parametric continuation
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Physical processes occurring in devices with distributed variables and a turbulent tide with a dispersion of mass and heat are often modeled using systems of nonlinear equations. Solving such a system is sometimes impossible in an analytical manner. The iterative methods, such as Newton’s method, are not always sufficiently effective in such cases. In this article, a combination of the homotopy method and the parametric continuation method was proposed to solve the system of nonlinear differential equations. These methods are symmetrical, i.e., the calculations can be made by increasing or decreasing the value of the parameters. Thanks to this approach, the determination of all roots of the system does not require any iterative method. Moreover, when the solutions of the system are close to each other, the proposed method easily determines all of them. As an example of the method use a mathematical model of a non-adiabatic catalytic pseudohomogeneous tubular chemical reactor with longitudinal dispersion was chosen.

Abstract In this paper, we consider the dynamics of a mobile vehicle moving under control on a perfectly rough horizontal plane. The vehicle consists of a horizontal platform and three omni-wheels that can rotate independently. An omni-wheel has freely rotating rollers on its rim [1]. We use its simplest model: an omni-wheel on a perfectly rough plane is modelled as a rigid disk with a constraint that its contact point velocity directed perpendicular to the disk's plane. The vehicle is controlled by three direct current motors in wheels' axes. Two terms model torques generated by motors: the rst one is proportional to the voltage, the second one is proportional to the value of the angular velocity of a wheel (counter-electromotive force). We study constant voltage dynamics and boundary-value problems for arbitrary initial and nal mass center coordinates, course angles and their derivatives using a piecewise constant control with one switching point. This problem is reduced to a system of algebraic equations for some specific (symmetric) vehicle model. We numerically model the system and analyze the possibility of optimization. For another vehicle configuration, we get the solution as numerical parametric continuation starting from the solution for the symmetric vehicle.

AbstractThe present study explores incompressible, steady power law nanoliquid comprising gyrotactic microorganisms flow across parallel plates with energy transfer. In which only one plate is moving concerning another at a time. Nonlinear partial differential equations have been used to model the problem. Using Liao's transformation, the framework of PDEs is simplified to a system of Ordinary Differential Equations (ODEs). The problem is numerically solved using the parametric continuation method (PCM). The obtained results are compared to the boundary value solver (bvp4c) method for validity reasons. It has been observed that both the results are in best settlement with each other. The temperature, velocity, concentration and microorganism profile trend versus several physical constraints are presented graphically and briefly discussed. The velocity profile shows positive response versus the rising values of buoyancy convection parameters. While the velocity reduces with the increasing effect of magnetic field, because magnetic impact generates Lorentz force, which reduces the fluid velocity.

Abstract The article explores the upshot of Hall current, thermal radiation and magnetic field on hybrid nanoliquid flow over the surface of a spinning disk. The motive of the present effort to upgrade the heat transmission rate for engineering and industrial purposes. The hybrid nanoliquids as comparative to the conventional fluids have higher thermal properties. A special class of nanoparticles known as carbon nanotubes (CNTs) and iron ferrite Fe3O4 are added to the base fluid. The system of modeled equations is depleted into dimensionless differential equations through similarity transformation. The transform equations are further solved through Parametric Continuation method (PCM). For parametric study the embedding flow factors on velocity, energy, mass transmission and motile microorganism’s concentration profiles have been sketched. The obtained results are compared with the existing literature, which shows best settlement. It concluded that the heat transmission rate reduces for Hall current and rises with radiative factor. The results perceived that the addition on CNTs in carrier fluid is more efficacious than any other types of nanoparticles, due to its C-C bond. CNTs nanoliquid can be more functionalized for the desired achievement, which can be utilize for a variety of applications by functionalization of non-covalent and covalent modification.

AbstractThe work relates to development and presentation a two-parameter continuation method for determining catastrophic sets of stationary states of a tubular chemical reactor with mass recycle. The catastrophic set is a set of extreme points occurring in the bifurcation diagrams of the reactor. There are many large IT systems that use the parametric continuation method. The most popular is AUTO’97. However, its use is sometimes not convenient. The method developed in this work allows to eliminate the necessity to use huge IT systems from the calculations. Unlike these systems, it can be inserted into the program as a short subroutine. In addition, this method eliminates time-consuming iterations from the calculations.

The present work explores the behavior of three-dimensional incompressible viscous fluid flow and heat transfer over the surface of a non-flat stretchable rotating disk. A variable thickness fluid is subjected under the influence of an external variable magnetic field and heat transfer. Navier–Stokes equation is coupled with Maxwell equations to examine the hydrothermal properties of fluid. The basic governing equations of motion are diminished to a system of nonlinear ordinary differential equations using appropriate similarity framework, which are further treated with numerical scheme known as parametric continuation method. The parametric continuation method has combined interesting characteristics of both shooting and implicit finite difference methods. For validity of the present numerical scheme, a comparison with the published work is performed and it is found that the results are in excellent agreement with each other. Numerical and graphical results for the velocity, temperature, and magnetic strength profiles as well as skin fractions and Nusselt number are presented and discussed in detail for various physical parameters. The heat transfer process is reduced with positive increment of no-flatness parameter [Formula: see text], while Prandtl number increases the heat transfer rate at the surface of the disk.