Modified Chebyshev collocation method for pantograph-type differential equations

2018 ◽  
Vol 134 ◽  
pp. 132-144 ◽  
Author(s):  
Changqing Yang
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation

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.

Fractional Chebyshev Collocation Method for Solving Linear Fractional-Order Delay-Differential Equations

2017 ◽  
Author(s):  
Arman Dabiri ◽  
Eric A. Butcher
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Orthogonal Relationship ◽  
Spectral Convergence ◽  
Discrete Orthogonal ◽  
Delay Differential

An efficient numerical method, the fractional Chebyshev collocation method, is proposed for obtaining the solution of a system of linear fractional order delay-differential equations (FDDEs). It is shown that the proposed method overcomes several limitations of current numerical methods for solving linear FDDEs. For instance, the proposed method can be used for linear incommensurate order fractional differential equations and FDDEs, has spectral convergence (unlike finite differences), and does not require a canonical form. To accomplish this, a fractional differentiation matrix is derived at the Chebyshev-Gauss-Lobatto collocation points by using the discrete orthogonal relationship of the Chebyshev polynomials. Then, using two proposed discretization operators for matrix functions results in an explicit form of solution for a system of linear FDDEs with discrete delays. The advantages of using the fractional Chebyshev collocation method are demonstrated in two numerical examples in which the proposed method is compared with the Adams-Bashforth-Moulton method.

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