Modified Results by Using New Generalized Definition of Fractional Derivative without Singular Kernel by Applying New Generalized Five Parameter Mittag-Leffler Function

In a series of papers [43-46] reviewed all results and generalized the existing results by modifications. In this article, a new approach of the derivative of arbitrary order (FD) with the kernel of the smooth type that gains different depictions for the temporal and spatial variables has been given. It first applies to the time variables and hence it is fit to us transform of Laplace type (LT). Secondly, a definition is linked to the spatial type variables, by a global derivative of arbitrary order (FD), for which we will apply the transform of Fourier type (FT). The courtesy for this new methodology with a kernel of regular type was native from the vision that there is a period of global systems, which can designate the material heterogeneities and the fluctuations of unlike scales, which cannot be well described by traditional local theories or by arbitrary order models with the kernel of singular type. In this endeavour we are introducing a new generalized five parameter Mittag-Leffler function which is used in the definition of fractional derivative.