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.
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.   In the paper, we present some applications and features related with the new notions of fractional derivatives with a time exponential kernel and with spatial Gauss kernel for gradient and Laplacian operators. Specifically, for these new mode ls we have proved the coherence with the thermodynamic laws. Hence, we have revised the standard linear solid of Zener within continuum mechanics and the model of Cole and Cole inside electromagnetism by these new fractional operators. Moreover, by the Gaussian fractional gradient and through numerical simulations, we have studied the bell shaped filtering effects comparing the results with exponential and Caputo kernel. 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. We introduce the fractional integral corresponding to the new concept of fractional derivative recently introduced by Caputo and Fabrizio and we study some related fractional differential equations.   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.  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.  