Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations. (English) Zbl 1464.65083
Commun. Nonlinear Sci. Numer. Simul. 70, 302-306 (2019); corrigendum ibid. 76, 138-139 (2019).
Summary: In the paper by A. Atangana and K. M. Owolabi [Math. Model. Nat. Phenom. 13, No. 1, Paper No. 3, 21 p. (2018; Zbl 1406.65045)], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations.
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
26A33 | Fractional derivatives and integrals |
34A08 | Fractional ordinary differential equations |
35B44 | Blow-up in context of PDEs |
35R11 | Fractional partial differential equations |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Citations:
Zbl 1406.65045Software:
MLReferences:
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