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On the fractional derivatives of radial basis functions: Theories and applications
Author(s) -
Saberi Zafarghandi Fahimeh,
Mohammadi Maryam,
Schaback Robert
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5618
Subject(s) - mathematics , fractional calculus , discretization , dimension (graph theory) , differential operator , gaussian , differential equation , order (exchange) , basis (linear algebra) , mathematical analysis , operator (biology) , radial basis function , type (biology) , pure mathematics , geometry , biochemistry , chemistry , physics , finance , repressor , quantum mechanics , machine learning , artificial neural network , transcription factor , computer science , economics , gene , ecology , biology
The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.