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


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.



[1] T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016 (2016), 18 pages.
[2] A. Alsaedi, D. Baleanu, S. Etemad, S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, J. Funct. Spaces, 2016 (2016), 8 pages.
[3] A. Atangana, B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439–4453.
[4] A. Atangana, B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: application of Caputo- Fabrizio derivative, Arab. J. Geosci., 1 (2016), 1–6.
[5] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[6] A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1–7.
[7] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[8] M. Caputo, M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1–11.
[9] H. S. Carslaw, J. C. Jaeger, Conduction of heat in solids, Reprint of the second edition, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, (1988).
[10] J. Crank, The mathematics of diffusion, Second edition, Clarendon Press, Oxford, (1975).
[11] M. H. Dodson, Closure temperature in cooling geochronological and petrological systems, Contrib. Mineral. Petrol., 40 (1973), 259–274.
[12] M. H. Dodson, Theory of cooling ages, Lectures in isotope geology, Springer, Berlin, Heidelberg, (1979), 194–202.
[13] A. Fabre, J. Hristov, On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity, Heat Mass Transf., 53 (2017), 177–204.
[14] J. Ganguly, Cation diffusion kinetics in aluminosilicate garnets and geological applications, Rev. Mineral. Geochem., 72 (2010), 559–601.
[15] J. Ganguly, M. Tirone, Diffusion closure temperature and age of a mineral with arbitrary extent of diffusion: theoretical formulation and applications, Earth Planet. Sci. Lett., 170 (1999), 131–140.
[16] J. F. Gómez-Aguilar, H. Yépez-Martínez, C. Calderón-Ramón, I. Cruz-Orduña, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel, Entropy, 17 (2005), 6289–6303.
[17] J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv. Difference Equ., 2017 (2017), 18 pages.
[18] T. R. Goodman, The heat balance integral and its application to problems involving a change of phase, Heat transfer and fluid mechanics institute, held at California Institute of Technology, Pasadena, Calif., June, Stanford University Press, Stanford, Calif., (1957), 383–400.
[19] T. R. Goodman, Application of integral methods to transient nonlinear heat transfer, Adv. heat transf., 1 (1964), 51–122.
[20] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, Udine, (1996), CISM Courses and Lect., Springer, Vienna, 378 (1997), 223–276.
[21] J. Hristov, The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and benchmark exercises, Therm. Sci., 13 (2009), 27–48.
[22] J. Hristov, Approximate solutions to time-fractional models by integral-balance approach, Fractional dynamics, De Gruyter Open, Berlin, (2015), 78–109.
[23] J. Hristov, Double integral-balance method to the fractional subdiffusion equation: approximate solutions, optimization problems to be resolved and numerical simulations, J. Vib. Control, (2015).
[24] J. Hristov, Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions, Heat Mass Transf., 52 (2016), 635–655.
[25] J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo- Fabrizio space-fractional derivative with Jefferys kernel and analytical solutions, Therm. Sci., 21 (2016), 827–839.
[26] J. Hristov, Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffreys kernel to the Caputo-Fabrizio time-fractional derivative, Therm. Sci., 20 (2016), 757–762.
[27] J. Hristov, Derivation of the Fractional Dodson Equation and Beyond: Transient diffusion with a non-singular memory and exponentially fading-out diffusivity, Progr. Fract. Differ. Appl., 3 (2017), 1–16.
[28] H. Jafari, A. Lia, H. Tejadodi, D. Baleanu, Analysis of Riccati differential equations within a new fractional derivative without singular kernel, Fund. Inform., 151 (2017), 161–171.
[29] I. Koca, A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana- Baleanu fractional derivatives, Therm. Sci., 00 (2016), 103–103.
[30] V. Krasil’nikov S. Savotchenko, Models of nonstationary diffusion over nonequilibrium grain boundaries in nanostructured materials, Tech. Phys., 60 (2015), 1031–1038.
[31] D. Kumar, J. Singh, M. Al Qurashi, D. Baleanu, Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel, Adv. Mech. Eng., 9 (2017), 1–8.
[32] D. Kumar, J. Singh, D. Baleanu, Modified Kawahara equation within a fractional derivative with non-singular kernel, Therm. Sci., (2017).
[33] Y. Liang, A simple model for closure temperature of a trace element in cooling bi-mineralic systems, Geochim. Cosmochim. Acta, 165 (2015), 35–43.
[34] J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92.
[35] B. I. McInnes, N. J. Evans, F. Q. Fu, S. Garwin, Application of thermochronology to hydrothermal ore deposits, Rev. Mineral. Geochem., 58 (2005), 467–498.
[36] S. L. Mitchell, T. G. Myers, Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev., 52 (2010), 57–86.
[37] S. L. Mitchell, T. G. Myers, Application of heat balance integral methods to one-dimensional phase change problems, Int. J. Differ. Equ., 2012 (2012), 22 pages.
[38] A. A. Nazarov, Grain-boundary diffusion in nanocrystals with a time-dependent diffusion coefficient, Phys. Solid State, 45 (2003), 1166–1169.
[39] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, (1999).
[40] N. A. Sheikh, F. Ali, M. Saqib, I. Khan, S. A. A. Jan, A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid, Eur. Phys. J., 132 (2017), 14 pages.
[41] Y.-X. Zhang, Diffusion in minerals and melts: theoretical background, Rev. Mineral. Geochem., 72 (2010), 5–59.