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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.
65L05Initial value problems for ODE (numerical methods)
34A08Fractional differential equations
34A34Nonlinear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
65L70Error bounds (numerical methods for ODE)
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