##
**The solution of direct and inverse fractional advection-dispersion problems by using orthogonal collocation and differential evolution.**
*(English)*
Zbl 07558594

Summary: 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.

PDF
BibTeX
XML
Cite

\textit{F. S. Lobato} et al., Soft Comput. 24, No. 14, 10389--10399 (2020; Zbl 07558594)

Full Text:
DOI

### References:

[1] | Aslefallah, M.; Rostamy, D., A numerical scheme for solving space-fractional equation by finite differences theta-method, Int J Adv Appl Math Mech, 1, 4, 1-9 (2014) · Zbl 1359.65146 |

[2] | Benítez, T.; Sherif, SA, Modeling spatial and temporal frost formation with distributed properties on a flat plate using the orthogonal collocation method, Int J Refrig, 76, 193-205 (2017) |

[3] | Benson, DA; Wheatcraft, SW; Meerschaert, MM, Application of a fractional advection-dispersion equation, Water Resour Res, 36, 6, 1403-1412 (2000) |

[4] | Demirci, E.; Ozalp, N., A method for solving differential equations of fractional order, J Comput Appl Math, 236, 2754-2762 (2012) · Zbl 1243.34003 |

[5] | Ebrahimi, AA; Ebrahim, HA; Jamshidi, E., Solving partial differential equations of gas-solid reactions by orthogonal collocation, Comput Chem Eng, 32, 1746-1759 (2008) |

[6] | Ebrahimzadeh, E.; Shahrak, MN; Bazooyar, B., Simulation of transient gas flow using the orthogonal collocation method, Chem Eng Res Des, 90, 1701-1710 (2012) |

[7] | Guo, B.; Xu, Q.; Yin, Z., Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions, Appl Math Mech, 37, 3, 403-416 (2016) · Zbl 1336.65135 |

[8] | Gupta, S.; Kumar, D.; Singh, J., Numerical study for systems of fractional differential equations via Laplace transform, J Egypt Math Soc, 23, 256-262 (2015) · Zbl 1330.65127 |

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

[10] | Jia, J.; Wang, H., Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions, J Comput Phys, 293, 359-369 (2015) · Zbl 1349.65561 |

[11] | Jiang, W.; Lin, Y., Approximate solution of the fractional advection-dispersion equation, Comput Phys Commun, 181, 557-561 (2010) · Zbl 1210.65168 |

[12] | Kashan, MH; Kashan, AH; Nahavandi, N., A novel differential evolution algorithm for binary optimization, Comput Optim Appl, 55, 481-513 (2013) · Zbl 1271.90052 |

[13] | Kirchner, JW; Feng, X.; Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments, Nature, 403, 524-526 (2000) |

[14] | Li, X.; Rui, H., A high-order fully conservative block-centered finite difference method for the time-fractional advection-dispersion equation, Appl Numer Math, 124, 89-109 (2018) · Zbl 1377.65107 |

[15] | Li, C.; Zhao, T.; Deng, W.; Wu, Y., Orthogonal spline collocation methods for the subdiffusion equation, J Comput Appl Math, 255, 1, 517-528 (2014) · Zbl 1291.65307 |

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

[17] | 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 · Zbl 1448.76109 |

[18] | Magin, RL, Fractional calculus in bioengineering (2006), New York: Begell House Publishers, New York |

[19] | Meerschaert, MM; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J Comput Appl Math, 172, 1, 65-77 (2004) · Zbl 1126.76346 |

[20] | Podlubny, I., Fractional differential equations (1999), San Diego: Academic Press, San Diego · Zbl 0918.34010 |

[21] | Rehman, M.; Khan, RA, A numerical method for solving boundary value problems for fractional differential equations, Appl Math Model, 36, 894-907 (2012) · Zbl 1243.65095 |

[22] | Risken, H., Fokker-Planck equation: methods of solution and applications (1984), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0546.60084 |

[23] | Sabatelli, L.; Keating, S.; Dudley, J.; Richmond, P., Waiting time distributions in financial markets, Eur Phys J B, 27, 273-275 (2002) |

[24] | Schumer, R.; Benson, DA; Meerschaert, MM; Baeumer, B., Multiscaling fractional advection-dispersion equation and their solutions, Water Resour Res, 39, 1022-1032 (2003) |

[25] | Singh, S.; Patel, V.; Singh, V., Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Appl Math Comput, 320, 407-424 (2018) · Zbl 1427.65301 |

[26] | Sonmezoglu, A., Exact solutions for some fractional differential equations, Adv Math Phys, 2015, 1-10 (2015) · Zbl 1375.35611 |

[27] | Storn, R.; Price, K., Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Int Comput Sci Inst, 12, 1-16 (1995) |

[28] | Storn, R.; Price, K.; Lampinen, JA, Differential evolution—a practical approach to global optimization. Springer—natural computing series (2005), Berlin: Springer, Berlin |

[29] | Szekeres, BJ; Izsák, F., A finite difference method for fractional diffusion equations with Neumann boundary conditions, Open Math, 13, 581-600 (2015) · Zbl 1515.35326 |

[30] | Villadsen, J.; Michelsen, ML, Solution of differential equation models by polynomial approximation (1978), Englewood Cliffs: Prentice-Hall, Englewood Cliffs · Zbl 0464.34001 |

[31] | Villadsen, JV; Stewart, WE, Solution of boundary-value problems by orthogonal collocation, Chem Eng Sci, 22, 1483-1501 (1967) |

[32] | Yang, X-J; Gao, F.; Machado, JAT; Baleanu, D., A new fractional derivative involving the normalized sinc function without singular kernel, Eur Phys J Spec Top, 226, 3567-3575 (2017) |

[33] | Yang, X-J; Machado, JAT; Baleanu, D., Exact traveling-wave solution for local fractional boussinesq equation in fractal domain, Fractals, 25, 4, 1740006 (2017) |

[34] | Yang, X-J; Machado, JAT; Nieto, JJ, A new family of the local fractional PDEs, Fundam Inform, 151, 63-75 (2017) · Zbl 1386.35461 |

[35] | Yang, X.; Zhang, H.; Xu, D., Orthogonal spline collocation method for the fourth-order diffusion system, Comput Math Appl, 75, 3172-3185 (2018) · Zbl 1409.65079 |

[36] | Yang, X-J; Gao, F.; Ju, Y.; Zhou, H-W, Fundamental solutions of the general fractional-order diffusion equations, Math Methods Appl Sci., 41, 9312-9320 (2018) · Zbl 1406.35480 |

[37] | Yang, X-J; Feng, Y-Y; Cattani, C.; Inc, M., Fundamental solutions of anomalous diffusion equations with the decay exponential kernel, Math Methods Appl Sci, 42, 1-7 (2019) · Zbl 1425.35228 |

[38] | Yıldırım, A.; Koçak, H., Homotopy perturbation method for solving the space-time fractional advection-dispersion equation, Adv Water Resour, 32, 1711-1716 (2009) |

[39] | Yuan, ZB; Nie, YF; Liu, F.; Turner, I.; Zhang, GY; Gu, YT, An advanced numerical modeling for Riesz space fractional advection-dispersion equations by a meshfree approach, Appl Math Model, 40, 7816-7829 (2016) · Zbl 1471.65167 |

[40] | Zhang, J., A stable explicitly solvable numerical method for the Riesz fractional advection-dispersion equations, Appl Math Comput, 332, 209-227 (2018) · Zbl 1427.65200 |

[41] | Zhang, X.; Liu, L.; Wu, Y.; Wiwatanapataphee, B., Nontrivial solutions for a fractional advection-dispersion equation in anomalous diffusion, Appl Math Lett, 66, 1-8 (2017) · Zbl 1364.35429 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.