Numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method

2022 ◽  
Author(s):  
Xiangcheng Zheng
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Collocation Method ◽  
Numerical Approximation ◽  
Variable Order ◽  
Fractional Differential

The formation of dynamic variable order fractional differential equation

2016 ◽  
Vol 27 (07) ◽  
pp. 1650074 ◽  
Author(s):  
S. Sahoo ◽  
S. Saha Ray ◽  
S. Das ◽  
R. K. Bera
Keyword(s):  
Differential Equation ◽  
Transfer Function ◽  
Fractional Differential Equation ◽  
Variable Order ◽  
Dynamic Variable ◽  
Fractional Model ◽  
Fractional Differential ◽  
Viscoelastic Oscillator

In this paper, the formation of variable order (VO) model is established for continuous order fractional model. We review the definitions and properties of VO operators given by many researchers. We use the VO operator to define the new transfer function and analyze the model of a dynamic viscoelastic oscillator.

scholarly journals An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Chebyshev Polynomial ◽  
Algebraic System ◽  
Operational Matrix ◽  
Main Idea ◽  
Operational Matrices ◽  
Variable Order ◽  
Fractional Differential ◽  
Variable Order Fractional Derivative

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

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