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Numerical approaches to fractional calculus and fractional ordinary differential equation. (English) Zbl 1218.65070
Summary: Nowadays, fractional calculus are used to model various different phenomena in nature, but due to the non-local property of the fractional derivative, it still remains a lot of improvements in the present numerical approaches. In this paper, some new numerical approaches based on piecewise interpolation for fractional calculus, and some new improved approaches based on the Simpson method for the fractional differential equations are proposed. We use higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and use the Simpson method to design a higher order algorithm for the fractional differential equations. Error analyses and stability analyses are also given, and the numerical results show that these constructed numerical approaches are efficient.

MSC:
65L05 Numerical methods for initial value problems
34A08 Fractional ordinary differential equations and fractional differential inclusions
34A34 Nonlinear ordinary differential equations and systems, general theory
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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