On the mistake in defining fractional derivative using a non-singular\n kernel

2019/12/09 by E. Capelas de Oliveira, de Oliveira, Edmundo Capelas, Stefânia Jarosz +3
Engineering · Mathematics · #26A33 #35E15 #44A10 #60J60 #Classical Analysis and ODEs (math.CA) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical Physics (math-ph) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1912.04422

openalex publication_date 2019/12/09 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28

Abstract

Definitions of fractional derivative of order \α (0 < \α \≤ 1)\nusing non-singular kernels have been recently proposed. In this note we show\nthat these definitions cannot be useful in modelling problems with a initial\nvalue condition (like, for example, the fractional diffusion equation) because\nthe solutions obtained for these equations do not satisfy the initial condition\n(except for the integer case \α = 1). In order to satisfy an arbitrary\ninitial condition the definitions of fractional derivative must necessarily\ninvolve a singular kernel.\n

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