Maple Software Research Articles

SDIR Mathematical Approach with Time Delay Factor to Reveal Indonesian Students' Interest in Government Organizations and Internships: A Case in FMIPA UNM

This study is to build a time-delay SDIR model on the case of student interest in Organizations and Government Internship Programs, analyze the model and conduct a model simulation to predict the level of student interest cases in Organizations. This study is a theoretical and applied study. Analysis of the time-delay SDIR model of the case of student interest in Organizations, while the model simulation uses Maple Software. The population of the study was active students of FMIPA UNM, with a sample size of 1029 students obtained using the Sloving technique. The results of this study are a mathematical model of SDIR on the case of student interest in Organizations which is a system of differential equations. The model analysis provides a value of the free equilibrium point of the case of student interest in Organizations and a stable endemic equilibrium point. The results also found that the basic reproduction number value for cases without a solution would produce R0n = 9.912507841, which means that in social cases where one individual can influence 9-10 people in their environment to hesitate to participate in organizational activities or internship programs, but on the other hand if the case is given a solution, it will produce R0s = 0.2737372211, which means that there is no psychological spread, where each individual does not influence other individuals.

Interactions of localized wave and dynamics analysis in the new generalized stochastic fractional potential-KdV equation.

In this paper, we investigate the new generalized stochastic fractional potential-Korteweg-de Vries equation, which describes nonlinear optical solitons and photon propagation in circuits and multicomponent plasmas. Inspired by Kolmogorov-Arnold network and our earlier work, we enhance the improved bilinear neural network method by using a large number of activation functions instead of neurons. This method incorporates the concept of simulating more complicated activation functions with fewer parameters, with more diverse activation functions to generate more complex and rare analytical solutions. On this basis, constraints are introduced into the method, reducing a significant amount of computational workload. We also construct neural network architectures, such as "2-3-1," "2-2-3-1," "2-3-3-1," and "2-3-2-1" using this method. Maple software is employed to obtain many exact analytical solutions by selecting appropriate parameters, such as the superposition of double-period lump solutions, lump-rogue wave solutions, and three interaction solutions. The results show that these solutions exhibit more complex waveforms than those obtained by conventional methods, which is of great significance for the electrical systems and multicomponent fluids to which the equation is applied. This novel method shows significant advantages when applied to fractional-order equations and is expected to be increasingly widely used in the study of nonlinear partial differential equations.

A Novel Method for Analyzing Global Bifurcation and Global Hysteresis of FGM Truncated Conical Shell Structure

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

Analogical elucidation of dusty‐hybrid nanofluid flow through the microchannel: An unsteady case

AbstractThis work is the exploration of unsteady flow of dusty fluid in the horizontal microchannel when the channel plates are susceptible to linear radiation. The flow of the dusty fluid is in the existence of magnetic field. Along with dusty fluid, there are immersions of two different types of nanoparticles namely ferrous oxide and silver . The study gives the comparison of nanofluid phase with dusty fluid phase; hybrid nanofluid phase with dusty fluid phase separately. The modeled partial differential equations are non‐dimensionalized and pdsolve command of Maple software tackles these formed partial differential equations directly. Comparative study of nanofluid‐dusty fluid and hybrid nanofluid‐dusty fluid for the flow in microchannel is to elucidate the fact of heat transfer enhancement upon appropriate insertion of nanoparticles in the base fluid; especially the nanoparticles possessing paramagnetic effect for the better performance on the application of the magnetic field. Results deduce that the fluid flow and heat transmission are known to progress with time. Also, the fluid phase is known to reach the steady state earlier than the dust phase. The nanoparticle volume fraction is known to deplete the temperature in both the hybrid fluid phase and nanofluid phase. The particle concentration parameter diminishes the velocity profile, both for fluid phase and dust phase.

Innovative approache for the nonlinear atangana conformable Klein-Gordon equation unveiling traveling wave patterns

The aim of current work is to establish novel traveling wave solutions of the nonlinear Atangana conformable Klein - Gordon equation using a new extended direct algebraic technique. The Klein - Gordon equation is the relativistic state of the Schrödinger equation with a second - order time derivative and zero spin. Complex wave variable transformation is used to convert Atangana conformable nonlinear differential equation into an ordinary differential equation. Using the proposed technique based on Maple software structure, various types of solutions, such as, generalized trigonometric, generalized hyperbolic, and exponential functions, are established. When special parameteric values are considered for this method, solitary wave solutions can be obtained through other methods, such as the (G′G)-expansion method, the modified Kudryashov method, the sub-equation method, and so forth. A physical explanation is provided for the solutions under consideration to enhance comprehension of the physical phenomena resulting from the obtained solutions, provided that the physical parameters are set appropriately using 3D, 2D, and contour simulations. The results demonstrated that the new extended direct algebraic method provides a more potent mathematical tool for solving numerous more nonlinear partial differential equations with the aid of symbolic computation.

Model development and heat transfer characteristics in renewable energy systems conveying hybrid nanofluids subject to nonlinear thermal radiation

PurposeIn today’s world, the demand for energy to power industrial and domestic activities is increasing. To meet this need and enhance thermal transport, solar energy conservation can be tapped into via solar collector coating for thermal productivity. Hybrid nanofluids (HNFs), which combine nanoparticles with conventional heat transfer fluids, offer promising opportunities for improving the efficiency and sustainability of renewable energy systems. Thus, this paper explores fluid modeling application techniques to analyze and optimize heat transfer enhancement using HNFs. A model comprising solar energy radiation with nanoparticles of copper (Cu) and alumina oxide (Al2O3) suspended in water (H2O) over an extending material device is developed.Design/methodology/approachThe model is formulated using conservation laws to build relevant equations, which are then solved using the Galerkin numerical technique simulated via Maple software. The computational results are displayed in various graphs and tables to showcase the heat transfer mechanism in the system.FindingsThe results reveal the thermal-radiation-boost heat transfer phenomenon in the system. The simulations of the theoretical fluid models can help researchers understand how HNFs facilitate heat transfer in renewable energy systems.Originality/valueThe originality of this study is in exploring the heat transfer properties within renewable energy systems using HNFs under the influence of nonlinear thermal radiation.

On the hydrothermal features of MHD nanofluid flow over a differently shaped thin needle influenced by Hall current with nonlinear thermal radiation

PurposeThe investigation has appraised the problem of an incompressible laminar steady magnetohydrodynamic (MHD) nanofluid stream over three distinct slendering thin isothermal needles of paraboloid, cylindrical and cone shapes. Water as a base liquid is assumed in this flow model. The influences of the Hall current and variable sorts of magnetic forces have enriched our investigation. Energy and concentration expressions consist of thermophoresis and Brownian migration phenomena. The analysis of thermal and mass slips of the presumed model has also been performed.Design/methodology/approachA relevant transformation is implemented for the alteration of the leading partial differential equations (PDEs) to the equations with nonlinear ordinary form. Due to the strong nonlinearity of the foremost equations, the problem is solved numerically by embedding the well-known RK-4 shooting practice. The software MAPLE 2017 has been exploited in reckoning the entire computation. To enunciate the investigated upshots, some graphical diagrams have been regarded here. According to technological interest, we measured the engineering quantities like the Sherwood number, the coefficient of drag friction and the Nusselt number in tabular customs.FindingsThe obtained consequences support that Hall current intensifies fluid movement when the needle is in a cone shape, while the superior velocity is noticed for cylindrical-shaped needles. The transference of heat responds inversely along with the growths of thermal and mass slip factors.Originality/valueNo work has been performed on the flow model of radiated nanofluid over a variable-shaped thin needle under Hall current, the variable magnetic field and different slip factors.

Optical soliton solutions of the stochastic resonant nonlinear Schrödinger equation with spatio temporal and inter-modal dispersion under generalized Kudryashov's law non-linearity

This article investigates optical soliton solutions within the stochastic resonant nonlinear Schrödinger equation (SRNLSE). This equation incorporates both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD), along with multiplicative white noise and generalized Kudryashov's law non-linearity. By applying the extended auxiliary equation method, we identify a range of soliton solutions, including bright, dark, and singular solitons. Additionally, we derive solutions that take the form of Jacobi elliptic functions, Weierstrass elliptic functions, and periodic wave functions. The study provides significant insights into soliton dynamics within nonlinear optical systems affected by stochastic influences and complex dispersion interactions. Specifically, it highlights how the interplay of STD and IMD, coupled with the presence of multiplicative noise, shapes the behavior of solitons. Moreover, we delve into the effects of multiplicative noise on the exact solutions of the NLSE using the Maple software. Our analysis reveals that multiplicative noise, interpreted in the Ito sense, plays a crucial role in stabilizing the soliton solutions, particularly maintaining their stability around the zero state. This finding underscores the importance of noise in influencing the stability and dynamics of solitons in optical systems.

Проблемы поиска и преследования беспилотных летательных аппаратов с использованием теоретико-игрового подхода

Currently, the application of mathematical modeling theory is widely used in various fields and, among other things, can be used to prevent illegal actions. This study examines scenarios involving a single pursuer tracking a single evader, as well as situations involving multiple pursuers pursuing multiple evaders. The authors formulate this problem as a search and pursuit problem for four-rotor unmanned aerial vehicles (UAVs) or quadcopters. The solution to the problem is based on game theory, as it provides a mathematical framework for modeling and studying strategic interactions involving multiple decision makers. The authors consider a game where the set of strategies of the runner is the set of possible combinations of speeds and directions of his movement, and the set of strategies of the pursuer is the set of all possible permutations of the elements of his speeds. The matrix of the resulting game consists of elements that are the time of capture. A mathematical problem on the optimal assignments of searching and pursuing unmanned aerial vehicles is formulated and solved. Purpose: optimizing the effectiveness of strategies for detecting and capturing four-rotor unmanned aerial vehicles (quadcopters). Methods: methods of mathematical modeling, game theory apparatus, Hungarian method of solving the problem of assignments, decision theory, the principle of dynamic programming, the Maple package for solving examples were used. Results: In order to fulfill the conditions for solving the problem of optimal assignments, it is necessary and sufficient that it is balanced. This assignment problem can be balanced by entering the required number of fictitious boats or escaping boats. After that, it is possible to formulate and solve the dual problem of optimal appointments. The resulting game can be solved by any method of solving matrix games. In this way, it is possible to determine the policy of harassment and search between drones. Practical importance: All escaped quadcopter UAVs can be chased and successfully intercepted using the developed models. Examples of the study of mathematical models using the Maple software package are given.

Using maple software to reduce problem solving errors in teaching mathematics to future computer science teachers

This article discusses some possible uses of Maple in mathematics. Mathematics is currently in a constant state of development in the application of computer technology in education and research. The use of algebraic systems in education is still relatively small, but growing research and interest in them point to their widespread use. There are complex and abstract ideas that present an insurmountable obstacle for many students in solving basic concepts and proving many mathematical concepts. One of the most powerful computer programs that helps in learning mathematics is Maple. Maple allows you to explain classic mathematical concepts in a practical and effective way. The effectiveness of Maple is due to its ability to visualize concepts and solve complex problems that are difficult to solve manually. Maple's features help students develop new methods and better understand, learn and teach complex concepts, advanced mathematical concepts and more. Ideal for students who are developing problem solving skills and advanced development skills. Allows students to solve mathematical problems step by step. Students can practice using various integration rules.

Aviation aspects of engine oil conveying copper tiny particles embedded in a rotating disk with a binary chemical reaction

Nanofluids have significant industrial applications and motivating heat transfer characteristics for various factors that contribute to the movement of nanofluids are essential significance. The aim of the present study focused on importance of heat transfer analysis on engine oil conveying copper nano particles embedded in a porous rotating disk with potential application in aerospace technology. In this model we considered magnetohydrodynamics, non-linear thermal radiation, thermophoresis and Brownian motion. The present investigation utilized copper nanoparticle and engine oil as a base fluid. The mathematical flow equations are transformed into ordinary differential equations (ODEs) by employing suitable self-similarity variables. The resultant ordinary differential equations are solved numerically by using the Midrich technique in the Maple software. The results are calculated to measure the impact of active parameters on velocity, temperature, concentration equations are presented graphically and in tabular form. Higher values of the magnetic field parameter the velocity profile decreased while the opposite tendency we noticed on energy profile. When increasing the radiation parameter values the energy profile increased. In both cases when increasing the magnetic field and radiation parameter values the Nusselt number profile increased. The research has important implications in a number of real-world situations. In particularly, the advancement of aircraft technology has presented manufacturers with new criteria and problems for the functioning of their devices. It is essential that, in order to guarantee the secure operation of aerospace machinery, the failure mechanisms be identified and the operational durability of critical structural components be improved as quickly as possible.

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.

Analytical model of friction at low shear rates for soft materials in 3D printing.

The biological properties of silicone elastomers such as polydimethylsiloxane (PDMS) have widespread use in biomedicine for soft tissue implants, contact lenses, soft robots, and many other small medical devices, due to its exceptional biocompatibility. Additive manufacturing of soft materials still has significant challenges even with major advancements that have occurred in development of these technologies for customized medical devices and tissue engineering. The aim of this study was to develop a mathematical model of tangential stress in relation to shear stress, shear rate, 3D printing pressure and velocity, for non-Newtonian gels and fluids that are used as materials for 3D printing. This study used FENE (finitely extensible nonlinear elastic model) model, for non-Newtonian gels and fluids to define the dependences between tangential stress, velocity, and pressure, considering viscosity, shear stress and shear rates as governing factors in soft materials friction and adhesion. Experimental samples were fabricated as showcases, by SLA and FDM 3D printing technologies: elastic polymer samples with properties resembling elastic properties of PDMS and thermoplastic polyurethane (TPU) samples. Experimental 3D printing parameters were used in the developed analytical solution to analyse the relationships between governing influential factors (tangential stress, printing pressure, printing speed, shear rate and friction coefficient). Maple software was used for numerical modelling. Analytical model applied on a printed elastic polymer, at low shear rates, exhibited numerical values of tangential stress of 0.208-0.216 N m - 2 at printing velocities of 0.9 to 1.2 mm s - 1, while the coefficient of friction was as low as 0.09-0.16. These values were in accordance with experimental data in literature. Printing pressure did not significantly influence tangential stress, whereas it was slightly influenced by shear rate changes. Friction coefficient linearly increased with tangential stress. Simple analytical model of friction for elastic polymer in SLA 3D printing showed good correspondence with experimental literature data for low shear rates, thus indicating possibility to use it for prediction of printing parameters towards desired dimensional accuracy of printed objects. Further development of this analytical model should enable other shear rate regimes, as well as additional soft materials and printing parameters.

Lax integrability and nonlinear dispersive wave phenomenon for the (3 + 1) dimensional Kudryashov–Sinelshchikov equation

A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear Bäcklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota’s bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes.

SIMPLIFYING IRRATIONAL EXPRESSIONS AND APPLICATION OF THE MAPLE COMPUTER PACKAGE

The article is devoted to the presentation of one of the main concepts of the school algebra course: “traditional” and “non-traditional” approaches of simplifying irrational expressions, as well as a description of the simplification of irrational expressions using the computer package MAPLE. The article examines irrational expressions, containing multiple roots of the 2nd degree, and presents both “traditional” approaches of simplifying them, using full quadratic division and formula, and “non-traditional” approach of noting-squaring. Specific examples are considered in order to more clearly describe the application of each of the above approaches, point out their differences, features, advantages and provide clear methodological instructions. In addition to the above approaches for simplifying irrational expressions, the article also describes the implementation of simplifying irrational expressions using the computer package MAPLE: the simplification of an irrational expression given in the article is also implemented in the MAPLE environment, bringing methodological instructions, as a result of which it becomes possible to instantly simplify an irrational expression.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Investigation of Space-Time Dynamics of Akbota Equation using Sardar Sub-Equation and Khater Methods: Unveiling Bifurcation and Chaotic Structure

This paper focuses on obtaining exact solutions of nonlinear Akbota equation through the application of the modified Khater method and Sardar sub-equation method. Renowned as one of the latest and precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The equation is crucial in the study of optical solitons, which are stable pulses of light that maintain their shape over long distances. The Akbota equation helps in understanding the behavior and stability of these solitons. The governing equation undergoes transformation into an ordinary differential equation through a well-suited wave transformation. This analytical simplification paves the way for the derivation of trigonometric, hyperbolic, and rational solutions through the proposed methods. To illuminate the physical behavior of the model, the study presents graphical plots of the selected solutions of Khater and Sardar sub-equation method. This visual representation, achieved by selecting appropriate values for arbitrary parameters, enhances the understanding of the system’s dynamics. All calculations in this study are meticulously conducted using the Mathematica and Maple software, ensuring accuracy and reliability in the analysis of the obtained solution. Furthermore we investigate the sensitivity analysis of the dynamical system.

Statistical analysis of topological indices in linear phenylenes for predicting physicochemical properties using algorithms

QSPR mathematically links physicochemical properties with the structure of a molecule. The physicochemical properties of chemical molecules can be predicted using topological indices. It is an effective method for eliminating costly and time-consuming laboratory tests. We established a QSPR between mev-degree and mve-degree-based indices and the physical properties of benzenoid hydrocarbons. To compute these indices, we designed a program using Maple software and the correlation between indices and physical properties was developed using the SPSS software. Our study reveals that the mve-degree-based sum-connectivity (χmve)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\chi ^{mve})$$\\end{document} and atom bond connectivity (ABCmve\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ABC^{mve}$$\\end{document}) index, mev-degree-based Randić (Rmev\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{mev}$$\\end{document}) and Zagreb (Mmev\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{mev}$$\\end{document}) index are the three most significant parameters and have good prediction ability for the physicochemical properties. We examined that Rmev\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{mev}$$\\end{document} predicts the molar refractivity and boiling point, χmve\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\chi }^{mve}$$\\end{document} predicts the LogP and enthalpy, ABCmve\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ABC^{mve}$$\\end{document} predicts the molecular weight, Mmev\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{mev}$$\\end{document} predicts the Gibb’s energy, Pie-electron energy and Henry’s law. Moreover, we computed the indices for the linear [n]-phenylen.

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Abstract In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.