Math. Model. Nat. Phenom.
Volume 13, Number 1, 2018
Theory and applications of fractional differentiation
|Number of page(s)||21|
|Published online||06 April 2018|
New numerical approach for fractional differential equations
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
9300, South Africa
* Corresponding author: firstname.lastname@example.org
Accepted: 8 November 2017
In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario.The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power α = 1. Numerical results are finally given to justify the effectiveness of the proposed schemes.
Mathematics Subject Classification: 26A33 / 34A34 / 35B44 / 65M06
Key words: Adams-Bashforth method / Atangana-Baleanu derivative / Caputo-Fabrizio derivative / fractional differential equation / numerical approximation
© EDP Sciences, 2018
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