteweg–de Vries equation. This article provides a rich graphical analysis of the solutions to the stochastic modified Korteweg–de Vries equation from multiple perspectives, and provides a fast and efficient solution method for the exact solution of the equation. The research method in this paper can also be used to study nonlinear partial differential equation in other fields.

Abundant optical solutions for the Sasa-Satsuma equation with M-truncated derivative

Here, we look at the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD). The analytical solutions in the form of trigonometric, hyperbolic, elliptic, and rational functions are constructed using the Jacobi elliptic function and generalizing Riccati equation mapping methods. Because the Sasa–Satsuma equation is applied to explain the propagation of femtosecond pulses in optical fibers, the acquired solutions can be employed to explain a wide range of important physical phenomena. Moreover, we apply the MATLAB tool to generate a series of graphs to address the effect of the M-truncated derivative on the exact solution of the SSE-MTD.

The Analytical Fractional Solutions for Coupled Fokas System in Fiber Optics Using Different Methods

The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend F-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is explained by the Fokas system, the derived solutions may be utilized to analyze a broad range of important physical processes. In order to comprehend the impacts of MTD on the solutions, the dynamic behavior of the various generated solutions are shown using 2D and 3D figures.

OPTICAL SOLITARY WAVES AND SOLITON SOLUTIONS OF THE (3 + 1)-DIMENSIONAL GENERALIZED KADOMTSEV–PETVIASHVILI–BENJAMIN–BONA–MAHONY EQUATION

In this investigation, an appropriate traveling wave transformation has been employed to analyze the fourth-order nonlinear (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation for an offshore structure. In addition to being integrable and having constant coefficients, the examined model represents fluid flow in the context of an offshore structure. The aforesaid nonlinear system is subjected to the initial application of two trustworthy and dependable approaches, namely the improved Bernoulli sub-equation function method and the modified extended-function method. Investigating and obtaining certain explicitly exact traveling waves, periodic waves, and soliton solutions is the major objective. The generated solutions take the form of trigonometric hyperbolic functions, exponential functions, rational functions, and multiple forms of trigonometric functions. The proposed solutions are both novel and important in that they provide light on the relevant aspects of the physical phenomena. The characteristics of the solutions have been displayed in a variety of figures, including two- and three-dimensional ones, for the best visual optical discernment.

The Extended Direct Algebraic Method for Extracting Analytical Solitons Solutions to the Cubic Nonlinear Schrödinger Equation Involving Beta Derivatives in Space and Time

In the field of nonlinear optics, quantum mechanics, condensed matter physics, and wave propagation in rigid and other nonlinear instability phenomena, the nonlinear Schrödinger equation has significant applications. In this study, the soliton solutions of the space-time fractional cubic nonlinear Schrödinger equation with Kerr law nonlinearity are investigated using an extended direct algebraic method. The solutions are found in the form of hyperbolic, trigonometric, and rational functions. Among the established solutions, some exhibit wide spectral and typical characteristics, while others are standard. Various types of well-known solitons, including kink-shape, periodic, V-shape, and singular kink-shape solitons, have been extracted here. To gain insight into the internal formation of these phenomena, the obtained solutions have been depicted in two- and three-dimensional graphs with different parameter values. The obtained solitons can be employed to explain many complicated phenomena associated with this model.