scholarly journals A SYSTEM OF DIFFERENTIAL EQUATIONS WITH A SMALL PARAMETER: A NUMERICAL SOLUTION BASED ON ASYMPTOTIC REPRESENTATIONS

2017 ◽  
Author(s):  
N. Yu. Petukhova ◽  
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Numerical Solution ◽  
System Of Differential Equations ◽  
Asymptotic Representations

The nonwandering set of a special system of differential equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 35-48
Author(s):  
V. A. Pliss
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
System Of Differential Equations ◽  
Special System ◽  
Non Linear ◽  
Nonwandering Set

SynopsisIn the theory of non-linear oscillations there occur systems with a small parameter in the derivatives and discontinuous forcing terms. Here we study such a system.

scholarly journals Численный метод решения краевой задачи пятого порядка для обыкновенных дифференциальных уравнений

2018 ◽  
Author(s):  
K.P. Pramod
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Numerical Technique ◽  
Boundary Value ◽  
Test Problems ◽  
Algebraic Equations ◽  
System Of Differential Equations ◽  
Differential Problem ◽  
Fifth Order

In this article we have proposed a technique for solving the fifth order boundary value problem as a coupled pair of boundary value problems. We have considered fifth order boundary value problem in ordinary differential equation for the development of the numerical technique. There are many techniques for the numerical solution of the problem considered in this article. Thus we considered the application of the finite difference method for the numerical solution of the problem. In this article we transformed fifth order differential problem into system of differential equations of lower order namely one and four. We discretized the system of differential equations into considered domain of the problem. Thus we got a system of algebraic equations. For the numerical solution of the problem, we have the system of algebraic equations. The solution of the algebraic equations is an approximate solution of the problem considered. Moreover we get numerical approximation of first and second derivative as a byproduct of the proposed method. We have shown that proposed method is convergent and order of accuracy of the proposed method is at lease quadratic. The numerical results obtained in computational experiment on the test problems approve the efficiency and accuracy of the method.

scholarly journals ON SOME ALGORITHMS FOR THE NUMERICAL SOLUTION OF THE CAUCHY PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER RUNGE - KUTT METHOD IN DELPHI

2014 ◽  
Vol 12 (04) ◽  
pp. 26-31
Author(s):  
Erik Nurlanovich Bayandiyev ◽  
◽  
Lyazat Rysbekovna Seytbekova ◽  
Aynur Tursynkhanovna Tolkynbayeva ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Numerical Solution ◽  
System Of Differential Equations ◽  
First Order ◽  
The Cauchy Problem

scholarly journals Mathematical modeling of film flow of a liquid on a surface of a body of a rotation

2018 ◽  
Author(s):  
I. S. Tonkoshkur
Keyword(s):  
Differential Equations ◽  
Film Thickness ◽  
Small Parameter ◽  
Rheological Model ◽  
The Body ◽  
Viscoplastic Fluid ◽  
Parameter Method ◽  
Order Partial Differential Equation ◽  
System Of Differential Equations ◽  
Partial Differential

The problem of the spatial nonwave stationary flow of the viscoplastic fluid on the surface of the body of rotation under the action of gravity is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the dis-tance along the normal to the surface. To describe the flow of a liquid film, a viscous in-compressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. For the closure of a system of differential equations, the Schulman rheological model is used, which is a gener-alization of the Ostwald-de-Ville power model and the Shvedov-Bingham viscoplastic model. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations according to the proposed method for the circular cone located at a certain angle to the vertical are presented. Calculations show that the parameters of nonlinearity and plasticity of this rheological model of a liquid can significantly affect the speed profiles and the distribution of the thickness of the viscous layer on the surface of the body

scholarly journals MODELING THE INTERACTION OF A FILM OF A NONLINEAR VISCOUS LIQUID WITH A GAS FLOW

2021 ◽  
pp. 203-209
Author(s):  
Illia Tonkoshkur
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Viscous Liquid ◽  
Gas Flow ◽  
Flat Surface ◽  
Liquid Films ◽  
System Of Differential Equations ◽  
Layer Film ◽  
Gas Layer ◽  
Analytical Expressions

The problem of the interaction of a two-layer film of a nonlinear viscous liquid flowing down a flat surface with a gas flow directed vertically up or down is considered. To simplify the initial system of differential equations, the method of a small parameter is used, for which the relative thicknesses of the films and the gas layer were chosen. Analytical expressions are obtained for the profiles of the velocities and thicknesses of liquid films.

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