Maple Software Research Articles

Dynamical property of interaction solutions to the Chafee-Infante equation via NMSE method

In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

STABILITY OF THE DYNAMIC MODEL OF SVPR SEXUAL VIOLENCE CASES

In this study, the stability of the equilibrium points of the Susceptible Violent Punishment Recovered (SVPR) model in the spread of sexual violence cases is analyzed. The model assumes that sexual violence can spread similarly to the transmission of infectious diseases. The constructed model is a system of nonlinear differential equations. The model has two equilibrium points: the equilibrium point with sexual violence-free and the endemic equilibrium point with sexual violence. Stability analysis is then carried out for both fixed points, indicating that the violence-free equilibrium is asymptotically stable if the basic reproduction number . Meanwhile, the endemic equilibrium is asymptotically stable if the condition and four other conditions are satisfied. Numerical simulations are required to observe the implementation of the model using MAPLE software. As for this research, we conclude a model that shows the spread of sexual violence is not widespread because it was found that .

Study of the payload extraction trajectory heavy class carrier rocket

Relevance. As the weight and complexity of the payload that needs to be launched into orbit increases, the relevance of rational trajectory selection to ensure maximum efficiency and minimum costs for delivering the payload to a given orbit increases. Rational choice of the trajectory of a heavy-class launch vehicle has a number of important practical applications. Firstly, it allows you to increase the payload capacity of the launch vehicle and reduce the cost of delivering payload to the target orbit. This is especially important in the context of the development of the space industry, when more and more companies and organizations are showing interest in launching their own satellites and other spacecraft in conditions of fierce economic competition. Choosing a rational trajectory for launching a payload into orbit will significantly reduce the cost of launches and make them available to a wider range of potential customers. Secondly, the choice of launch vehicle trajectory parameters is important for ensuring safety and minimizing risks during spacecraft launches. Thanks to the rational choice of trajectory, it is possible to reduce adverse impacts on the environment and eliminate the possibility of emergency situations associated with loss of control over the flight of the launch vehicle. Rational selection of launch vehicle trajectory parameters is a complex task that requires comprehensive research and consideration of various factors, such as aerodynamic parameters of the atmosphere, mass and characteristics of the payload (spacecraft), engine operating parameters, characteristics of the target orbit, features of the launch of the launch vehicle and many other factors. A more thorough and systematic study of the influence of these parameters will significantly improve the efficiency and reliability of launching spacecraft into orbit. Thus, the choice of rational parameters for the launch vehicle trajectory is a relevant and important topic for scientific research. Increasing the rocket's payload capacity, reducing the cost of delivering a spacecraft to a given orbit, and ensuring launch safety are tasks that depend on the chosen shape and parameters of the rocket's trajectory. Such research has important practical significance and can become the basis for the development of new technologies and methods in the space industry. The purpose of the study is to study and select rational parameters for the trajectory of a heavy-class launch vehicle when launching a payload. The main task is to determine the flight path parameters that will allow achieving maximum efficiency and accuracy in delivering the payload to a given orbit. To achieve the goal of the study, an analysis of various factors influencing the launch parameters of the spacecraft is required, such as the structural and aerodynamic characteristics of the rocket, the influence of aerodynamic factors and the Earth’s gravitational field on the flight path. Taking these factors into account, numerical calculations were carried out on the basis of a system of differential equations of motion using a computer program created in the MAPLE software package. Based on calculations, modeling of the shape and parameters of the launch vehicle flight path was carried out. Research results. During the study, the rational parameters of the trajectory of a heavy-class launch vehicle were selected. Calculations were carried out using numerical modeling of the parameters of payload launch trajectories, and an analysis of the resulting trajectories was carried out. Minimizing the rocket's flight time was identified as the main criterion for the rational choice of trajectory, which allows increasing launch efficiency and saving energy resources. An increase in payload mass and minimization of fuel consumption were adopted as additional criteria. Conclusion. The procedure for choosing rational parameters for the trajectory of a heavy-class launch vehicle proposed in this work will improve the delivery accuracy and reliability of spacecraft launches at the stage of ballistic analysis when designing rockets. The results of the study have practical significance for the development of future heavy-duty launch vehicle missions and improving the efficiency of space launches.

Travelling wave solutions and conservation laws of the (2+1)-dimensional new generalized Korteweg–de Vries equation

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

Darcy-Forchheimer MHD micropolar water based hybrid nanofluid flow, heat and mass transfer features past on stretching/shrinking surface with slip and radiation effects

Hybrid nanofluids (HNF) play a vital role in enhancing the heat transfer characteristics of all types of traditional fluids, both in industrial and experimental applications. In this regard, the laminar two-dimensional (2D) boundary layer magnetohydrodynamic (MHD) Darcy-Forchheimer flow, heat transfer, and mass transfers characteristics of Cu–MoS2/micropolar water-based hybrid nanofluid have been considered over a stretching/shrinking surface. The thermal radiation and partial slip effects are considered in a porous medium. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The equations are numerically solved using the shooting method in Maple software, and dual solutions are obtained for different ranges of the applied parameters. The effects of the different physical parameters and nanoparticle volume fractions on velocity, microrotation, temperature, and concentration profiles along with skin friction, couple stress, Nusselt and Sherwood numbers are examined. The main findings of this study show that the velocity profiles decrease with an increase in the suction, the Darcy-Forchheimer number, velocity slip, magnetic and micromaterial parameters, while oppositely, it increases with an increase in nanoparticle volume fractions. Moreover, an increase in the magnetic field, nanoparticle volume fractions, and thermal radiation increases the temperature profiles, while an increase in the Prandtl number, suction, and thermal slip parameters decreases it. An increase in nanoparticle volume fractions decreases skin friction, the couple stress coefficient, and the Nusselt number, but increases the Sherwood number with the variation of suction.

Dynamic Feedback Linearization of Control Systems with Symmetry

Control systems of interest are often invariant under Lie groups of transformations. For such control systems, a geometric framework based on Lie symmetry is formulated, and from this a sufficient condition for dynamic feedback linearizability obtained. Additionally, a systematic procedure for obtaining all the smooth, generic system trajectories is shown to follow from the theory. Besides smoothness and the existence of symmetry, no further assumption is made on the local form of a control system, which is therefore permitted to be fully nonlinear and time varying. Likewise, no constraints are imposed on the local form of the dynamic compensator. Particular attention is given to the consideration of geometric (coordinate independent) structures associated to control systems with symmetry. To show how the theory is applied in practice we work through illustrative examples of control systems, including the vertical take-off and landing system, demonstrating the significant role that Lie symmetry plays in dynamic feedback linearization. Besides these, a number of more elementary pedagogical examples are discussed as an aid to reading the paper. The constructions have been automated in the Maple package DifferentialGeometry.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Analysis of Fractional Advection Equation with Improved Homotopy Analysis Method

The enhanced homotopy analysis method was employed to solve the nonlinear time-fractional advection equation, resulting in the derivation of a series of solutions. The objective of this study is to minimize the absolute error by identifying the ideal value for a certain parameter, indicated as h, by the utilization of the residual error (RE) function associated with that parameter. The 3-dimensional graphs illustrating the absolute discrepancies between the solutions obtained from the exact approach and the modified homotopy analysis method have been made using the MAPLE software. Hence, the proposed approach demonstrates efficacy and appropriateness in tackling fractional partial differential equations.

Arithmetic of D-algebraic functions

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.

A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation

It is true that my previous series of papers about the Riemann Hypothesis has raised lots of discussions or the concerns among the mathematics society. Indeed, some professionals may think that the Riemann Zeta function at s = 1 or ∑_(n=1)^∞▒1/n is actually divergent and may tend to an infinity, how it can be acted as a denominator in a proof or something like 1/∞ ? This author’s idea is there may be no need to evaluate such kind of summation as one may consider the sum just as an (imaginary) constant (but NOT the complex number). Or we may only consider such summation as a kind of improper integrals with a black-boxed answer. In other words, the true final value of the summation ∑_(n=1)^∞▒1/n is NOT an concerning serious issue. The function ∑_(n=1)^∞▒1/n^s is in practice a smooth function and equals to log N or the logarithmic function for s = 1 or have the infinite differentiable properties except n = 0, that is why the commercial software Maple Soft can compute the Zeta function’s associated Taylor series successfully. In fact, from the above series together with the harmonic properties, one may calculate the artifical Zeta function’s root model equation such as the 0.5 +/- β*cot(ln(x))/(αx+1)n (will be discussed in my next paper of the series). Indeed, this is NOT the best model. Actually, the true Riemann Zeta non-trivial Root model equation is: {z = x + yI | z∈0.5±(y_1⩽y⩽y_2 )I} where y1= (±cot(k))/ln(k) , y2 = (±tan(k))/ln(k) with 0<k<2π. Certainly, this author cannot avoid mistakes. Some of the defects in the previous paper such as one in the linguistic proof of the RH will be stated and amended. Lastly, this author wants to remind that the present series of paper is first to establish a model equation for zeta function’s roots and the Plank’s like constant. This author then applies the telescopic and logarithmic methods for a 3 cases proof of the RH. Also, we have a Matlab code for the evaluation of RH complex contour integral. Finally, the RH problem will be transformed into a linguistic one and hence we may have solved the Riemann Hypothesis issue completely through a 4 cases of truth table collaborations with the logical inference [7] & [15] to the causality (structural) organization of the sentences [16] or just named as “Logical & Organized Context”.

Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques

This study aims to explore the precise resolution of the nonlinear Benjamin Bona Mahony Burgers (BBMB) equation, which finds application in a variety of nonlinear scientific disciplines including fluid dynamics, shock generation, wave transmission, and soliton theory. Within this paper, we employ two versatile methodologies, specifically the extended exp(-Ψ(χ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\Psi (\\chi ))$$\\end{document} expansion technique and the novel Kudryashov method, to identify the exact soliton solutions of the nonlinear BBMB equation. The solutions we discovered involve trigonometric functions, hyperbolic functions, and rational functions. The uniqueness of this research lies in uncovering the bright soliton, kink wave solution, and periodic wave solution, and conducting stability analysis. Furthermore, the solutions’ graphical characteristics were explored through the utilization of the mathematical software Maple 2022 (https://maplesoft.com/downloads/selectplatform.aspx?hash=61ab59890f2313b2241fde3423fd975e). The system’s physical interpretation is defined through various types of graphs, including contour graphs, 3D-surface graphs, and line graphs, which use appropriate parameter values. These recommended techniques hold significant importance and are applicable in diverse nonlinear evolutionary equations found in the field of nonlinear sciences for illustrating nonlinear physical models.

Calculation of the stress-strain state of a three-layer cylindrical shell using approximating functions

A calculation of the stress-strain state of a three-layer cylindrical shell is proposed using the MAPLE software package and approximating functions. Keywords:three-layer structure, equilibrium equations, cylindrical shell, boundary conditions, approximating functions, MAPLE computer algebra system hairulla213@mail.ru, alexbild_58@mail.ru, yarahmedov94@mail.ru

Formation of solitary waves solutions and dynamic visualization of the nonlinear schrödinger equation with efficient techniques

This article investigates the non-linear generalized geophysical KdV equation, which describes shallow water waves in an ocean. The proposed generalized projective Riccati equation method and modified auxiliary equation method extract a more efficient and broad range of soliton solutions. These include U-shaped, W-shaped, singular, periodic, bright, dark, kink-type, breather soliton, multi-singular soliton, singular soliton with high amplitude, multiple periodic, multiple lump wave soliton, and flat kink-type soliton solutions. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Maple and Wolfram Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. Sensitivity and modulation instability analyses were performed to offer insights into the dynamics of the examined model. The observed dynamics of the proposed model were presented, revealing quasi-periodic chaotic, periodic systems, and quasi-periodic behaviour. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations.

Exploration of uniform and sudden lifting on hybrid nanofluid flow induced by the ramped motion of magnetized porous plate suspended by dust particles

This study concentrates on heat transmission and momentum in the hydromagnetic flow of hybrid nano-dusty fluid over the thermal plate with radiation, Newtonian heating wall conditions, and heat sink/source. The outcome of this investigation finds application in wastewater management, groundwater treatment, soil remediation, magnetic separation, air pollution, biomedical diagnostics, continuous casting, and environmental engineering. MAPLE Software is used to solve flow-controlling equations numerically. The influence of physical parameters on the dust and fluid phases due to uniform lifting and sudden lifting of the thermal plate are compared by preparing graphs and tables. The obtained outcomes are validated. This study concludes that the fluid velocity and temperature are more significant in uniform lifting than the sudden lifting of the thermal plate. An increase in particle concentration, magnetic field, particle relaxation time, and porosity parameter results in growth in shearing stress with time. The fluid phase has a higher flow velocity and temperature than the dust phase. The temperature field is more in the fluid phase than the dust phase and is augmented with nanoparticle enhancement.

Qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative

The focus of the article studies the qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative. Firstly, the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative is transformed into two-dimensional planar dynamic system through complex exponential wave transformation. Secondly, phase portrait and its orbit of the system are analyzed through the theory of dynamical system. By using Maple software the phase portrait, Poincaré section and sensitivity analysis are compared and displayed. Finally, parameter conditions for stable systems are provided through modulation instability.

A variety of soliton solutions of time M-fractional: Non-linear models via a unified technique.

This work explores diverse novel soliton solutions of two fractional nonlinear models, namely the truncated time M-fractional Chafee-Infante (tM-fCI) and truncated time M-fractional Landau-Ginzburg-Higgs (tM-fLGH) models. The several soliton waves of time M-fractional Chafee-Infante model describe the stability of waves in a dispersive fashion, homogeneous medium and gas diffusion, and the solitary waves of time M-fractional Landau-Ginzburg-Higgs model are used to characterize the drift cyclotron movement for coherent ion-cyclotrons in a geometrically chaotic plasma. A confirmed unified technique exploits soliton solutions of considered fractional models. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. Keeping special values of the constraint, this inquisition achieved kink shape, the collision of kink type and lump wave, the collision of lump and bell type, periodic lump wave, bell shape, some periodic soliton waves for time M-fractional Chafee-Infante and periodic lump, and some diverse periodic and solitary waves for time M-fractional Landau-Ginzburg-Higgs model successfully. The required solutions in this work have many constructive descriptions, and corporal behaviors have been incorporated through some abundant 3D figures with density plots. We compare the m-fractional derivative with the beta fractional derivative and the classical form of these models in two-dimensional plots. Comparisons with others' results are given likewise.

Design of a Circular Micro-strip Patch Antenna for Improved Directivity and Gain of Mobile Communication Base Station

This study presents the design of Circular Micro-strip Patch Antenna (CMPA) for improved directivity and gain of mobile communication base station. The design incorporates a Rogger/5880 substrate material with a permittivity of 2.2, tangent angle of 0.00009 and a substrate height (thickness) of 0.98μm at a frequency of 1.65THz. Diversity Antenna Array Processing (DAAP) and Chebyshev Antenna Array (CAA) methods were used in the design. AutoCAD software and 2D data plotter were used for the diagrams and charts while Maple software was used for simulation. For angles of 0o, 30o, 60o, 90o, 120o, 150o and 180o, the array factors are 7.84dB, 7.85dB, 7.87dB, 7.88dB, 7.87dB, 7.85dB and 7.84db respectively. The results from the design demonstrated a remarkably low side lobe level of 0.01dB, indicating excellent suppression of unwanted radiation in undesired directions. Additionally, the antenna exhibits a Voltage Standing Wave Ratio (VSWR) of unity, ensuring maximum power transfer efficiency from the source to the load. Furthermore, the antenna shows a high directivity and gain of approximately 9dB, which enhances the signal strength and coverage range. The Half Power Beam-width (HPBW) and First Null Beam-width (FNBW) was measured as 300 and 600 respectively. These parameters depict the angular width of the main radiation beam and the sharpness of the beam pattern’s nulls. In all, the designed CMPA with its CAA based antenna array holds great potential for communication networks and for mobile communication base station, offering enhanced performance in terms of radiation characteristics and signal quality.

Breather wave solutions for an integrable (3+1)-dimensional combined pKP–BKP equation

This work is about the retrieval of breather wave solutions for the integrable (3+1)-dimensional combined potential Kadomtsev–Petviashvili (pKP) equation with B-type Kadomtsev–Petviashvili (BKP) equation, called pKP–BKP equation. We also provide breather wave solutions for two other (3+1)-dimensional reduced forms of the aforementioned pKP–BKP equation. Moreover, we furnish a variety of other solutions that include kink, periodic, and exponential solutions for these two reduced equations. The derived solutions for all examined models are all depicted through the Maple software.

Dynamical behaviors, chaotic pattern and multiple optical solitons for coupled stochastic Schrödinger–Hirota system in magneto-optic waveguides with multiplicative white noise via Itô calculus

The primary focus of this study was on exploring the optical soliton solutions and chaotic patterns in magneto-optic waveguides through the coupled stochastic Schrödinger–Hirota equation with multiplicative white noise. Firstly, by means of traveling wave transformations and homogeneous balance principle, the coupled stochastic Schrödinger–Hirota equation in magneto-optic waveguides is transformed into ordinary differential equation. By selecting some suitable parameters, phase diagrams are plotted with the help of the mathematical software Maple. Secondly, the optical soliton solutions of the coupled stochastic Schrödinger–Hirota equation corresponding to phase orbits can be easily deduced through the method of dynamical systems. In addition, chaotic behavior for the coupled stochastic Schrödinger–Hirota system with perturbation term has been discussed in detail. Finally, the two-dimensional and three-dimensional graphs of the stochastic Schrödinger–Hirota equation are drawn, which further explain the propagation of the coupled stochastic Schrödinger–Hirota equation in nonlinear optics.

On the implication of exponentially decaying internal heat generation on mixed convection flow from a vertical porous plate influenced by second‐order thermal and momentum slips

AbstractThis article contains vital information on the internal heat generation effect on mixed convection slip flow past a vertical plate. The partial differential equations were first simplified to ordinary differential equations through similarity transformation. The simplified first‐order differential equations were integrated with Maple Software 2022 after utilizing the shooting technique. One of the most important discoveries is that the internal heat generation acts as a barrier to stop heat from flowing from the left to right plate edge. However, this could be averted by considering strong mixed convection to covet away the heat conducted through the left surface of the plate and the internal heat generated. However, for weak mixed convection, the movement of the fluid from the left to the right surface of the plate is achievable even with minimal internal heat generation. Since the right surface temperature is warmer (higher) than the left surface, the flow properties, in this case, are also affected by the internal heat generation, which also causes the reverse of heat flow from the plate. Particle injection could also be used to avert unwanted reversed flow for various momentum slip conditions considered. There is a marginal reverse flow at the plate surface, which may be related to the flow's mixed convective characteristics. Due to the plate surface's reinforcement, the rise in flow formation caused by the momentum slip parameter is more substantial close to the plate.