Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium


  • A. Souigat Ecole Normale Superieure de Ouargla
  • Z. Korichi Ecole Normale Superieure de Ouargla
  • M. T. Meftah Kasdi Merbah University, Ouargla, Algeria



Fractional derivative; Caputo-Fabrizio; diffusion equation; arsenic; germanium; simulation


In this work, we focused on solving the space-time fractional diffusion equation with an application on the intrinsic arsenic diffusion in germanium. At first we have treated the differential equation in a semi-infinite medium by using Caputo-Fabrizio fractional derivative. We have introduced the Laplace transform to solve this type of equations. Secondly, Based on the obtained solution, we have simulated an profile of arsenic diffusion in germanium under intrinsic conditions. Accurate simulations have been achieved showing that the fractional derivative orders affect on the estimation of the diffusion coefficient, where increasing the time fractional derivative order α reduces the value of the diffusion coefficient, while increasing the space fractional derivative order β increases the value of the diffusion coefficient.


N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A. 268 (2000) 298,

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation J. Math. Phys. 47 (2006) 082104,

J. Dong and M. Xu, Some solutions to the space fractional Schrödinger equation using momentum representation method, J. Math. Phys. 48 (2007) 072105,

R. Herrmann, (2012), arXiv:math-ph/1210. 4410v2[math-ph] (Preprint)

N. Laskin, Fractional quantum mechanics, Phys. Rev. E. 62 (2000) 3135,

J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. E. 15 (1977) 319,

J. Lega, J. V. Moloney, and A. C. Newell, Swift-Hohenberg Equation for Lasers, Phys Rev Lett. 73 (1994) 2978 ,

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier, Amsterdam, 2006)

Z. Korichi, and M. T. Meftah, Quantum statistical systems in Ddimensional space using a fractional derivative, Theor. Math. Phys. 186 (2016) 374,

H. Sakaguchi, and H. R. Brand, Localized patterns for the quintic complex Swift-Hohenberg equation, Physica D. 117 (1998) 95,

M.V. Bartucelli, On the asymptotic positivity of solutions for the extended Fisher-Kolmogorov equation with nonlinear diffusion, Math Method Appl Sci. 25 (2002) 701,

W. Shaowei and X. Mingyu, Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative, Acta Mech. 187 (2006) 103,

S. K. Luo, X. T. Zhang, and J. M. He, A general method of fractional dynamics, i.e., fractional Jacobi last multiplier method, and its applications, Acta Mech. 228 (2017) 157,

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, (Wiley, New York, 1993)

I. Podlubny, Fractional Differential Equations, (Academic Press, New York, 1999)

K. Diethelm, The analysis of Fractional Differential Equations, (Springer–Verlag, Germany, 2010)

M. Caputo, Geophys. J. Int. 13 (1967) 529,

Y. Ying, Y. Lian, S. Tang, and W.K. Liu, Enriched reproducing kernel particle method for fractional advection-diffusion equation, Acta Mech Sin. 34 (2018) 515,

D. Zhao, X. J. Yang, and H. M. Srivastava, Some fractal heat-transfer problems with local fractional calculus, Therm. Sci. 19 (2015)1867,

X. J. Yang, J. A. T. Machado, and H. M. Srivastava, A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach, Appl. Math. Comput. 274 (2016) 143,

H. Jafari, H. Tajadodi, and S. J. Johnston, A decomposition method for solving diffusion equations via local fractional time derivative, Therm. Sci. 19 (2015) 123,

M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1 (2015) 73,

J. Losada and J. J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1 (2015) 87,

D. Kumar, J.Singh, Al, M. Qurashi, and D. Baleanu, Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel, Adv. Mech. Eng. 9 (2017) 1,

E. F. D. Goufo, Application of the Caputo-Fabrizio Fractional Derivative without Singular Kernel to Korteweg-De VriesBurgers Equation, Math. Model. Anal. 21 (2016) 188,

A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Applied Mathematics and Computation, Appl. Math. Comput. 273 (2016) 948-956,

M. Abdullah, A. R. Butt, and N. Raza, Heat transfer analysis of Walters’-B fluid with Newtonian heating through an oscillating vertical plate by using fractional Caputo-Fabrizio derivatives, Mech. Time-Dep. Mater. 23 (2019) 33,

F. E. Bouzenna, M. T. Meftah, and M. Difallah, The effect of non-local derivative on Bose-Einstein condensation, Condens. Matter Phys. 24 (2021) 13002,

F. Ali, M. Saqib, I. Khan, and N. A. Sheikh, Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model, Eur. Phys. J. Plus. 131 (2016) 377,

S. Qureshi, N. A. Rangaig, and D. Baleanu, New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator, Mathematics. 7 (2019) 374,

T. M. Atanackovic, M. Janev, and S. Pilipovi´c, Wave equation in fractional Zener-type viscoelastic media involving CaputoFabrizio fractional derivatives M, Meccanica. 54 (2019) 155,

S. Mondal, A. Sur, and M. Kanoria, Magneto-thermoelastic interaction in a reinforced medium with cylindrical cavity in the context of Caputo-Fabrizio heat transport law, Acta Mech. 230 (2019) 4367,

P.P. Kostrobij, B. M. Markovych, Viznovych, and M.V. Tokarchuk, Generalized transport equation with nonlocality of space-time. Zubarev’s NSO method, Physica A. 514 (2019) 63,

J. Crank, The Mathematics of Diffusion, 2nd edition. (Oxford University Press, Oxford, 1975)

H. Mehrer, Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes, (Springer, Berlin, Heidelberg, 2007)

E. N. Sgourou, et al., Appl Sci. 9 (2019) 2454,

U. Otuonye, H. W. Kim, and W. D. Lua, Ge nanowire photodetector with high photoconductive gain epitaxially integrated on Si substrate, Appl. Phys. Lett. 110 (2017) 173104,

E. Simoen and C. Claeys, Germanium Based Technologies: From Materials to Devices, (Elsevier, Amsterdam, 2007)

R. P. Smith, A. A. Hwang, T. Tobias, and E. Helgren, Am. J. Phys. 86 (2018) 740,

S. Brotzmann and H. Bracht, Intrinsic and extrinsic diffusion of phosphorus, arsenic, and antimony in germanium, J Appl Phys. 103 (2008) 033508,

A. Cheng and P. Sidauruk, Approximate inversion of the Laplace transform, The Mathematical Journal 4 (1994) 76

M. Wojcik, M. Szukiewicz, and P. Kowalik, Application of Numerical Laplace Inversion Methods in Chemical Engineering with Maple, J. Appl. Comput. Sci. Methods. 7 (2015) 5 - 15,

Z. Krougly, and M. Davison, The Role of High Precision Arithmetic in Calculating Numerical Laplace and Inverse Laplace Transforms, S. Aiyar, Appl. Math. 8 (2017), 562,




How to Cite

A. Souigat, Z. Korichi, and M. T. Meftah, “Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium”, Rev. Mex. Fís., vol. 70, no. 1 Jan-Feb, pp. 010501 1–, Jan. 2024.