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Axioms 2018, 7(3), 46; https://doi.org/10.3390/axioms7030046

Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems

Dept. SBAI, Università di Roma “La Sapienza”, Via Antonio Scarpa 16, 00161 Roma, Italy
Received: 14 May 2018 / Revised: 17 June 2018 / Accepted: 22 June 2018 / Published: 2 July 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Abstract

Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation–Galerkin method are also displayed. View Full-Text
Keywords: B-spline; optimal basis; fractional derivative; Galerkin method; collocation method B-spline; optimal basis; fractional derivative; Galerkin method; collocation method
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Pitolli, F. Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems. Axioms 2018, 7, 46.

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