Abstract
The major aim of this paper is the presentation of Aboodh transform of the Atangana–Baleanu fractional differential operator both in Caputo and Riemann–Liouville sense by using the connection between the Laplace transform and the Aboodh transform. Moreover, we aim to obtain the approximate series solutions for the time-fractional differential equations with an Atangana–Baleanu fractional differential operator in the Caputo sense using the Aboodh transform iterative method, which is the modification of the Aboodh transform by combining it with the new iterative method. The relation between the Laplace transform and the Aboodh transform is symmetrical. Some graphical illustrations are presented to describe the effect of the fractional order. The outcome reveals that Aboodh transform iterative method is easy to implement and adequately captures the behavior and the fractional effect of the fractional differential equation.
Highlights
The role of fractional differential operations in evaluating and simulating history dependent evolution models in physics and engineering cannot be over emphasized because of their properties [1,2,3,4,5]
If M(ψ) is the Aboodh transform of Q(t) ∈ C and Q(s) is the Laplace transform of Q(t)∈ C, the Aboodh transform of Atangana–Baleanu fractional derivative according to the Caputo sense is derived as follows
Assume that M(ψ) is the Aboodh transform of Q(t) ∈ C and Q(s) is the Laplace transform of Q(t) ∈ C, the Aboodh transform of Atangana–Baleanu fractional derivative according to the Riemann–Liouville sense is derived as follows
Summary
The role of fractional differential operations in evaluating and simulating history dependent evolution models in physics and engineering cannot be over emphasized because of their properties [1,2,3,4,5]. Atangana and Baleanu presented a new fractional differential operator which utilizes the Mittag–Leffler function as the kernel to replace the exponential function kernel of the Caputo-Fabrizo fractional differential operator [11,12]. The novelty of this paper is the establishment of the Aboodh transform of the Atangana– Baleanu fractional differential operator both in the Caputo and Riemman–Liouville sense using the connection between the Laplace transform and the Aboodh transform. We validate the Aboodh transform iterative method [4] for the solution of Atangana–Baleanu fractional differential equation.
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