Maple Software Research Articles

Bifurcation, phase portrait and traveling wave solution of time-fractional thin-film ferroelectric material equation with beta fractional derivative

In this paper, the bifurcation, phase portrait and traveling wave solution of time-fractional thin-film ferroelectric material equation with beta fractional derivative are studied, which are also the biggest innovations of this article compared with existing literature. Firstly, the time-fractional thin-film ferroelectric material equation with beta fractional derivative is converted into the nonlinear ordinary differential equation by using the traveling wave transformation. Secondly, the nonlinear ordinary differential equation is regarded as two-dimensional planar dynamical system. What's more, the bifurcation and phase portraits of two-dimensional planar dynamical system are plotted by using the Maple software. Finally, the traveling wave solutions of time-fractional thin-film ferroelectric material equation with beta fractional derivative are constructed by using the planar dynamical system method.

Exact solutions and invariant subspaces to the nonlinear dissipative–dispersive equation

In this paper, we performed Lie symmetry analysis and applied [Formula: see text] expansion method on the nonlinear dissipative–dispersive equation. The purpose of this research is to find the vector fields and transform the nonlinear dissipative–dispersive equation into simpler forms. The Maple software was used to obtain the vector field and similarity reductions for nonlinear dissipative–dispersive equations. In addition, we obtained exact solutions based on the [Formula: see text] expansion method and power series method, including the hyperbolic functions, the trigonometric functions and the rational functions. The method we used is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations. Furthermore, the invariant subspaces of the nonlinear dissipative–dispersive equation were identified using the refined invariant subspaces method. The invariant subspaces of solutions to linear ordinary differential equations were used to prove that nonlinear dissipative–dispersive equation admits subspaces. The exact solutions were obtained by using generalized separated variables.

Mathematical model for the dynamics of bacterial meningitis (Meningococcal meningitis): a case study of Yobe State Specialist Hospital, Damaturu, Nigeria

A model for bacterial meningitis was created by adding a class of transporters to the basic Susceptible Carrier Infected and Recovered (SCIR) model, since vaccination and treatment are the best methods of controlling the transmission of most overpowering sicknesses. Immunization assists helpless people with building either a drawn out invulnerability or transient resistance while treatment decreases the quantity of sickness actuated passing and the quantity of irresistible people locally or country. This study comprises of a mathematical model for bacterial meningitis dynamics that can be used to a wide range of mathematical modeling problems. In this exploration, a nonlinear deterministic model with time reliance controls has been proposed to depict the elements of bacterial meningitis in a populace. We discovered that the (EEP) and (DFE) are locally asymptotically stable in our study. We now advise the researcher to determine whether it is globally asymptotically stable in order to achieve optimal disease control. The presence of an endemic harmony and the calculation of the reproduction number . The mathematical arrangement shows that vaccinating vulnerable people will prompt disposal of the infection in the public and furthermore it will lessen the weight on wellbeing suppliers. Numerical simulations were presented to explain the parameters in the end path in the model were carried out by using MAPLE software. Most likeable of this research work shows that the rate at which treatment and vaccination rate increases to the higher value the recovered compartment increases to the peak point. This means treatment and vaccination has an impact on reducing the case of bacterial meningitis in a population. Now we conclude that a high infection transmission rate requires a high vaccine and treatment rate on the effect of vaccination against meningitis.

Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

New traveling solutions, phase portrait and chaotic pattern for the generalized (2+1)-dimensional nonlinear conformable fractional stochastic Schrödinger equations forced by multiplicative Brownian motion

The paper mainly studies the traveling wave solutions, phase portrait and chaotic pattern of the generalized fractional stochastic Schrödinger equations forced by multiplicative Brownian motion. By using the polynomial complete discrimination method, the hyperbolic functions, rational functions and Jacobi elliptic function solutions are obtained. Moreover, the physics properties of the obtained solutions are shown through three-dimensional, two-dimensional and contour graphs. In order to help better analyze dynamic behavior, phase diagrams, chaos behavior and sensitivity are plotted by using Maple software.

On the solution of random linear difference equations with Laplace transform method

In this study, Laplace transformation, which is very important for solutions to initial value problems, is examined. To solve the initial value problem of a discrete‐time equation, Laplace implements the conversion method. Here, Laplace transformation is used to obtain an approach to the solutions of random difference equations formed by randomizing components of deterministic difference equations. For random behavior of linear difference equations under random effects, uniform, geometric, binomial, Poisson, and Bernouilli distributions are used, and approximate expected value, variance, standard deviation, and confidence interval of equations obtained by Laplace transformation are calculated. The results were obtained through the Maple package program.

Aspects of active-passive controls of nanoparticles of chemically reactive and radiative nanofluid flow past a frequently moving thin needle with thermal and velocity slip: A numerical framework

This communication unfolds the flow features and distribution of mass, heat of a steady MHD nanofluidic stream over an axially moved thin needle with passive and active controls of tiny particles. A linear sort of heat radiation with magnetic influence and a higher-order chemical reaction are addressed in this study. The investigation has been enriched by the consideration of thermophoretic forces with the Brownian movement of fluid. The statement of the problem is formed on the basis of Buongiorno’s model, which introduced the existence of thermal & velocity slips. A suitable set of conversions was utilized to transfigure the foremost equations into nondimensional ordinary ones. Due to the extreme nonlinearity of this problem, we avoid the analytical scheme of solution. Hence, we have implemented the RK4 procedure in shooting approach through the algebra software MAPLE 2017, to solve it in a numerical manner. Numerous graphical diagrams and tables are buildups to discuss our attained impacts. Also, it is ensured the validation of these results with the study which was published earlier. Investigation depicts that the order of chemical reaction intensifies fluid density for active control, while the reverse is recorded in the passive strategy. Velocity slip amplifies the drag friction but reduces the transportation of heat. In the active control system, advanced chemical reaction offers more transmission of heat, but the opposite happens in another control.

Electroosmotically actuated peristaltic-ciliary flow of propylene glycol + water conveying titania nanoparticles

The main focus of this article is to mathematically formulate the microfluidics-based mechanical system for nanofluids. A 50:50 mixture of propylene glycol (PG) and water is used as a heat transfer fluid because of its tremendous anti-freezing properties, and nontoxicity and it is safe to be utilized at the domestic level. Titanium dioxide (titania) nanoparticles are suspended in the working fluid to enhance its heat transfer ability. The fluid flow is induced by electroosmosis in a microtube, which is further assisted by cilia beating. The impacts of Joule heating and non-linear thermal radiation are also considered. The simplification of the dimensionless system is done under lubrication theory and the Debye-Hückel linearization principle. The nonlinear system of equations is executed for a numerical solution by adopting the symbolic mathematical software Maple 17 using the command “dsolve” along with the additional command “numeric” to get the numerical solution. This command utilizes a low-ordered method along with accuracy-enhancing schemes such as the deferred correction technique and Richardson extrapolation to get a numerical answer of desired accuracy, where we can choose the accuracy level and mesh points according to our requirements. The detailed analysis of results obtained from the numerical treatment of the considered problem indicates that the efficiency of the PG + water enhances due to the suspension of the nanoparticles and heat is rapidly removed from the system. Further, the velocity of the fluid is augmented by decreasing the thickness of the electric double layer and raising the strength of the electric field in the forwarding direction.

Dynamics and Embedded Solitons of Stochastic Quadratic and Cubic Nonlinear Susceptibilities with Multiplicative White Noise in the Itô Sense

The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and its perturbation system are obtained by using a traveling wave transformation. Secondly, the phase portraits of the two-dimensional dynamics system are plotted. Furthermore, the chaotic behavior, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed via Maple software. Finally, the embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimensional solitons of stochastic quadratic and cubic nonlinear susceptibilities are plotted.

Interaction solutions for the (2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation in incompressible fluid

This paper aims to search for the solutions of the (2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation. Lump solutions, breather solutions, mixed solutions with solitons, and lump-breather solutions can be obtained from the N-soliton solution formula by using the long-wave limit approach and the conjugate complex method. We use both specific circumstances and general higher-order forms of the hybrid solutions as examples. With the help of maple software, we create density and 3D graphs to summarize the dynamic properties of these solutions. Additionally, it is possible to observe how the solutions’ trajectory, velocity, and shape vary over time.

Bifurcation, chaotic pattern and optical soliton solutions of generalized nonlinear Schrödinger equation

This article studies the generalized nonlinear Schrödinger equation, which is used to simulate the propagation model of optical pulses in Non-Kerr medium. Building upon the traveling wave transformation, the generalized nonlinear Schrödinger equation is simplified to an ordinary differential equation. By employing the two-dimensional planar dynamic system to analyze, the bifurcation, phase portraits and chaotic behaviors of the system is presented. Furthermore, 2D and 3D phase portraits of the dynamic system with perturbation term are plotted with the Maple software. The optical soliton solutions of the generalized nonlinear Schrödinger equation are constructed by using the polynomial complete discriminant method.

Newton’s forward interpolation method for solving nonlinear algebraic equation

As we know, the interpolation is one of the most basic and most useful numerical techniques in Mathematics. Newton’s forward interpolation method is one of most important of these methods. Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist to find roots. But in this paper, we introduce the interpolation technique for this purpose. The proposed method derived from the newton forward interpolation method and we compared the results with another existing method (Bisection Method (BM), Regula-Falsi Method (RFM), Secant Method (SM), Newton Raphson Method (NRM)) and the method proposed by J. Sanaullah (SJM). It’s observed that the proposed method has fast convergence but it has same order of convergence of the method (SJM). Maple software is used to solve problems by different methods.

Qualitative analysis of stochastic Schrödinger–Hirota equation in birefringent fibers with spatiotemporal dispersion and parabolic law nonlinearity

In this paper, the stochastic Schrödinger–Hirota equation in birefringent fibers with spatiotemporal dispersion and parabolic law nonlinearity is studied, which is usually used to describe the mathematical model of optical soliton propagation in dispersive optical fibers from the field of nonlinear optics. Firstly, the stochastic Schrödinger–Hirota equation is converted into two-dimensional planar dynamic system by using the traveling wave transformation. Secondly, the qualitative analysis of the stochastic Schrödinger–Hirota equation is discussed by using the theory of planar dynamical systems. Finally, the chaos pattern of the stochastic Schrödinger–Hirota equation with perturbed system is considered. What is more, phase portraits and sensitivity analysis of the perturbed system are plotted by using the Maple software.

Design and implementation of symbolic algorithms for the computation of generalized asymptotes

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, mathscr {C}, implicitly or parametrically defined. The asymptotes of a curve mathscr {C} reflect the status of mathscr {C} at points with sufficiently large coordinates. It is well known that an asymptote of a curve mathscr {C} is a line such that the distance between mathscr {C} and the line approaches zero as they tend to infinity. However, a curve mathscr {C} may have more general curves than lines describing the status of mathscr {C} at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field.

Entropy generation on MHD flow of Williamson hybrid nanofluid over a permeable curved stretching/shrinking sheet with various radiations

The research aims to scrutinize the significance of MHD, radiation, Joule heating, porous medium, and heat generation impacts on the convective boundary flow. It contains blood as the base fluid, silver (Ag), and aluminum oxide (Al2O3 ) as the nanoparticles over a curved stretching and shrinking sheet. To investigate the rheology of blood, we used a Williamson fluid model with comparisons for both curved stretching/shrinking sheets for velocity and a comparison between linear and nonlinear thermal radiation for temperature, entropy production, and Bejan number. This model is used to study physiological systems, surgical procedures, and nano-pharmacological delivery systems. The governing nonlinear coupled PDEs are transformed into nonlinear associated ODEs via the similarity transformation in the Maple software solver using the Homotopy Perturbation Method. The Homotopy Perturbation Method (HPM) results are compared with the Numerical Method (NM) results, where HPM gives reliable results. The results and discussion provide graphical results with several parameters for velocity, temperature, entropy production, Bejan number, Nusselt number, and skin friction. As the magnetic field parameter (M) increases, the velocity profile decreases for curved stretching sheet. We observe the opposite behavior of the velocity profile for a curved shrinking sheet. As the volume fractions increase, the temperature profile increases for linear and non-linear thermal radiation. The temperature profile rises due to the thermal conductivity of nanoparticles, which is amplified by estimations of the nanoparticle volume fractions.

Dynamics and optical soliton solutions in birefringent fibers for the coupled Kaup–Newell equation via planar dynamical system analysis method

In this paper, the dynamical behavior and optical soliton of the coupled Kaup–Newell equation are studied. Firstly, the phase portraits of the coupled Kaup–Newell equation are drawn by using Maple software. Secondly, the optical soliton solutions of the coupled Kaup–Newell equation are construct by the bifurcation theory of planar dynamical system together with the “three-step method” of Professor Li’s method. Finally, in order to better explain the wave propagation in birefringent fiber, the three-dimensional graphic, the two-dimensional graphic, contour plot and polar plot of some optical soliton solutions are drawn.

A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics

This study proposes a fractional-order mathematical model for malaria and COVID-19 co-infection using the Atangana–Baleanu Derivative. We explain the various stages of the diseases together in humans and mosquitoes, and we also establish the existence and uniqueness of the fractional order co-infection model solution using the fixed point theorem. We conduct the qualitative analysis along with an epidemic indicator, the basic reproduction number R0 of this model. We investigate the global stability at the disease and endemic free equilibrium of the malaria-only, COVID-19-only, and co-infection models. We run different simulations of the fractional-order co-infection model using a two-step Lagrange interpolation polynomial approximate method with the aid of the Maple software package. The results reveal that reducing the risk of malaria and COVID-19 by taking preventive measures will reduce the risk factor for getting COVID-19 after contracting malaria and will also reduce the risk factor for getting malaria after contracting COVID-19 even to the point of extinction.

Exploration of New Solitons for the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model

The key objective of the current manuscript was to investigate the exact solutions of the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model. For this purpose, we applied two reliable and efficient approaches; specifically, the modified simple equation (MSE) and exponential rational function (ERF) techniques. The methods considered in this paper offer solutions for problems in nonlinear theory and mathematical physics practice. We also present solutions obtained graphically with the Maple package program.

Operations for D-Algebraic Functions

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass elliptic functions, and holonomic or D-finite functions are D-algebraic. These functions form a field, and are closed under composition, taking functional inverse, and derivation. We present implementation for each underlying operation. We also give a systematic way for computing an algebraic differential equation from a linear differential equation with D-finite function coefficients. Each command is a feature of our Maple package NLDE available at https://mathrepo.mis.mpg.de/DAlgebraicFunctions/NLDEpackage.

Functional Education as a Nexus between Agricultural and Industrial Revolution: An Epidemiological Modeling Approach

Education has been regarded as a major route to economic prosperity. Its potentials, if properly exploited, have the tendency to revolutionize a nation’s productivity, income, and development. [Objetive] In this work, a mathematical model was formulated, and an epidemiological modeling approach was explored to derive the necessary and sufficient conditions for agricultural and industrial transformation in Nigeria in terms of education. The use of mathematics to quantify such a phenomenon is relatively new. [Methodology] Ample mathematics theorems were employed to test for the existence, boundedness, and positivity of the model’s solutions which are basic features of a valid epidemiological model. The model was solved to derive the equilibrium points, and the analytic threshold that governed revolution in industry and agriculture in terms of education was derived. Stability and sensitivity analyses were conducted via the stability theory of nonlinear differential equations and normalized forward sensitivity index, respectively. Simulations were later conducted using the Maple software to validate the analytical results. [Results] A government expenditure of 25% on education with or without corruption did not make education functional. Also, an increase in expenditure on education to 65% but with corruption of more than 30% retarded functional education. In general, education became functional when government investment in education attained 60%, with corruption levels lower than 20%. [Conclusions] Both the analytical and numerical results linked agricultural transformation and industrial revolution through functional education to massive investment in education and victory over corruption.