scholarly journals A collocation method for the Williams equation with Chebyshev polynomials

2021 ◽  
Vol 2056 (1) ◽  
pp. 012005
Author(s):  
O V Germider ◽  
V N Popov
Keyword(s):  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Basic Equation ◽  
Gas Flow ◽  
Diffuse Reflection ◽  
Constant Pressure ◽  
Plane Channel ◽  
Reflection Model ◽  
Linearized Problem ◽  
The Given

Abstract The linearized problem of gas flow in plane channel with infinite walls has been solved in the kinetic approximation. The flow in the channel is caused by a constant pressure gradient parallel to the walls of the channel. The Williams equation has been used as a basic equation, and the boundary condition has been set in terms of the diffuse reflection model. The collocation method for Chebyshev polynomials has been applied to construct the solution of the equation of Williams with the given boundary conditions. The mass flux of the gas in the channel has been calculated.

scholarly journals Неизотермическое течение газа в эллиптическом канале с внутренним круговым цилиндрическим элементом в свободномолекулярном режиме

2019 ◽  
Vol 89 (1) ◽  
pp. 27
Author(s):  
О.В. Гермидер ◽  
В.Н. Попов
Keyword(s):  
Mass Flow ◽  
Basic Equation ◽  
Gas Flow ◽  
Diffuse Reflection ◽  
Boltzmann Kinetic Equation ◽  
Molecular Gas ◽  
Pressure Drops ◽  
Reflection Model ◽  
Linearized Problem ◽  
Temperature And Pressure

AbstractThe linearized problem of free-molecular gas flow in a long elliptic channel with a circular cylindrical element inside has been solved in the kinetic approximation. The flow in the channel is caused by temperature and pressure drops between its ends. The Boltzmann kinetic equation for collisionless gas has been used as a basic equation, and the boundary condition has been set in terms of the diffuse reflection model. The distribution of the mass velocity of the gas over the cross section of the channel has been obtained. The mass flow rate of the gas in the channel versus the temperature and pressure drops between its ends has been calculated. It has been found that the mass flow of the gas substantially depends on the radius of the inner cylinder.

scholarly journals Analytical solution to the problem of diffusion of the light component of the binary gas mixture in a plane channel

2021 ◽  
Vol 2056 (1) ◽  
pp. 012004
Author(s):  
V N Popov ◽  
I V Popov
Keyword(s):  
Analytical Solution ◽  
Flow Rate ◽  
Diffuse Reflection ◽  
Plane Channel ◽  
Boltzmann Kinetic Equation ◽  
Heavy Component ◽  
Light Component ◽  
Reflection Model ◽  
Binary Gas Mixture ◽  
Gas Component

Abstract Within the framework of the kinetic approach, an analytical solution to the problem of diffusion of the light component of a binary mixture in a flat channel with infinite parallel walls is constructed. It is assumed that the mass of light component molecules and their concentration is much less than the mass of molecules and the concentration of heavy components. The flow rate of the heavy component is assumed to be zero. The change in the state of a light gas component is described on the basis of the BGK (Bhatnagar, Gross, Kruk) model of the Boltzmann kinetic equation. The diffuse reflection model is used as a boundary condition on the channel walls. The mass velocity profile of the light gas component is constructed. The flow rate of the light gas component per unit channel width is calculated. A comparison with similar results presented in open sources was done.

scholarly journals Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 181
Author(s):  
Yalçın ÖZTÜRK
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Matrix Equation ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Solution Function ◽  
The Given

In this paper, we use the collocation method together with Chebyshev polynomials to solve system of Lane–Emden type (SLE) equations. We first transform the given SLE equation to a matrix equation by means of a truncated Chebyshev series with unknown coefficients. Then, the numerical method reduces each SLE equation to a nonlinear system of algebraic equations. The solution of this matrix equation yields the unknown coefficients of the solution function. Hence, an approximate solution is obtained by means of a truncated Chebyshev series. Also, to show the applicability, usefulness, and accuracy of the method, some examples are solved numerically and the errors of the solutions are compared with existing solutions.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

Simulation of transient gas flow using the orthogonal collocation method

2012 ◽  
Vol 90 (11) ◽  
pp. 1701-1710 ◽  
Author(s):  
Edris Ebrahimzadeh ◽  
Mahdi Niknam Shahrak ◽  
Bahamin Bazooyar
Keyword(s):  
Collocation Method ◽  
Gas Flow ◽  
Orthogonal Collocation

Productivity Analysis of Transverse Horizontal Well Commingling in Stripped Gas Reservoir with Bottom Water

2013 ◽  
Vol 868 ◽  
pp. 657-663
Author(s):  
Wen Qi Zhao ◽  
Lian Yu ◽  
Lun Zhao ◽  
Li Chen ◽  
Song Chen
Keyword(s):  
Gas Flow ◽  
Horizontal Well ◽  
Bottom Water ◽  
Constant Pressure ◽  
Production Condition ◽  
Productivity Analysis ◽  
Gas Reservoir ◽  
Constant Rate ◽  
Seepage Theory ◽  
The Green Function

Banded gas reservoir with bottom water is a typical gas reservoir. Based on the development characteristics of formation, the further study on commingling production by transverse horizontal well is done, and also combined with seepage theory including the Green function, Duhamel method etc. the dimensionless definitions of relative parameter are given. Whats more, the derivation and formula of production can be achieved respectively under the constant pressure condition and constant production condition. Meanwhile, take commingling production with three banded gas layers for example. Whether the initial pseudo pressure for these layers are equal or not, in both of these cases, the variation curve of gas productivity under the constant pressure and constant rate conditions is described separately. And the law of variation of gas flow backward for some layer when the physical properties of these layers are significantly different.

scholarly journals Numerical Solution of Nonlinear Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
R. C. Mittal ◽  
Rachna Bhatia
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Gordon Equation ◽  
Spline Collocation ◽  
Sine Gordon Equation ◽  
B Spline ◽  
Spline Collocation Method ◽  
B Splines ◽  
Spline Basis ◽  
The Given

Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.

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