Laguerre Matrix-Collocation Method to Solve Systems of Pantograph Type Delay Differential Equations

2020 ◽  
pp. 121-132
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Delay Differential

scholarly journals A new stable collocation method for solving a class of nonlinear fractional delay differential equations

2020 ◽  
Vol 85 (4) ◽  
pp. 1123-1153
Author(s):  
Lei Shi ◽  
Zhong Chen ◽  
Xiaohua Ding ◽  
Qiang Ma
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Orthonormal Bases ◽  
Fractional Delay ◽  
Nonlinear Condition ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

AbstractIn this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of $W^{1}_{2,0}$ W 2 , 0 1 . Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of $W_{2,0}^{1}$ W 2 , 0 1 . To overcome the nonlinear condition, we make use of Newton’s method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

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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