A comparative study of the Chebyshev collocation method and the finite difference method for solving fourth-order partial differential equations

2019 ◽  
Author(s):  
Somruthai Apornsaengsawang ◽  
Khomsan Neamprem ◽  
Farida Chamchod
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Comparative Study ◽  
Collocation Method ◽  
Fourth Order ◽  
Difference Method ◽  
Chebyshev Collocation Method ◽  
The Finite Difference Method

scholarly journals Modified Chebyshev Collocation Method for Solving Differential Equations

CAUCHY ◽  
2015 ◽  
Vol 3 (4) ◽  
pp. 208
Author(s):  
M Ziaul Arif ◽  
Ahmad Kamsyakawuni ◽  
Ikhsanul Halikin
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Collocation Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Polynomial Collocation

This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

Transients During a Railway Air Brake Release Demand

1990 ◽  
Vol 204 (1) ◽  
pp. 31-38 ◽  
Author(s):  
M A Murtaza ◽  
S B L Garg
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Laboratory Data ◽  
One Dimensional ◽  
The Finite Difference Method ◽  
Air Brake System ◽  
Good Agreement

This paper deals with the simulation of railway air brake release demand of a twin-pipe graduated release railway air brake system based on the solution of partial differential equations governing one-dimensional flow by the finite difference method supported by extrapolation/interpolation. Air brake release demand is simulated as an exponential input of pressure. The analysis incorporates the corrections needed to be used for various restrictions in the brake pipeline. Results are in good agreement with the laboratory data.

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Timoshenko Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Functionally Graded ◽  
Timoshenko Beams ◽  
Partial Differential ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded non-prismatic Timoshenko beams (AFGNPTB) using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, shear moduli, and mass density) of axially functionally graded beams vary along the longitudinal axis. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation considerations. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating differential or partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, and brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. Vibration analysis of AFGNPTB was conducted with this model, and natural frequencies were determined. Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFGNPTB, considering the damping. The efforts and displacements could be determined at any time.

Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations

1993 ◽  
Vol 60 (1) ◽  
pp. 167-174 ◽  
Author(s):  
N. S. Abhyankar ◽  
E. K. Hall ◽  
S. V. Hanagud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Chaotic Dynamics ◽  
Partial Differential ◽  
Galerkin Approximations ◽  
The Finite Difference Method

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Partial Differential ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential or partial differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating differential or partial differential equations with finite differences and introducing new points (additional or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, and brutal change of stiffness). The introduction of additional points allows satisfying boundary and continuity conditions. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, the direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The efforts and displacements could be determined at any time.

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