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.

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.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

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 Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

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.

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