International Scientific Journal


The fractional advection-diffusion equations are obtained from a fractional power law for the matter flux. Diffusion processes in special types of porous media which has fractal geometry can be modelled accurately by using these equations. However, the existing nonlocal fractional derivatives seem complicated and also lose some basic properties satisfied by usual derivatives. For these reasons, local fractional calculus has recently been emerged to simplify the complexities of fractional models defined by nonlocal fractional operators. In this work, the conformable, a local, well-behaved and limit-based definition, is used to obtain a local generalized form of advection-diffusion equation. In addition, this study is devoted to give a local generalized description to the combination of diffusive flux governed by Fick’s law and the advection flux associated with the velocity field. As a result, the constitutive conformable advection-diffusion equation can be easily achieved. A Dirichlet problem for conformable advection-diffusion equation is derived by applying fractional Laplace transform with respect to time t and finite sin-Fourier transform with respect to spatial coordinate x. Two illustrative examples are presented to show the behaviours of this new local generalized model. The dependence of the solution on the fractional order of conformable derivative and the changing values of problem parameters are validated using graphics held by MATLcodes.
PAPER REVISED: 2016-05-30
PAPER ACCEPTED: 2016-06-15
CITATION EXPORT: view in browser or download as text file
  1. Kaviany, M., Principles of Heat Transfer in Porous Media, Springer, New York, USA, 1995
  2. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J., Fractional calculus: Models and numerical methods. Series on Complexity, Nonlinearity and Chaos- Vol.3, World Scientific Publishing Co. Pte. Ltd, Singapore, 2012.
  3. Chen, W., et al., Anomalous Diffusion Modeling by Fractal and Fractional Derivatives, Comput. Math. Appl., 59 (2010), 5, pp. 1754-1758
  4. Gorenflo, R., Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order, in: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinetti, F. Mainardi), Springer-Verlag, New York, 1997, pp. 223-276.
  5. Povstenko, Y., Space-Time-Fractional Advection Diffusion Equation in a Plane, in: Advances in Modeling and Control of Non-integer Order Systems, Lecture Notes in Electrical Engineering (Eds. K.J. Latawiec et al.), Springer, Switzerland, 2015, Vol. 320, pp. 275-284
  6. Metzler, R., Klafter, J., The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Phys. Rep., 339 (2000), 1, pp. 1-77.
  7. Benson, D.A., et al., Application of A Fractional Advection- dispersion Equation, Water Resour. Res., 36 (2000), 6, pp. 1403-1412
  8. Povstenko, Y., Fundamental Solutions to Time-Fractional Advection Diffusion Equation in A Case of Two Space Variables, Math. Probl. Eng. 2014 (2014), Article ID 705364, 7 pages
  9. Povstenko, Y., Theory of Diffusive Stresses Based on The Fractional Advection-Diffusion Equation, in: Fractional Calculus: Applications (Eds. R. Abi Zeid Daou, X. Moreau), Nova Science Publishers, New York, 2015, pp. 227-241
  10. Povstenko, Y., Fractional Thermoelasticity, Springer, New York, USA, 2015
  11. Liu, F, et al., Time-Fractional Advection-Dispersion Equation, J. Appl. Math. Comput., 13 (2003), 1-2, pp. 233-245
  12. Huang, F, Liu, F., The Time Fractional Diffusion Equation and The Advection-Dispersion Equation. ANZIAM J., 46 (2005), 3, pp. 317-330
  13. Hristov, J., Approximate solutions to time-fractional models by integral balance approach, Chapter 5, In: Fractional Dynamics , C. Cattani, H.M. Srivastava, Xia-Jun Yang, (eds), De Gruyter Open, 2015 , pp.78-109.
  14. Hristov, J., Diffusion models with weakly singular kernels in the fading memories: How the integral-balance method can be applied? Thermal Science, 19 (2015), 3, pp. 947-957
  15. Wei, S., et al., Implicit Local Radial Basis Function Method for Solving Two- Dimensional Time Fractional Diffusion Equations, Thermal Science, 19 (2015), Suppl. 1, pp. S59-S67
  16. Povstenko, Y., Klekot, J., The Dirichlet Problem for The Time-Fractional Advection-Diffusion Equation in A Line Segment, Boundary Value Problems, 89 (2016), DOI 10.1186/s13661-016-0597-4
  17. Khalil, R., et al., M., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014), pp. 65- 70
  18. Abdeljawad, T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015), pp. 57-66
  19. Atangana, A., et al., New Properties of Conformable Derivative, Open Math., 13 (2015), 1, pp. 889-898
  20. Abu Hammad, I., Khalil, R., Fractional Fourier Series with Applications, Am. J. Comput. Appl. Math., 4 (2014), 6, pp. 187-191
  21. Abu Hammad, I., Khalil, R., Conformable Fractional Heat Differential Equation, Int. J. Pure Appl. Math., 94 (2014), 2, pp. 215-221
  22. Khalil, R., Abu-Shaab, H., Solution of Some Conformable Fractional Differential Equations, Int. J. Pure Appl. Math., 103 (2015), 4, pp. 667-673
  23. Avcı, D., Eroğlu, B. B. İ, Özdemir, N., Conformable Heat Problem in A Cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 2016, pp. 572-58
  24. Iyiola, O. S., Nwaeze, E. R., Some New Results on The New Conformable Fractional Calculus with Application using D'Alembert Approach, Progr. Fract. Differ. Appl., 2 (2016), 2, pp.1-7
  25. Neamaty, A., et al., On The Determination of The Eigenvalues for Airy Fractional Differential Equation with Turning Point, TJMM, 7 (2015), 2, pp. 149-153
  26. Eroğlu, B. B. İ, Avcı, D., Özdemir, N., Optimal Control Problem For A Conformable Fractional Heat Equation, ICCESEN 2016, Antalya, Turkey, 2016 (Accepted)
  27. Atangana, A., Derivative with A New Parameter: Theory, Methods and Applications, Elsevier, London, UK, 2015
  28. Jaiswal, D. K., et al., Analytical Solution to The One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients, Journal of Water Resource and Protection, 3 (2011), pp. 76-84
  29. Banks, D. S., Fradin, C., Anomalous Diffusion of Proteins Due to Molecular Crowding, Biophysical Journal, 89 (2005), pp. 2960-2971
  30. Wu, J., Berland, K.M., Propagators and Time-Dependent Diffusion Coefficients for Anomalous Diffusion, Biophysical Journal, 95 (2008), pp. 2049-2052
  31. Sneddon, I. N., The Use of Integral Transforms, McGraw-Hill, New York, USA, 1972

© 2017 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence