Skip to main content
SpringerLink
Log in

The solution of direct and inverse fractional advection–dispersion problems by using orthogonal collocation and differential evolution

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

The advection–dispersion phenomenon can be observed in various fields of science. Mathematically, this process can be studied by considering empirical models, high-order differential equations, and fractional differential equations. In this paper, a fractional model considered to represent the transport of passive tracers carried out by fluid flow in a porous media is studied both in the direct and inverse contexts. The studied mathematical model considers a one-dimensional fractional advection–dispersion equation with fractional derivative boundary conditions. The solutions of both direct and inverse problems are obtained by using the orthogonal collocation method and the differential evolution optimization algorithm approaches, respectively. In this case, the source term along the spatial and time coordinates is taken as a design variable. The obtained results with the solution of the direct problem are compared with those determined by using an implicit finite difference scheme. The results indicate that the proposed approach characterizes a promising methodology to solve the direct and inverse fractional advection–dispersion problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Aslefallah M, Rostamy D (2014) A numerical scheme for solving space-fractional equation by finite differences theta-method. Int J Adv Appl Math Mech 1(4):1–9

    MATH  Google Scholar 

  • Benítez T, Sherif SA (2017) Modeling spatial and temporal frost formation with distributed properties on a flat plate using the orthogonal collocation method. Int J Refrig 76:193–205

    Google Scholar 

  • Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation. Water Resour Res 36(6):1403–1412

    Google Scholar 

  • Demirci E, Ozalp N (2012) A method for solving differential equations of fractional order. J Comput Appl Math 236:2754–2762

    MathSciNet  MATH  Google Scholar 

  • Ebrahimi AA, Ebrahim HA, Jamshidi E (2008) Solving partial differential equations of gas–solid reactions by orthogonal collocation. Comput Chem Eng 32:1746–1759

    Google Scholar 

  • Ebrahimzadeh E, Shahrak MN, Bazooyar B (2012) Simulation of transient gas flow using the orthogonal collocation method. Chem Eng Res Des 90:1701–1710

    Google Scholar 

  • Guo B, Xu Q, Yin Z (2016) Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions. Appl Math Mech 37(3):403–416

    MathSciNet  MATH  Google Scholar 

  • Gupta S, Kumar D, Singh J (2015) Numerical study for systems of fractional differential equations via Laplace transform. J Egypt Math Soc 23:256–262

    MathSciNet  MATH  Google Scholar 

  • Hernández-Calderón OM, Rubio-Castro E, Rios-Iribe EY (2014) Solving the heat and mass transfer equations for an evaporative cooling tower through an orthogonal collocation method. Comput Chem Eng 71:24–38

    Google Scholar 

  • Jia J, Wang H (2015) Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions. J Comput Phys 293:359–369

    MathSciNet  MATH  Google Scholar 

  • Jiang W, Lin Y (2010) Approximate solution of the fractional advection–dispersion equation. Comput Phys Commun 181:557–561

    MathSciNet  MATH  Google Scholar 

  • Kashan MH, Kashan AH, Nahavandi N (2013) A novel differential evolution algorithm for binary optimization. Comput Optim Appl 55:481–513

    MathSciNet  MATH  Google Scholar 

  • Kirchner JW, Feng X, Neal C (2000) Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403:524–526

    Google Scholar 

  • Li X, Rui H (2018) A high-order fully conservative block-centered finite difference method for the time-fractional advection–dispersion equation. Appl Numer Math 124:89–109

    MathSciNet  MATH  Google Scholar 

  • Li C, Zhao T, Deng W, Wu Y (2014) Orthogonal spline collocation methods for the subdiffusion equation. J Comput Appl Math 255(1):517–528

    MathSciNet  MATH  Google Scholar 

  • Liang X, Yang Y-G, Gao F, Yang X-J, Xue Y (2018) Anomalous advection–dispersion equations within general fractional-order derivatives: models and series solutions. Entropy 20(1):78–85

    Google Scholar 

  • Liu T, Hou M (2017) A fast implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. Adv Math Phys, 1–8

  • Magin RL (2006) Fractional calculus in bioengineering. Begell House Publishers, New York

    Google Scholar 

  • Meerschaert MM, Tadjeran C (2004) Finite difference approximations for fractional advection–dispersion flow equations. J Comput Appl Math 172(1):65–77

    MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  • Rehman M, Khan RA (2012) A numerical method for solving boundary value problems for fractional differential equations. Appl Math Model 36:894–907

    MathSciNet  MATH  Google Scholar 

  • Risken H (1984) Fokker-Planck equation: methods of solution and applications. Springer, Berlin, Heidelberg

    MATH  Google Scholar 

  • Sabatelli L, Keating S, Dudley J, Richmond P (2002) Waiting time distributions in financial markets. Eur Phys J B 27:273–275

    MathSciNet  Google Scholar 

  • Schumer R, Benson DA, Meerschaert MM, Baeumer B (2003) Multiscaling fractional advection–dispersion equation and their solutions. Water Resour Res 39:1022–1032

    Google Scholar 

  • Singh S, Patel V, Singh V (2018) Application of wavelet collocation method for hyperbolic partial differential equations via matrices. Appl Math Comput 320:407–424

    MathSciNet  MATH  Google Scholar 

  • Sonmezoglu A (2015) Exact solutions for some fractional differential equations. Adv Math Phys 2015:1–10

    MathSciNet  MATH  Google Scholar 

  • Storn R, Price K (1995) Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces. Int Comput Sci Inst 12:1–16

    Google Scholar 

  • Storn R, Price K, Lampinen JA (2005) Differential evolution—a practical approach to global optimization. Springer—natural computing series. Springer, Berlin

    MATH  Google Scholar 

  • Szekeres BJ, Izsák F (2015) A finite difference method for fractional diffusion equations with Neumann boundary conditions. Open Math 13:581–600

    MathSciNet  MATH  Google Scholar 

  • Villadsen J, Michelsen ML (1978) Solution of differential equation models by polynomial approximation. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Villadsen JV, Stewart WE (1967) Solution of boundary-value problems by orthogonal collocation. Chem Eng Sci 22:1483–1501

    Google Scholar 

  • Yang X-J, Gao F, Machado JAT, Baleanu D (2017a) A new fractional derivative involving the normalized sinc function without singular kernel. Eur Phys J Spec Top 226:3567–3575

    Google Scholar 

  • Yang X-J, Machado JAT, Baleanu D (2017b) Exact traveling-wave solution for local fractional boussinesq equation in fractal domain. Fractals 25(4):1740006

    MathSciNet  Google Scholar 

  • Yang X-J, Machado JAT, Nieto JJ (2017c) A new family of the local fractional PDEs. Fundam Inform 151:63–75

    MathSciNet  MATH  Google Scholar 

  • Yang X, Zhang H, Xu D (2018a) Orthogonal spline collocation method for the fourth-order diffusion system. Comput Math Appl 75:3172–3185

    MathSciNet  MATH  Google Scholar 

  • Yang X-J, Gao F, Ju Y, Zhou H-W (2018b) Fundamental solutions of the general fractional-order diffusion equations. Math Methods Appl Sci. 41:9312–9320

    MathSciNet  MATH  Google Scholar 

  • Yang X-J, Feng Y-Y, Cattani C, Inc M (2019) Fundamental solutions of anomalous diffusion equations with the decay exponential kernel. Math Methods Appl Sci 42:1–7

    MathSciNet  MATH  Google Scholar 

  • Yıldırım A, Koçak H (2009) Homotopy perturbation method for solving the space-time fractional advection–dispersion equation. Adv Water Resour 32:1711–1716

    Google Scholar 

  • Yuan ZB, Nie YF, Liu F, Turner I, Zhang GY, Gu YT (2016) An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach. Appl Math Model 40:7816–7829

    MathSciNet  MATH  Google Scholar 

  • Zhang J (2018) A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations. Appl Math Comput 332:209–227

    MathSciNet  MATH  Google Scholar 

  • Zhang X, Liu L, Wu Y, Wiwatanapataphee B (2017) Nontrivial solutions for a fractional advection–dispersion equation in anomalous diffusion. Appl Math Lett 66:1–8

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful for the financial support provided to the present research effort by CAPES, CNPq (574001/2008-5, 304546/2018-8, 439126/2018-5 and 431337/2018-7), FAPEMIG (TEC-APQ-3076-09, TEC-APQ-02284-15, TEC-APQ-00464-16, and PPM-00187-18), and CAPES through the INCT-EIE.

Author information

Authors and Affiliations

  1. School of Chemical Engineering, Federal University of Uberlândia, Uberlândia, MG, Brazil

    F. S. Lobato, A. Ap. Cavalini Jr. & V. Steffen Jr.

  2. School of Industrial Mathematics, Mathematics and Technology Institute, Federal University of Goiás, Catalão, GO, Brazil

    W. J. Lima & R. A. Borges

Authors
  1. F. S. Lobato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. J. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. A. Borges
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Ap. Cavalini Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V. Steffen Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Borges.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Communicated by V. Loia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lobato, F.S., Lima, W.J., Borges, R.A. et al. The solution of direct and inverse fractional advection–dispersion problems by using orthogonal collocation and differential evolution. Soft Comput 24, 10389–10399 (2020). https://doi.org/10.1007/s00500-019-04541-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04541-y

Keywords

Search

Navigation