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Zbl 1218.65070
Li, Changpin; Chen, An; Ye, Junjie
Numerical approaches to fractional calculus and fractional ordinary differential equation. (English)
[J] J. Comput. Phys. 230, No. 9, 3352-3368 (2011). ISSN 0021-9991

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 2000:: *65L05 Initial value problems for ODE (numerical methods)
34A08
34A34 Nonlinear ODE and systems, general
65L20 Stability of numerical methods for ODE
65L70 Error bounds (numerical methods for ODE)

Keywords: numerical results; fractional calculus; fractional differential equations; piecewise interpolation; Simpson method; fractional derivatives; fractional integral; error analysis; stability; numerical results

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Zentralblatt MATH Berlin [Germany]