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Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

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Abstract

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.

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References

  1. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamic approach. Phys. Rep. 339(1), 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Razminia, K., Razminia, A., Tenreiro Machado, J.A.: Analysis of diffusion process in fractured reservoirs using fractional derivative approach. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3161–3170 (2014)

    Article  MathSciNet  Google Scholar 

  3. Raghavan, R.: Fractional derivatives: application to transient flow. J. Pet. Sci. Eng. 80(1), 7–13 (2011)

    Article  Google Scholar 

  4. Tian, J., Tong, D.K.: The flow analysis of fluids in fractal reservoir with the fractional derivative. J. Hydrodyn. Ser. B 18(3), 287–293 (2006)

    Article  MATH  Google Scholar 

  5. Raghavan, R., Chen, C.: Fractured-well performance under anomalous diffusion. SPE Reserv. Eval. Eng. 16(3), 237–245 (2013)

    Article  Google Scholar 

  6. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach Science Publishers, London (1993)

    MATH  Google Scholar 

  7. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Albinali, A., Ozkan, E.: Analytical modeling of flow in highly disordered, fractured nano-porous reservoirs. SPE Western Regional Meeting, Anchorage, Alaska, USA, SPE-180440-MS (2016)

  10. Abiola, O.D., Enamul, H.M., Kassem, M., Sidqi, A.A.: A modified memory-based mathematical model describing fluid flow in porous media. Comput. Math. Appl. 73(6), 1385–1402 (2017)

    Article  MathSciNet  Google Scholar 

  11. Raghavan, R., Chen, C., DaCunha, J.J.: Nonlocal diffusion in fractured rocks. SPE Reservoir Evaluation & Engineering, SPE-184404-PA (2016)

  12. Garra, R., Salusti, E.: Application of the nonlocal Darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media. Phys. D 250, 52–57 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sapora, A., Cornetti, P., Chiaia, B., Lenzi, E.K.: Nonlocal diffusion in porous media: a spatial fractional approach. J. Eng. Mech. ASCE 143(5), D4016007 (2017)

    Article  Google Scholar 

  14. Caffarelli, L., Vazquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration Mech. Anal. 202(2), 537–565 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Raghavan, R.: Fractional diffusion: performance of fractured wells. J. Pet. Sci. Eng. 92–93, 167–173 (2012)

    Article  Google Scholar 

  16. Nakhushev, A.M.: Fractional Calculus and Its Applications. FIZMATLIT, Moscow (2003). (in Russian)

    MATH  Google Scholar 

  17. Nakhushev, A.M.: About equations of states for continuous one-dimensional systems and their analogues in fractional calculus. Reports of Adyghe (Circassian) International Academy of Sciences, 22–26 (1994) (in Russian)

  18. Nahusheva, V.A.: One class of substance equation of state. Reports of Adyghe (Circassian) International Academy of Sciences 7(2), 101–108 (2005). (in Russian)

    Google Scholar 

  19. Meilanov, R.P., Magomedov, R.A.: Thermodynamics in fractional calculus. J. Eng. Phys. Thermophys. 87(6), 1521–1531 (2004)

    Article  Google Scholar 

  20. Magomedov, R.A., Meilanov, R.P., Akhmedov, E.N., Aliverdiev, A.A.: Calculation of multicomponent compound properties using generalization of thermodynamics in derivatives of fractional order. J. Phys. Conf. Ser. 774(1), 012025 (2016)

    Article  Google Scholar 

  21. Prieur, F., Holm, S.: Nonlinear acoustic wave equations with fractional loss operators. J. Acoust. Soc. Am. 130(3), 1125–1132 (2011)

    Article  Google Scholar 

  22. Ovsyannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)

    MATH  Google Scholar 

  23. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)

    Book  MATH  Google Scholar 

  24. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  25. Ibragimov N.H.: CRC Handbook of Lie group analysis of differential equations. Vol. 1. Symmetries, exact solutions and conservation laws (1994). Vol. 2. Application in engineering and physical sciences (1995). Vol. 3. New trends in theoretical developments and computational methods (1996). CRC Press Inc., Boca Raton, Florida (1994-1996)

  26. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Continuous transformation groups of fractional differential equations. Vestn. UGATU 9, 125–135 (2007). (in Russian)

    MATH  Google Scholar 

  27. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Fractional differential equations: change of variables and nonlocal symmetries. Ufa Math. J. 4(4), 54–67 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Lukashchuk, SYu.: Constructing conservation laws for fractional-order integro-differential equations. Theor. Math. Phys. 184(2), 1049–1066 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Symmetry properties of fractional diffusion equations. Phys. Scr. 136, 014016 (2009)

    Article  Google Scholar 

  30. Lukashchuk, SYu., Makunin, A.V.: Group classification of nonlinear time-fractional diffusion equation with a source term. Appl. Math. Comput. 257, 335–343 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Lukashchuk, SYu.: Conservation laws for time-fractional subdiffusion and diffusion-wave equations. Nonlinear Dyn. 80(1–2), 791–802 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lukashchuk, SYu.: Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term. Ufa Math. J. 8(4), 111–122 (2016)

    Article  MathSciNet  Google Scholar 

  33. Elwakil, S.A., Elhanbaly, S., Abdou, M.A.: Adomian decomposition method for solving fractional nonlinear differential equations. Appl. Math. Comput. 182(1), 313–324 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Duan, J.-S., Rach, R., Baleanu, D., Wazwaz, A.-M.: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Frac. Calc. 3(2), 73–99 (2012)

    Google Scholar 

  35. El-Ajou, A., Abu Arqub, O., Momani, S.: Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Abu Arqub, O., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Abu Arqub, O.: Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time- fractional partial differential equations subject to initial and Neumann boundary conditions. Comput. Math. Appl. 73(6), 1243–1261 (2017)

    Article  MathSciNet  Google Scholar 

  38. Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Num. Simulat. 14(3), 674–684 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34(1), 24–35 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yulita Molliq, R., Noorani, M.S.M., Hashim, I.: Variational iteration method for fractional heat- and wave-like equations. Nonlinear Anal. Real World Appl. 10(3), 1854–1869 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, G.C.: A fractional variational iteration method for solving fractional nonlinear differential equations. Appl. Math. Comput. 61(8), 2186–2190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365(5–6), 345–350 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tarasov, V.E., Zaslavsky, G.M.: Dynamics with low-level fractionality. Phys. A 368(2), 399–415 (2006)

    Article  Google Scholar 

  44. Tofighi, A., Golestani, A.: A perturbative study of fractional relaxation phenomena. Phys. A 387(8–9), 1807–1817 (2008)

    Article  Google Scholar 

  45. Tofighi, A.: An especial fractional oscillator. Int. J. Stat. Mech. 2013, 175–273 (2013)

    Article  Google Scholar 

  46. Lukashchuk, SYu.: An approximate solution method for ordinary fractional differential equations with the Riemann–Liouville fractional derivatives. Commun. Nonlinear Sci. Num. Simul. 19(2), 390–400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gazizov, R.K., Lukashchuk, SYu.: Approximations of fractional differential equations and approximate symmetries. IFAC PapersOnLine 50(1), 14022–14027 (2017)

    Article  Google Scholar 

  48. Nayfeh, A.H.: Perturbation Methods. Wiley-VCH, Weinheim (2000)

    Book  MATH  Google Scholar 

  49. Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Approximate symmetries. Math. USSR Sb. 64(2), 427–441 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  50. Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Perturbation methods in group analysis. J. Sov. Math. 55(1), 1450–1490 (1991)

    Article  MATH  Google Scholar 

  51. Gazizov, R.K.: Lie algebras of approximate symmetries. Nonlinear Math. Phys. 3(1–2), 96–101 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ibragimov, N.H., Kovalev, V.F.: Approximate and Renormgroup Symmetries. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  53. Ibragimov, N.H.: Transformation Groups and Lie Algebras. World Scientific, New Jersey (2013)

    Book  MATH  Google Scholar 

  54. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media. Southern Methodist University, Dallas (2006)

    Book  MATH  Google Scholar 

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Acknowledgements

This work is financially supported by the Ministry of Education and Science of the Russian Federation (State task No. 1.3103.2017/4.6).

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  1. Laboratory “Group Analysis of Mathematical Models in Natural and Engineering Sciences”, Ufa State Aviation Technical University, 12, K. Marx Str., Ufa, Russia, 450000

    Stanislav Yu. Lukashchuk & Regina D. Saburova

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  1. Stanislav Yu. Lukashchuk
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  2. Regina D. Saburova
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Correspondence to Stanislav Yu. Lukashchuk.

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Lukashchuk, S.Y., Saburova, R.D. Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type. Nonlinear Dyn 93, 295–305 (2018). https://doi.org/10.1007/s11071-018-4192-3

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