Maple Software Research Articles

The Effect of MHD on Marangoni Boundary Layer of Hybrid Nanofluid Flow Past a Permeable Stretching Surface

Numerous researchers studied on the properties of hybrid copper-alumina/water nanofluid in order to have better thermal efficiency that could be used in industrial applications. Therefore, the present study accentuates the effect of MHD on Marangoni boundary layer of hybrid copper-alumina/water nanofluid flow over a stretching surface. Furthermore, the governing boundary layer equations which is partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs) using similarity transformations. Then, these problems are solved numerically using shooting method through Maple software. It is shows that the rate of heat transfer decreases with the increase in magnetic parameter. The velocity and temperature profiles, as well as local Nusselt number are observed with the specific parameter namely, magnetic parameter, nanoparticles volume fraction and suction. By increasing the volume fraction for copper and magnetic parameter, the velocity profiles decrease while the temperature profiles increase.

Production Inventory Optimization Considering Direct and Indirect Carbon Emissions under a Cap-and-Trade Regulation

Background: The latest global agreement on net-zero emissions encourages new studies on production inventory optimization that promote carbon emissions reduction without harming a company’s profit performance, particularly because certain carbon-pricing regulations bind manufacturing companies. Methods: This study aims to develop a production inventory model that considers direct and indirect emissions in three emission scopes. It incorporates emissions from production, material handling, transportation, and waste disposal for further treatment under a carbon cap-and-trade regulation. With the help of Maple software, a convex total cost function was solved. Results: The results show that the optimum production quantity depends on the values of demand, setup cost, holding cost, fixed cost per delivery, fixed cost for waste disposal, and other parameters related to carbon prices. This study also found that the total cost was highly dependent on the values of the carbon cap, carbon price, and delivery distance. Meanwhile, changes in the delivery distance and fuel emissions standard significantly impacted total emissions. Conclusions: The proposed model can guide manufacturing companies in setting the optimum production quantity per cycle. Moreover, they must carefully manage the delivery and setting of the carbon cap and carbon price from the government.

Analytical Solutions of the Fractional Complex Ginzburg-Landau Model Using Generalized Exponential Rational Function Method with Two Different Nonlinearities

The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order α . Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model.

Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers

In this paper, the bifurcation, stationary optical solitons and exact solutions in optical fiber propagation are studied. The propagation model is described by a generalized nonlinear Schrödinger equation with Kudryashov’s quintuple power law of refractive index. First, the equation is transformed into a second-order nonlinear ordinary differential equation by an appropriate transformation. Then, the bifurcation of the phase diagram of the obtained ordinary differential equation is established using planar dynamic system theory, and the types of optical solitons and exact solutions are determined. What is more, the stationary optical solitons and exact solutions are obtained by the polynomial complete discriminant system, and are demonstrated in 3D and 2D graphics by using the symbolic computation software Maple. Finally, the obtained stationary optical solitons and exact solutions include the Jacobi elliptic function solution, hyperbolic function solution, rational function solution, trigonometric function solution and exponential function solution. Moreover, the results provide a method to further study stationary optical solitons and exact solutions of propagation pulses in optical fibers.

Localized wave solutions for (2+1)-dimensional third-order Fokou-Kofane-Mohamadou-Yomba equation

Analytical and numerical investigations of localized wave solutions for a nonlinear evolution of shallow water waves with surface tension, which is described by the (2+1)-dimensional third-order Fokou-Kofane-Mohamadou-Yomba (FKMY) equation are performed. Furthermore, we show, mainly by Maple software and the Hirota bilinear method, that appropriated ansatzes can be used to generate new large families of traveling localized structures such as lump, soliton, periodic soliton, quasi-periodic soliton, and quasi-periodic breather solutions. Interactions between those soliton solutions and their dependence on the system physical parameters have been carefully analysed. We show that the soliton solutions of the 2D third-order FKMY display a very rich spectrum of dynamical behaviour when its parameters are varied. The present results could be applicable in explaining the basic features of localized disturbances in many fields of science where the 2D third-order FKMY equation appears.

Optical solitons of the cubic–quartic-nonlinear Schrödinger’s equation having cubic–quintic–septic–nonic form of self-phase modulation

This paper is object to construct the optical solitons of the cubic–quartic-nonlinear Schrödinger’s equation having cubic–quintic–septic–nonic form of self-phase modulation. All optical solitons, including the rational function solution, the trigonometric function solution, the exponential function solution, the solitary wave solution, the elliptic double periodic function solution and the Jacobian function solution, are acquired via the complete discrimination system technique. Optical solitons are treasure-trove in the field of telecommunications. The results of this paper will be beneficial to further acquaintance with the complete picture of soliton dynamics system. Moreover, for the purpose of physical illustration presenting in visual form, we represent some of the obtained solutions graphically with Maple software.

Boundary Layer Flow of Williamson Hybrid Ferrofluid over A Permeable Stretching Sheet with Thermal Radiation Effects

This research investigated the convective boundary layer flow and heat transfer of Williamson hybrid ferrofluid over a permeable stretching sheet with thermal radiation effects. Human blood is employed as a based fluid while magnetite (Fe3O4) and copper (Cu) are taken as the hybrid ferroparticle. The study started with transforming the non-linear partial differential equation system that governed the model to a more convenience non-linear dimensionless ordinary differential equations using the similarity transformation. The transformed equations obtained then are solved numerically using the Runge-Kutta-Fehlberg (RKF45) method in Maple software. The characteristics and effects of stretching parameter, permeability parameter, thermal radiation parameter as well as the ferroparticle volume fraction in the Williamson hybrid ferrofluid towards the temperature profiles, velocity profiles as well as the Nusselt number and the skin friction coefficient are analysed and discussed. The result of this research for various pertinent parameter varies differently. It can be concluded that the increase in magnetic parameter, the Williamson parameter, the stretching parameter, and the permeability rate parameter increase the skin friction and reduced the velocity profile. Furthermore, the increase in stretching parameter, thermal radiation parameter and the permeability rate results to the increase in the Nusselt number.

On exact solutions of the (2+1)-dimensional time conformable Maccari system

For the ([Formula: see text])-dimensional time conformable Maccari system with beta-derivative, exact solutions are obtained in this paper using the improved sub-equation method. This technique has been used to find rational solutions, generalized hyperbolic function solutions and generalized trigonometric function solutions. The new structures of the solutions are shown after the proper parameter values have been given. It is easier to understand the intricate physical processes of these dynamical models by viewing the physical surfaces of some obtained solutions in various forms. The outcomes demonstrate the efficiency of the applied strategy, which may be used to solve a number of nonlinear physical problems. ([Formula: see text])-dimensional time conformable Maccari system is considered and the graphs of solutions are sketched using MAPLE software after its exact solutions which are acquired using the improved sub-equation method.

Analysis and Simulation of Mathematical Model of COVID-19 Incorporated with Vaccination and Media Induced Fear

COVID-19 pandemic is now increasing concern to authorities and public health officials. The study is aimed to investigate the impact of vaccination in the absence of media induced fear. In this research, analysis and simulation of a mathematical model of COVID-19 incorporated with media induced fear and vaccination was made, the total population is divided into 5 sub-population classes; Susceptible, Exposed, Infected, Quarantine and Recovery. The disease free and the endemic equilibrium of the model were carried out and the basic reproduction number was obtained using the next generation matrix. The stability analysis of the model was done and it was ascertained that the disease free equilibrium of the biological model is stable. Due to its efficiency and accuracy in handling nonlinear coupled ordinary differential equations, the homotopy perturbation method is applied to obtain the approximate solution of the mathematical model and the obtained results was simulated using the computation software Maple 18 to study the impact of vaccination in each compartment of the model when the media induced fear. The outcome of the simulation process were presented graphically and interpreted accordingly and it was discovered that in eradicating the spread of COVID-19 in a society where there is no fear, vaccination is an alternative and better measure. HIGHLIGHTS Model formulation with vaccination Stability analysis of the disease free equilibrium and endemic equilibrium Numerical simulation of the mathematical model using the homotopy perturbation method Analysis of the impact of vaccination in controlling the spread of COVID-19 in the absence of media induced fear GRAPHICAL ABSTRACT

Optimization and Numerical Modeling of TCO/SnO2/CdS/CdTe Solar Cells

Due to the excellent performance of the CdTe solar cells, research is ongoing to increase the efficiency of these cells. The first purpose of this study is to increase the accuracy of the physical parameters of a solar cell in the electron ̶hole production rate equation. In previous studies, this section was neglected because of using only ready-made software. Simulations were performed using a one-dimensional diffusion model in MATLAB and Maple software. Then, in theory, we simulated cadmium telluride-based layered solar cells for the first time without using ready-made software and with coding in MATLAB and Maple software. We designed and optimized the thickness of the layers in solar cells in detail. Then we studied the effect of layer thickness on the short-circuit current (Jsc), open-circuit voltage (Voc), filling factor (FF), and its efficiency. It is found that the efficiency of solar cells layered with TCO/SnO2/CdS/CdTe layers is as follows: the thickness of the TCO layer is 0.1 μm, that of the SnO2 layer is equal to 0.1 μm, that of the CdS layer as the window layer is 0.1 μm and the thickness of the CdTe layer as the absorber layer is 3.9 μm. The efficiency of the solar cell with the TCO/SnO2/CdS/CdTe structure increases significantly and reaches a maximum value of more than 20%.

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.

On the Solution of Fractional Biswas–Milovic Model via Analytical Method

Through the use of a unique approach, we study the fractional Biswas–Milovic model with Kerr and parabolic law nonlinearities in this paper. The Caputo approach is used to take the fractional derivative. The method employed here is the homotopy perturbation transform method (HPTM), which combines the homotopy perturbation method (HPM) and Yang transform (YT). The HPTM combines the homotopy perturbation method, He’s polynomials, and the Yang transform. He’s polynomial is a wonderful tool for dealing with nonlinear terms. To confirm the validity of each result, the technique was substituted into the equation. The described techniques can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give a precise solution. Graphs are used to show the derived numerical results. The maple software package is used to carry out the numerical simulation work. The results of this research are highly positive and demonstrate how effective the suggested method is for mathematical modeling of natural occurrences.

Magnetohydrodynamic hybrid nanofluid flow over a decelerating rotating disk with Soret and Dufour effects

PurposeThe investigation of fluid flow over a rotating disk has been increasing due to the spread of machine technology. Because of this development, we scrutinized the Magnetohydrodynamic (MHD) flow of hybrid nanofluid caused by a decelerating rotating disk with Ohmic heating, Soret and Dufour effects. The disk's angular velocity is taken to be an inversely time-dependent linear function. Moreover, the temperature-dependent viscosity of hybrid nanofluid is incorporated in the present investigation. Methanol is considered as base fluid, while copper oxide (CuO) and magnesium oxide (MgO) are nanoparticles.Design/methodology/approachEstimated fundamental partial differential equations of flow problems are altered as a dimensionless system of ordinary differential equations using appropriate similarity transformation and solved using a numerical technique: BVP Midrich scheme in Maple software. The impression of emerging non-dimensional parameters is portrayed graphically. All outcomes are shown in the velocity, temperature and concentration profiles.FindingsThe developed flow problem involves a non-dimensional parameter (A) that reveals the deceleration of the disk. For larger values of A, the disk decelerates faster and for some fixed time, the fluid surrounding the disk revolves more rapidly than the disk itself. The radial velocity of fluid diminishes and axial velocity becomes uniform when the disk is subjected to wall suction velocity (B).Originality/valueThis analysis is significant in biomedical engineering, cancer therapeutic, manufacturing industries and nano-drug suspension in pharmaceuticals. The novelty of the current study is the hybrid nanofluid flow with Ohmic heating, Soret and Dufour effects on a decelerating rotating disk. To the best of the author's knowledge, no such consideration has been published in the literature.

Study of MHD nanofluid flow with fuzzy volume fraction in thermal field-flow fractionation

This paper discusses the magnetohydrodynamic Jeffery–Hamel nanofluid flow (MHD-JHNF) between two rigid non-parallel plane walls. It enhances the study of simple Jeffery–Hamel flow problems to the MHD nanofluid flow problems with uncertain volume fractions. It examines the impact of various parameters, such as channel angles and Reynolds number with magnetic field and nanoparticles, on the fuzzy velocity profiles. The nanoparticle volume fraction is considered an uncertain parameter by using a triangular fuzzy number ranging from 0.0 to 0.2. A novel double parametric form-based homotopy analysis approach with its convergence analysis is introduced to examine the fuzzy velocity profiles at distinct convergent and divergent channel positions and fuzzy velocity boundaries at other channels and illustrates the efficiency of the process. Finally, Maple software is used to make the numerical simulations, and the results are checked with the available results for specific cases in crisp environments.

New single traveling wave solution of the Fokas system via complete discrimination system for polynomial method

<abstract> <p>In this paper, the traveling wave solution of the Fokas system which represents the irregular pulse propagation in monomode optical fibers is studied by using the complete discriminant system method of polynomials. Firstly, the Fokas system is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the Jacobian function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solution of Fokas system are obtained by using the complete discriminant system method of polynomials. Finally, in order to show the propagation of Fokas system in monomode optical fibers, three-dimensional diagram, two-dimensional diagram, contour plot and density plot of some solutions are drawn by using Maple software.</p> </abstract>

"TA’LIM JARAYONIDA BIRINCHI TARTIBLI CHIZIQLI ODDIY DIFFERENSIAL TENGLAMALARNING YECHIMINI MAPLE DASTURIDA TOPISH"

"TA’LIM JARAYONIDA BIRINCHI TARTIBLI CHIZIQLI ODDIY DIFFERENSIAL TENGLAMALARNING YECHIMINI MAPLE DASTURIDA TOPISH"

On the Stability and Boundedness of Solutions to Certain Second Order Differential Equation

In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.

ANALYSIS OF (1 + n)-DIMENSIONAL GENERALIZED CAMASSA–HOLM KADOMTSEV–PETVIASHVILI EQUATION THROUGH LIE SYMMETRIES, NONLINEAR SELF-ADJOINT CLASSIFICATION AND TRAVELLING WAVE SOLUTIONS

In this paper, the nonlinear [Formula: see text]-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (g-CH-KP) equation is examined using Lie theory. Lie point symmetries of the equation are computed using MAPLE software and are generalized for the case of any dimension. Moreover, the equation is transformed into a nonlinear ordinary differential equation using the Abelian subalgebra. The nonlinear self-adjoint classification of the equation under consideration is accomplished with the help of which conservation laws for a particular dimension are calculated. Moreover, the new extended algebraic approach is used to compute a wide range of solitonic structures using different set of parameters. Graphic description of some specific applicable solutions for certain physical parameters is portrayed.

Utilization of the ExactMPF package for solving a discrete analogue of the nonlinear Schrödinger equation by the inverse scattering transform method

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

HERMITE WAVELET METHOD FOR APPROXIMATE SOLUTION OF HIGHER ORDER BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper, Hermite wavelet method (HWM) is considered for numerical solution of 12- and 13-order boundary value problems (BVPs) of ordinary differential equations (ODEs). The proposed algorithm for HWM developed in Maple software converts the ODEs into an algebraic systems of equations. These algebraic equations are then solved by evaluating the unknown constants present in the system of equations and the approximate solution of the problem is obtained. Test problems are considered and their solutions are investigated using HWM-based algorithm. The obtained results from the test problems are compared with exact solution, and with other numerical methods solution in the existing literature. Results comparison are presented both graphically and in tabular form showing close agreement with exact solution, and greater accuracy than homotopy perturbation method (HPM) and differential transform method (DTM).