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:
| [1] | Atangana, A.; Owolabi, K. M., New numerical approach for fractional differential equations, Math Model Nat Phenom, 13, 1, 3 (2018) · Zbl 1410.65323 |
| [2] | Deng, W., Short memory principle and a predictor-corrector approach for fractional differential equations, J Comput Appl Math, 206, 1, 174-188 (2007) · Zbl 1121.65128 |
| [3] | Diethelm, K., Smoothness properties of solutions of Caputo-type fractional differential equations, Fract Calc Appl Anal, 10, 2, 151-160 (2007) · Zbl 1136.26302 |
| [4] | Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam, 29, 1-4, 3-22 (2002) · Zbl 1009.65049 |
| [5] | Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer Algorithms, 36, 1, 31-52 (2004) · Zbl 1055.65098 |
| [6] | Dixon, J., On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions, BIT, 25, 4, 624-634 (1985) · Zbl 0584.65091 |
| [7] | Ford, N. J.; Simpson, A. C., The numerical solution of fractional differential equations: speed versus accuracy, Numer Algorithms, 26, 4, 333-346 (2001) · Zbl 0976.65062 |
| [8] | Garrappa, R., Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J Numer Anal, 53, 3, 1350-1369 (2015) · Zbl 1331.33043 |
| [9] | Garrappa, R., Trapezoidal methods for fractional differential equations: theoretical and computational aspects, Math Comput Simulation, 110, 96-112 (2015) · Zbl 1540.65194 |
| [10] | Garrappa, R., Numerical solution of fractional differential equations: a survey and a software tutorial, Mathematics, 6, 2, 16 (2018) · Zbl 06916890 |
| [11] | Giusti, A., A comment on some new definitions of fractional derivative, Nonlinear Dyn, 93, 3, 1757-1763 (2018) · Zbl 1398.28001 |
| [12] | Lubich, C., Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math Comp, 41, 163, 87-102 (1983) · Zbl 0538.65091 |
| [13] | Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering, 198 (1999), Academic Press, Inc.: Academic Press, Inc. San Diego, CA · Zbl 0918.34010 |
| [14] | Stynes, M., Too much regularity may force too much uniqueness, Fract Calc Appl Anal, 19, 6, 1554-1562 (2016) · Zbl 1353.35306 |
| [15] | Tarasov, V. E., Local fractional derivatives of differentiable functions are integer-order derivatives or zero, Int J Appl Comput Math, 2, 2, 195-201 (2016) · Zbl 1420.26006 |
| [16] | Tarasov, V. E., No nonlocality. No fractional derivative, Commun Nonlinear Sci Numer Simul, 62, 157-163 (2018) · Zbl 1470.26014 |
| [17] | Young, A., Approximate product-integration, Proc R Soc London Ser-A, 224, 552-561 (1954) · Zbl 0055.36001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
