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Algorithm for solution of systems of singularly perturbed differential equations with adifferential turning point

Sobchuk Valentyn, Zelenska Iryna, Laptiev Oleksandr

Bulletin of the Polish Academy of Sciences. Technical Sciences

2023

Vol. 71, nr 3

art. no. e145682

The dynamic development of science requires constant improvement of approaches to modeling physical processes and phenomena. Practically all scientific problems can be described by systems of differential equations. Many scientific problems are described by systems of differential equations of a special class, which belong to the group of so-called singularly perturbed differential equations. Mathematical models of processes described by such differential equations contain a small parameter near the highest derivatives, and it was the presence of this small factor that led to the creation of a large mathematical theory. The work proposes a developed algorithm for constructing uniform asymptotics of solutions to systems of singularly perturbed differential equations.

The limit properties diffusion process in a semi-Markov environment

Chabanyuk Y., Rosa W.

Journal of Applied Mathematics and Computational Mechanics

2018

Vol. 17, nr 1

5--14

In this paper we consider the stochastic diffusion process with semi-Markov switchings in an averaging scheme. We present results and conditions on convergence to the classic diffusion process, in case with semi-Markov process perturbation is uniformly ergodic. We used small parameter scheme to get the main result.

Inertia effects in the flow of a micropolar fluid in a slot between rotating surfaces of revolution

Walicka A.

International Journal of Applied Mechanics and Engineering

2001

Vol. 6, no 3

731-790

At present there exist several approaches to the formulation of fluids that contain structures. These fluids have various names such as simple microfluids, micropolar fluids, deformable directed fluids, polar fluids, anisotropic fluids, etc. In this paper the steady laminar flow of micropolar fluid in a slot between rotating surfaces of revolution, having a common axis of symmetry, is considered. To solve this problem the boundary layer equations for micropolar fluid are used and expressed for the axially symmetric case in the intrinsic curvilinear orthogonal coordinate system _ . The method of small parameter is used to solve the boundary layer equations. As a result one obtains the formulae for the velocity field and pressure. The solution to the equations of motion have been illustrated by plots of velocity components _ microrotation _ and pressure p.