Analytical properties and solution methods of differential equations involving Atangana–baleanu fractional operators
DOI:
https://doi.org/10.29070/vqgmg105Keywords:
Atangana–Baleanu fractional derivative, Mittag-Leffler kernel, fixed-point theorem, Laplace transform, Ulam–Hyers stability, Fractional calculusAbstract
The fractional calculus is known as a powerful mathematical tool for the description of systems with memory, hereditary properties and non-locality. The Atangana–Baleanu (AB) fractional operators based on the non-singular, non-local Mittag-Leffler kernel has attracted much research attention of the fractional operators developed to overcome the singular-kernel limitations of the classical Riemann–Liouville and Caputo derivatives. In this paper, a systematic analytical study of differential equations with Atangana-Baleanu fractional operators are presented. We introduce an extended modified Atangana-Baleanu Caputo derivative together with its associated integral operator and prove their main properties including linearity, boundedness, non-locality and consistency with the classical derivatives. A generalised Laplace transform of the extended operator is derived and employed to develop methods for analytical solutions of linear and nonlinear Atangana-Baleanu fractional differential equations (AB-FDEs) including initial value and boundary value problems. Existence, uniqueness and Ulam-Hyers stability of solutions are proved via Banach and Krasnoselskii type fixed point theorems. Furthermore, within the extended AB framework we obtain new fractional derivative relations for the Gamma, Beta, hypergeometric and orthogonal polynomial functions. Numerical examples are provided to validate the theoretical results and a comparative discussion is provided to place the Atangana–Baleanu operator with regard to classical fractional derivatives. The results provide analytical tools that help in the treatment of memory dependent differential equations arising in applied mathematics, physics and engineering.
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