Analysis of Laminated Anisotropic Cylindrical Shell by Chebyshev Collocation Method

2003 ◽  
Vol 70 (3) ◽  
pp. 391-403 ◽  
Author(s):  
C.-H. Lin ◽  
M.-H. R. Jen
Keyword(s):  
Exact Solution ◽  
Cylindrical Shell ◽  
Collocation Method ◽  
Anisotropic Shell ◽  
Difficult Problem ◽  
Governing Equations ◽  
Chebyshev Collocation Method ◽  
Independent Variables ◽  
Chebyshev Collocation ◽  
Anisotropic Cylindrical Shell

The governing equations of a laminated anisotropic cylindrical shell problem are a system of partial differential equations. The boundary conditions will complicate the problem. Thus, it is hard to handle the governing equations in the form of functions of independent variables. Herein, Chebyshev collocation method is proposed to achieve the exact solution theoretically of such a difficult problem. Finally, two examples with numerical results are presented. The preciseness and efficiency of the proposed Chebyshev collocation method for laminated anisotropic shell problem are highlighted.

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.

An embedded formula of the Chebyshev collocation method for stiff problems

2017 ◽  
Vol 351 ◽  
pp. 376-391 ◽  
Author(s):  
Xiangfan Piao ◽  
Sunyoung Bu ◽  
Dojin Kim ◽  
Philsu Kim
Keyword(s):  
Collocation Method ◽  
Stiff Problems ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation

scholarly journals A Spectral Method for Two-Dimensional Ocean Acoustic Propagation

2021 ◽  
Vol 9 (8) ◽  
pp. 892
Author(s):  
Xian Ma ◽  
Yongxian Wang ◽  
Xiaoqian Zhu ◽  
Wei Liu ◽  
Qiang Lan ◽  
...  
Keyword(s):  
Numerical Calculation ◽  
Spectral Method ◽  
Collocation Method ◽  
Approximation Error ◽  
Sound Field ◽  
Acoustic Propagation ◽  
Two Dimensional ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Wide Range

The accurate calculation of the sound field is one of the most concerning issues in hydroacoustics. The one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it is difficult to solve actual ocean acoustic fields using this model due to its application conditions and approximation error. Therefore, it is necessary to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation without using simplified models. Here, two commonly used spectral methods, Chebyshev–Galerkin and Chebyshev–collocation, are used to correctly solve the two-dimensional Helmholtz model equation. Since Chebyshev–collocation does not require harsh boundary conditions for the equation, it is then used to solve ocean acoustic propagation. The numerical calculation results are compared with analytical solutions to verify the correctness of the method. Compared with the mature Kraken program, the Chebyshev–collocation method exhibits higher numerical calculation accuracy. Therefore, the Chebyshev–collocation method can be used to directly solve the representative two-dimensional ocean acoustic propagation equation. Because there are no model constraints, the Chebyshev–collocation method has a wide range of applications and provides results with high accuracy, which is of great significance in the calculation of realistic ocean sound fields.

scholarly journals The impact of the Chebyshev collocation method on solutions of the time-fractional Black–Scholes

2020 ◽  
Author(s):  
H. Mesgarani ◽  
A. Beiranvand ◽  
Y. Esmaeelzade Aghdam
Keyword(s):  
Collocation Method ◽  
Linear Interpolation ◽  
Stability And Convergence ◽  
Finite Domain ◽  
European Options ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Black Scholes ◽  
Temporal Derivative ◽  
The Impact

AbstractThis paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the $$(2-\alpha)$$ ( 2 - α ) -order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

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