Available online at www.isr-publications.com/mns Math. Nat. Sci., 1 (2017), 1–17 Research Article Journal Homepage:www.isr-publications.com/mns The non-linear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization Jordan Hristov Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia 1756, 8 Kliment Ohridsky, blvd. Bulgaria. Communicated by O. K. Matthew Abstract The Dodson mass diffusion equation with exponentially diffusivity is analyzed through approximate integral solutions. Integral-balance solutions were developed to integer-order versions as well as to formally fractionalized models. The formal fractionalization considers replacement of the time derivative with a fractional version with either singular (Riemann-Liouville or Caputo) or non-singular fading memory. The solutions developed allow seeing a new side of the Dodson equation and to separate the formal fractional model with Caputo-Fabrizio time derivative with an integral-balance allowing relating the fractional order to the physical relaxation time as adequate to the phenomena behind. c 2017 All rights reserved. Keywords: Non-linear diffusion, singular fading memory, non-singular fading memory, formal fractionalization, Caputo-Fabrizio derivative, integral balance approach. 2010 MSC: 47H10, 54H25. 1. Introduction 1.1. Dodson diffusion equation The Dodson diffusion equation (1.1) is related to diffusion in minerals [11, 12] related to cooling history in geology ∂C(x, t) ∂t = D 0 exp(-βt) ∂ 2 C(x, t) ∂x 2 , β = 1/τ, (1.1) where the diffusion in solids is thermally promoted process [15, 41]. The same equation is relevant to other solid diffusion process in modern materials [30, 38]. Moreover, this is a common approach to model diffusion in solids [30, 38]. In solid diffusion, the dependence of the diffusion coefficient on the absolute temperature T can be simply expressed by the Arrhenius equation (1.2) D(T )= D 0 exp  - E RT  . (1.2) Email address: jordan.hristov@mail.bg (Jordan Hristov) doi:10.22436/mns.01.01.01 Received 2017-05-16