The nonlinear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization

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Authors
Jordan Hristov
 Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia 1756, 8 Kliment Ohridsky, blvd. Bulgaria
Abstract
The Dodson mass diffusion equation with exponentially diffusivity is analyzed through approximate integral solutions.
Integralbalance solutions were developed to integerorder versions as well as to formally fractionalized models. The formal
fractionalization considers replacement of the time derivative with a fractional version with either singular (RiemannLiouville
or Caputo) or nonsingular fading memory. The solutions developed allow seeing a new side of the Dodson equation and
to separate the formal fractional model with CaputoFabrizio time derivative with an integralbalance allowing relating the
fractional order to the physical relaxation time as adequate to the phenomena behind.
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ISRP Style
Jordan Hristov, The nonlinear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization, Mathematics in Natural Science, 1 (2017), no. 1, 117
AMA Style
Hristov Jordan, The nonlinear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization. Math. Nat. Sci. (2017); 1(1):117
Chicago/Turabian Style
Hristov, Jordan. "The nonlinear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization." Mathematics in Natural Science, 1, no. 1 (2017): 117
Keywords
 Nonlinear diffusion
 singular fading memory
 nonsingular fading memory
 formal fractionalization
 integral balance approach
References
[1] T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete MittagLeffler kernels, Adv. Difference Equ., 2016 (2016), 18 pages.
[2] A. Alsaedi, D. Baleanu, S. Etemad, S. Rezapour, On coupled systems of timefractional 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 KellerSegel 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 nonsingular 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 integralbalance approach to the transient heat conduction with linearly temperaturedependent 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. GómezAguilar, H. YépezMartínez, C. CalderónRamón, I. CruzOrduña, R. F. EscobarJiménez, V. H. OlivaresPeregrino, Modeling of a massspringdamper system by fractional derivatives with and without a singular kernel, Entropy, 17 (2005), 6289–6303.
[17] J. F. GómezAguilar, H. YépezMartínez, J. TorresJiménez, T. CórdovaFraga, R. F. EscobarJiménez, V. H. OlivaresPeregrino, 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 heatbalance 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 timefractional models by integralbalance approach, Fractional dynamics, De Gruyter Open, Berlin, (2015), 78–109.
[23] J. Hristov, Double integralbalance 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 powerlaw diffusivity: a semiinfinite medium with fixed boundary conditions, Heat Mass Transf., 52 (2016), 635–655.
[25] J. Hristov, Steadystate heat conduction in a medium with spatial nonsingular fading memory: derivation of Caputo Fabrizio spacefractional derivative with Jefferys kernel and analytical solutions, Therm. Sci., 21 (2016), 827–839.
[26] J. Hristov, Transient heat diffusion with a nonsingular fading memory: from the Cattaneo constitutive equation with Jeffreys kernel to the CaputoFabrizio timefractional derivative, Therm. Sci., 20 (2016), 757–762.
[27] J. Hristov, Derivation of the Fractional Dodson Equation and Beyond: Transient diffusion with a nonsingular memory and exponentially fadingout 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 CattaneoHristov model of elastic heat diffusion with CaputoFabrizio 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 nonsingular kernel, Adv. Mech. Eng., 9 (2017), 1–8.
[32] D. Kumar, J. Singh, D. Baleanu, Modified Kawahara equation within a fractional derivative with nonsingular kernel, Therm. Sci., (2017).
[33] Y. Liang, A simple model for closure temperature of a trace element in cooling bimineralic 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 onedimensional Stefan problems, SIAM Rev., 52 (2010), 57–86.
[37] S. L. Mitchell, T. G. Myers, Application of heat balance integral methods to onedimensional phase change problems, Int. J. Differ. Equ., 2012 (2012), 22 pages.
[38] A. A. Nazarov, Grainboundary diffusion in nanocrystals with a timedependent 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 AtanganaBaleanu and CaputoFabrizio 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.