The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

The research of single and multi-wall carbon nanotubes (SWCNTs/MWCNTs) mixed in sodium alginate-based nanofluid with MHD stagnation-point flow on a convective heated stretching disk with viscous dissipation and suction effects is being done with the intention of decoding the heat and mass transmission mechanism. It is possible to transform PDEs that govern the boundary layer into ODEs. MATLAB’s Bvp5c is used to numerically solve the revised equations. The Yamada-Ota model and the Buongiorno model are used in this work to scrutinize the flow, heat, and mass transfer parameters. The following parameters were brought up for discussion: volume fraction nanoparticle, magnetic parameter, suction, Brownian motion, thermophoresis, Lewis number, Eckert number, Biot number, stretching, and thermophoresis. This study found that nanofluid (SWCNT/sodium alginate) has a superior flow, heat, and mass transfer rate than nanofluid (MWCNT/sodium alginate). Graphical representations of the effects of various factors are shown, and a comparison of current and prior findings is given in a table. A comparison of current and previous findings reveals a 0% relative inaccuracy. The velocity ratio parameter has solutions that look close to the separation value. The performance of heat and mass transfer operations may be improved by increasing suction parameters. Increases in Brownian motion [Formula: see text] and suction decrease the temperature profile, whereas increases in velocity ratio and magnetic parameters increase velocity. This research is critical for estimating flow, temperature, and concentration behavior for CNTs with incorporated physical properties.

In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equationwhere the coefficients α i , β i ∈ (0, ∞), i = 0, 1, 2, 3, 4, and p, q, r, and s are positive integers. The initial conditions y −s ,. . . ,y −r ,. . . , y −q ,. . . , y −p ,. . . ,y −1 ,y 0 are arbitrary positive real numbers such that p < q < r < s. Some numerical examples will be given to illustrate our result.

This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space R3. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points O,P,Q,R,S,T,U,V and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters α,β,γ,δ,ζ,η,μ,ε,υ,ρ,σ,ω∈R+ and initial conditions x0,y0,z0 are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits.

The utilization of hybrid nanofluids (HNs) to boost heat transfer is a promising area of study, and thus, numerous scientists, researchers, and academics have voiced their admiration and interest in this area. One of the main functions of nanofluids is their dynamic role in cooling small electrical devices such as microchips and associated gadgets. The major goal of this study is to perform an analysis of the buoyancy flow of a shrinking/stretching sheet, whilst considering the fascinating and practical uses of hybrid nanofluids. The influence of a nonlinear heat source/sink induced by a micropolar fluid is also inspected. Water-based alumina and copper nanoparticles are utilized to calculate the fine points of the fluid flow and the features of heat transfer. The governing equations are framed with acceptable assumptions and the required similarity transformations are used to turn the set of partial differential equations into ordinary differential equations. The bvp4c technique is used to solve the simplified equations. Dual solutions are presented for certain values of stretching/shrinking parameters as well as the mixed convective parameter. In addition, the shear stress coefficient in the first-branch solution (FBS) escalates and decelerates for the second-branch solution (SBS) with the superior impact of the magnetic parameter, the mass transpiration parameter, and the solid nanoparticles volume fraction, while the contrary behavior is seen in both (FB and SB) solutions for the larger values of the material parameter.