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Integral Balance Approach to 1-D Space-Fractional Diffusion Models

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Mathematical Methods in Engineering

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 23))

Abstract

This chapter summarizes the recent results on approximate analytical integral-balance solutions of initial-boundary value problems of spatial-fractional partial differential diffusion equation with Riemann–Liouville fractional derivative in space. The approximate method is based on two principal steps: the integral-balance method and a series expansion of an assumed parabolic profile with undefined exponent. The spatial correlation of the superdiffusion coefficient in two power-law forms has been discussed. The law of the spatial and temporal propagation of the solution is the primary issue. Approximate solutions based on assumed parabolic profile with unspecified exponent have been developed.

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References

  1. Aguilar, J.F.G., Hernadez, M.M.: Space-time fractional diffusion-advection equation with Caputo derivative. Abstr. Appl. Anal. 2014, Article ID 283019, 8 pp (2014). https://doi.org/10.1155/2014/283019

  2. Bakunin, O.G.: Diffusion equations and turbulent transport. Plasma Phys. Rep. 1, 955–970 (2003)

    Article  Google Scholar 

  3. Bakunin, O.G.: Description of the anomalous diffusion of fast electrons by a kinetic equation with a fractional spatial derivative. Plasma Phys. Rep. 30, 338–342 (2004)

    Article  Google Scholar 

  4. Basu, T.S., Wang, H.: A fast second-order finite difference method for space-fractional diffusion equations. Int. J. Numer. Anal. Model. 9, 658–666 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Baumann, G., Sudland, N., Nonnenmacher, T.F.: Anomalous relaxation and diffusion processes in complex systems. Transp. Theory Stat. Phys. 29, 151–171 (2000)

    Article  MathSciNet  Google Scholar 

  6. Cobble, M.H.: Non-linear heat transfer if solids in orthogonal coordinate systems. Int. J. Non Linear Mech. 2, 417–4126 (1967)

    Article  Google Scholar 

  7. Das, S., Gupta, P.K., Ghosh, P.: An approximate analytical solution of nonlinear fractional diffusion equation. Appl. Math. Model. 35, 4071–4076 (2011)

    Article  MathSciNet  Google Scholar 

  8. El-Kady, M., El-Sayed, S.M., Salem, H.S., El-Gendi, S.E.: Nodal Galerkin method for solving linear and nonlinear partial fractional space equations. Int. J. Latest Res. Sci. Technol. 2, 10–17 (2013)

    Google Scholar 

  9. Ervin, V.J., Heur, N., Roop, J.: Numerical approximation of a time dependent, nonlinear, spacefractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2008)

    Article  Google Scholar 

  10. Fa, K.S., Lenzi, E.K.: Power law diffusion coefficient and anomalous diffusion: analysis of solutions and the first passage time. Phys. Rev. E 67, Article 061105 (2003). https://doi.org/10.1103/PhysRevE.67.061105

    Article  Google Scholar 

  11. Fa, K.S., Lenzi, E.K.: Time-fractional diffusion equation with time dependent diffusion coefficient. Physica A 72, Article 011107 (2005). https://doi.org/10.1103/PhysRevE.72.011107

    Google Scholar 

  12. Fabre, A., Hristov, A.J.: On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity. Heat Mass Transfer 53, 177–204 (2016)

    Article  Google Scholar 

  13. Goodman, T.R.: Application of integral methods to transient nonlinear heat transfer. In: Irvine, T.F., Hartnett, J.P. (eds.) Advances in Heat Transfer, vol. 1, p. 51. Academic, San Diego (1964)

    Google Scholar 

  14. Hamed, S., Lebedeff, S.A.: Application of the integral method to heat conduction in nonhomogeneous media. J. Heat Transfer 97, 304–305 (1975)

    Article  Google Scholar 

  15. He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    Article  MathSciNet  Google Scholar 

  16. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Hackensack (2000)

    Book  Google Scholar 

  17. Hristov, J.: The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and benchmark exercises. Therm. Sci. 13 22–48 (2009)

    Google Scholar 

  18. Hristov, J.: Heat-balance integral to fractional (half-time) heat diffusion sub-model. Therm. Sci. 14, 291–316 (2010)

    Article  Google Scholar 

  19. Hristov, J.: Approximate solutions to fractional subdiffusion equations: the heat-balance integral method. Eur. Phys. J. Spec. Top. 193, 229–243 (2011)

    Article  Google Scholar 

  20. Hristov, J.: Approximate solutions to time-fractional models by integral balance approach. In: Cattani, C., Srivastava, H.M., Yang, X.-J. (eds.) Fractals and Fractional Dynamics, p. 78. De Gruyter Open, Berlin (2015)

    Google Scholar 

  21. Hristov, J.: Double integral-balance method to the fractional subdiffusion equation: approximate solutions, optimization problems to be resolved and numerical simulations. J. Vib. Control 23, 2795–2818 (2015). https://doi.org/10.1177/1077546315622773

    Article  MathSciNet  Google Scholar 

  22. Hristov, J.: An approximate solution to the transient space-fractional diffusion equation: integral-balance approach, optimization problems and analyzes. Therm. Sci. 21, 309–321 (2017)

    Article  Google Scholar 

  23. Hristov, J.: Transient space-fractional diffusion with a power-law superdiffusivity: approximate integral-balance approach. Fundam. Inform. 151, 371–388 (2017)

    Article  MathSciNet  Google Scholar 

  24. Hristov, J.: Space-fractional diffusion with a potential power-law coefficient: transient approximate solution. Prog. Fract. Differ. Appl. 3, 19–39 (2017)

    Article  Google Scholar 

  25. Hristov, J.: Subdiffusion model with time-dependent diffusion coefficient: integral-balance solution and analysis. Therm. Sci. 21, 69–80 (2017)

    Article  Google Scholar 

  26. Huang, F., Liu, F.: The space-time fractional diffusion equation with Caputo derivatives. J. Appl. Math. Comput. 19, 179–190 (2005)

    Article  MathSciNet  Google Scholar 

  27. Huang, F., Liu, F.: The fundamental solution of the space-time fractional advection-diffusion equation. J. Appl. Math. Comput. 19(1–2), 339–350 (2005)

    Article  Google Scholar 

  28. Langford, D.: The heat balance integral method. Int. J. Heat Mass Transfer 16, 2424–2428 (1973)

    Article  Google Scholar 

  29. Li, C.P., Zhao, Z.G., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia, Bulgaria

    Jordan Hristov

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  1. Jordan Hristov
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Correspondence to Jordan Hristov .

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Editors and Affiliations

  1. Department of Mathematics, Çankaya University, Ankara, Turkey

    Kenan Taş

  2. Department of Mathematics, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

  3. Instituto Superior de Engenharia do Porto, Porto, Portugal

    J. A. Tenreiro Machado

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Hristov, J. (2019). Integral Balance Approach to 1-D Space-Fractional Diffusion Models. In: Taş, K., Baleanu, D., Machado, J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-91065-9_5

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  • DOI: https://doi.org/10.1007/978-3-319-91065-9_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91064-2

  • Online ISBN: 978-3-319-91065-9

  • eBook Packages: EngineeringEngineering (R0)

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