A new derivative with normal distribution kernel: Theory, methods and applications

• Published in: Physica A Vol. 476; pp. 1 - 14
• Main Authors: Atangana, Abdon, Gómez-Aguilar, J.F.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.06.2017
• Subjects: Continuous probability distribution; Exponential decay function; Mittag-Leffler function; Numerical approximations; Numerical approximations; Exponential decay function; Mittag-Leffler function; Continuous probability distribution
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2017.02.016

New approach of fractional derivative with a new local kernel is suggested in this paper. The kernel introduced in this work is the well-known normal distribution that is a very common continuous probability distribution. This distribution is very important in statistics and also highly used in natural science and social sciences to portray real-valued random variables whose distributions are not known. Two definitions are suggested namely Atangana–Gómez Averaging in Liouville–Caputo and Riemann–Liouville sense. We presented some relationship with existing integrals transform operators. Numerical approximations for first and second order approximation are derived in detail. Some Applications of the new mathematical tools to describe some real world problems are presented in detail. This is a new door opened the field of statistics, natural and socials sciences. •New differential operator was constructed using the normal distribution as kernel.•A new integral operator has been obtained.•Relations of new operators and existing integrals transform have been established.•The numerical approximation of the new operators has been suggested.•The new operators were applied to model three real world problems.