The Galerkin Method for Solving Radiation Transfer in Plane-Parallel Participating Media

1982 ◽  
Vol 104 (2) ◽  
pp. 351-354 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
Y. Yener
Keyword(s):  
Galerkin Method ◽  
Integral Form ◽  
Computer Time ◽  
Incident Radiation ◽  
Algebraic Equations ◽  
Participating Media ◽  
Plane Parallel ◽  
Reflecting Boundaries ◽  
Expansion Coefficients ◽  
The Galerkin Method

The Galerkin method is applied to solve radiative heat transfer in an isotropically scattering, absorbing, and emitting plane-parallel medium with diffusely reflecting boundaries. In this approach, the integral form of the equation of radiative transfer is transformed into a set of algebraic equations for the determination of the expansion coefficients associated with the representation of the incident radiation in a power series in the space variable. The method is easy and straightforward to apply and requires relatively little computer time for the computations, since explicit analytical expressions are obtainable for the expansion coefficients.

The Application of the Galerkin Method to the Solution of the Symmetric and Balanced Counterflow Regenerator Problem

1985 ◽  
Vol 107 (1) ◽  
pp. 214-221 ◽  
Author(s):  
B. S. Baclic
Keyword(s):  
Galerkin Method ◽  
Space Variable ◽  
Analytical Formula ◽  
Algebraic Equations ◽  
Higher Order Terms ◽  
The Matrix ◽  
Expansion Coefficients ◽  
The Galerkin Method ◽  
Analytical Expressions

The Galerkin method is applied to solve the symmetric and balanced counterflow thermal regenerator problem. In this approach, the integral equation, expressing the reversal condition in periodic equilibrium of regenerator matrix, is transformed into a set of algebraic equations for the determination of the expansion coefficients associated with the representation of the matrix temperature distribution at the start of cold period in a power series in the space variable. The method is easy and straightforward to apply and leads to the explict analytical expressions for expansion coefficients. As explicit analytical formula for regenerator effectiveness is derived and the corresponding numerical values are computed. An excellent agreement is found between the present results and those reported in the literature by different numerical methods. The convergence towards the exact results by carrying out the computations to higher order terms, as well as the extension of this method to the more general counterflow regenerator problem is discussed.

A Source Function Expansion in Radiative Transfer

1980 ◽  
Vol 102 (4) ◽  
pp. 715-718 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
W. H. Sutton
Keyword(s):  
Radiative Transfer ◽  
Source Function ◽  
Transfer Problem ◽  
Integral Form ◽  
Computer Time ◽  
Algebraic Equations ◽  
Function Expansion ◽  
Expansion Technique ◽  
Reflecting Boundaries ◽  
Special Case

The radiative heat transfer problem for an isotropically scattering slab with specularly reflecting boundaries is reduced to the solution of a set of algebraic equations by expanding the source function in Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The lowest order S-1 analysis requires very little computer time for calculations, is easy to apply and yields results which are sufficiently accurate. For an absorbing, emitting, isotropically scattering medium with small and intermediate optical thickness (i.e., τ = 2), which is of great interest in engineering applications, and for which the P-1 and P-3 solutions of the P-N method are not sufficiently accurate, the S-1 solution yields highly accurate results. In the case of a slab with diffusely reflecting boundaries, the problem is split up into a set of simpler problems each of which is solved with the source function expansion technique as a special case of the general problem considered.

A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

2003 ◽  
Author(s):  
Jose´ Antunes ◽  
Miguel Moreira ◽  
Philippe Piteau
Keyword(s):  
Fourier Series ◽  
Galerkin Method ◽  
Finite Length ◽  
Reynolds Equation ◽  
Double Fourier Series ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Squeeze Film Dampers ◽  
Galerkin Solution ◽  
The Galerkin Method

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.

Wavelets Galerkin Method for the Fractional Subdiffusion Equation

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. H. Heydari
Keyword(s):  
Galerkin Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Practical Importance ◽  
Time Derivative ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Equivalent Problem ◽  
Subdiffusion Equation ◽  
The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

scholarly journals Dynamic Behavior of a Cylindrical Shell with a Liquid under the Action of Nonstationary Pressure Wave

TEM Journal ◽  
2021 ◽  
pp. 815-819
Author(s):  
Boris A. Antufev ◽  
Vasiliy N. Dobryanskiy ◽  
Olga V. Egorova ◽  
Eduard I. Starovoitov
Keyword(s):  
Cylindrical Shell ◽  
Galerkin Method ◽  
Pressure Wave ◽  
Moving Load ◽  
Algebraic Equations ◽  
Thin Cylindrical Shell ◽  
Incompressibility Condition ◽  
Nonlinear Algebraic Equations ◽  
Deformed State ◽  
The Galerkin Method

The problem of axisymmetric hydroelastic deformation of a thin cylindrical shell containing a liquid under the action of a moving load is approximately solved. It is reduced to the equation of bending of the shell and the condition of incompressibility of the liquid in the cylinder. The deflections of the shell and the level of lowering of the liquid are unknown. For solution, the Galerkin method is used and the problem is reduced to a system of nonlinear algebraic equations. A simpler solution is considered without taking into account the incompressibility condition. Here, in addition to the deformed state of the shell, the critical speeds of the moving load are determined analytically.

scholarly journals The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 486 ◽  
Author(s):  
Neslihan Ozdemir ◽  
Aydin Secer ◽  
Mustafa Bayram
Keyword(s):  
Fractional Derivative ◽  
Galerkin Method ◽  
Collocation Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Coupled System ◽  
Algebraic Equations ◽  
Burgers Equations ◽  
Time Fractional Derivative ◽  
The Galerkin Method

In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.

scholarly journals An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Galerkin Method ◽  
Operational Matrix ◽  
Computational Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Dispersion Parameters ◽  
Wavelet Galerkin Method ◽  
Operational Matrix Of Integration ◽  
Nonlinear Physical Phenomena ◽  
The Galerkin Method

Abstract In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

2020 ◽  
pp. 107754632092761 ◽  
Author(s):  
Mahmood Fakher ◽  
Shahrokh Hosseini-Hashemi
Keyword(s):  
Galerkin Method ◽  
Nonlinear Vibration ◽  
Nonlocal Elasticity ◽  
Integral Form ◽  
Mode Shapes ◽  
Two Phase ◽  
Size Dependency ◽  
Linear Mode ◽  
Size Dependent ◽  
The Galerkin Method

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

Ritz Legendre Multiwavelet Method for the Damped Generalized Regularized Long-Wave Equation

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
S. A. Yousefi ◽  
Z. Barikbin
Keyword(s):  
Wave Equation ◽  
Galerkin Method ◽  
Numerical Method ◽  
Variable Coefficient ◽  
Algebraic Equations ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
The Galerkin Method

In this paper, a numerical method is proposed to approximate the solution of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The method is based upon Ritz Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the nonlinear DGRLW equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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