Initial value problem involving Atangana-Baleanu derivative is considered. An Explicit solution of the given problem is obtained by reducing the differential equation to Volterra integral equation of second kind and by using Laplace transform. To find the solution of the Volterra equation, the successive approximation method is used and a lemma simplifying the resolvent kernel has been presented. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of series expansion using orthogonal basis obtained by separation of variables.
Recently, two newly definitions of fractional derivative without singular kernel were suggested, namely, Caputo-Fabrizio fractional derivative @cite_5 and Atangana-Baleanu fractional derivative @cite_2 . These new derivatives have been applied to real life problems, for example, in the fields of thermal science, material sciences, groundwater modelling and mass-spring system @cite_12 , @cite_10 , @cite_4 , @cite_11 , @cite_2 and have been considered in a number of other recent work, see for example, @cite_13 , @cite_9 , @cite_8 , @cite_0 , @cite_3 , @cite_6 . The main difference between these two definitions is that Caputo-Fabrizio derivative is based on exponential kernel while Atangana-Baleanu definition used Mittag-leffler function as a non-local kernel. The non-locality of the kernel gives better description of the memory within structure with different scale. These two new derivatives are defined as follows and the Atangana-Baleanu fractional derivative is given by where @math denotes a normalization function such that @math and @math is the Mittag-leffler function of one parameter @cite_7 .
@cite_13 @cite_4 @cite_7 @cite_8 @cite_9 @cite_3 @cite_6 @cite_0 @cite_2 @cite_5 @cite_10 @cite_12 @cite_11
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Abstract Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative. In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model. In this work, we consider a number of boundary-value problems for time-fractional heat equation with the recently introduced Caputo-Fabrizio derivative. Using the method of separation of variables, we prove a unique solvability of the stated problems. Moreover, we have found an explicit solution to certain initial value problem for Caputo-Fabrizio fractional order differential equation by reducing the problem to a Volterra integral equation. Different forms of solution were presented depending on the values of the parameter appeared in the problem. Abstract In order to control the movement of waves on the area of shallow water, the newly derivative with fractional order proposed by Caputo and Fabrizio was used. To achieve this, we first proposed a transition from ordinary to fractional differential equation. We proved the existence and uniqueness of the coupled solutions of the modified system using the fixed-point theorem. We derive the special solution of the modified system using an iterative method. We proved the stability of the used method and also the uniqueness of the special solution. We presented the numerical simulations for different values of alpha.