New numerical approach for fractional differential equations

• Published in: Mathematical modelling of natural phenomena Vol. 13; no. 1; p. 3
• Main Authors: Atangana, Abdon, Owolabi, Kolade M.
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 2018
• Subjects: 26A33; 34A34; 35B44; 65M06; Adams-Bashforth method; Atangana-Baleanu derivative; Caputo-Fabrizio derivative; Differential equations; fractional differential equation; Kernels; Nonlinear systems; Nonlinearity; numerical approximation; Power law
• ISSN: 0973-5348; 1760-6101
• DOI: 10.1051/mmnp/2018010

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.