Neural Computing and Applications

, Volume 30, Issue 3, pp 917–924 | Cite as

A length factor artificial neural network method for the numerical solution of the advection dispersion equation characterizing the mass balance of fluid flow in a chemical reactor

  • Neha Yadav
  • Kevin Stanley McFall
  • Manoj Kumar
  • Joong Hoon Kim
Original Article


In this article, a length factor artificial neural network (ANN) method is proposed for the numerical solution of the advection dispersion equation (ADE) in steady state that is used extensively in fluid dynamics and in the mass balance of a chemical reactor. An approximate trial solution of the ADE is constructed in terms of ANN using the concept of the length factor in a way that automatically satisfies the desired boundary conditions, regardless of the ANN output. The mathematical model of ADE is presented adopting a first-order reaction, and the steady-state case for the same is examined by estimating the numerical solution using the ANN technique. Numerical simulations are performed by choosing the best ANN ensemble, based on a combination of numerous design parameters, random starting weights, and biases. The solution obtained using the ANN method is compared to the existing finite difference method (FDM) to test the reliability and effectiveness of the proposed approach. Three cases of ADE are considered in this study for different values of advection and dispersion. The numerical results show that the ANN method exhibits a higher accuracy than the FDM, even for the smaller number of training points in the domain, and eliminates the instability issues for the case where advection dominates dispersion.


Advection Steady state Dispersion Finite difference method Artificial neural network 

Mathematics Subject Classification




This work was supported by National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIP) (NRF-2013R1A2A1A01013886) and the Brain Korea 21 (BK-21) fellowship from the Ministry of Education of Korea.


  1. 1.
    Himmelblau DM (1967) Basic principles and calculations in chemical engineering, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  2. 2.
    Freijera JI, Veling EJM, Hassanizadeh SM (1998) Analytical solutions of the convection–dispersion equation applied to transport of pesticides in soil columns. Environ Model Softw 13(2):139–149CrossRefGoogle Scholar
  3. 3.
    O’Loughlin EM, Bowner KH (1975) Dilution and decay of aquatic herbicides in flowing channels. J Hydrol 26(34):217–235CrossRefGoogle Scholar
  4. 4.
    Hossain MA, Yonge DR (1999) On Galerkin models for transport in ground water. Appl Math Comput 100(2–3):249–263MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kumar N (1983) Unsteady flow against dispersion in finite porous media. J Hydrol 63(3–4):345–358CrossRefGoogle Scholar
  6. 6.
    Guvanasen V, Volker R (1983) Numerical solutions for solute transport in unconfined aquifers. Int J Numer Methods Fluids 3(2):103–123CrossRefzbMATHGoogle Scholar
  7. 7.
    van Genuchten MT, Alves WJ (1982) Analytical solutions of the one dimensional convective dispersive solute transport equations. US Dep Agric Tech Bull 1661:151Google Scholar
  8. 8.
    Ataie-Ashtiani B, Hosseini SA (2005) Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption. Environ Model Softw 20(7):817–826CrossRefGoogle Scholar
  9. 9.
    Ataie-Ashtiani B, Hosseini SA (2005) Error analysis of finite difference methods for two-dimensional advection dispersion reaction equation. Adv Water Resour 28(8):793–806CrossRefGoogle Scholar
  10. 10.
    Sheu TWH, Chen YH (2002) Finite element analysis of contaminant transport in groundwater. Appl Math Comput 127(1):23–43MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zheng C, Bennett GD (2002) Applied contaminant transport modelling. Wiley, New YorkGoogle Scholar
  12. 12.
    Kojouharov HV, Chen BM (1999) Nonstandard methods for the convective–dispersive transport equation with nonlinear reactions. Numer Methods Part Differ Equ 15(6):617–624MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kadalbajoo MK, Yadaw AS (2008) B-Spline collocation method for a two-parameter singularly perturbed convection–diffusion boundary value problems. Appl Math Comput 201(1–2):504–513MathSciNetzbMATHGoogle Scholar
  14. 14.
    Thongmoon M, McKibbin R (2006) A comparison of some numerical methods for the advection diffusion equation. Res Lett Inf Math Sci 10:49–62Google Scholar
  15. 15.
    Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000CrossRefGoogle Scholar
  16. 16.
    Yadav N, Yadav A, Kumar M (2015) An introduction to neural network methods for differential equations, Springer briefs in applied sciences and technology. Springer, NetherlandsGoogle Scholar
  17. 17.
    Raja MAZ, Samar R (2013) Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem. Neural Comput Appl 23(7):2199–2210CrossRefGoogle Scholar
  18. 18.
    Malek A, Beidokhti RS (2006) Numerical simulation for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183(1):260–271MathSciNetzbMATHGoogle Scholar
  19. 19.
    Smaoui N, Al-Enezi S (2004) Modelling the dynamics of nonlinear partial differential equation using neural networks. J Comput Appl Math 170(1):27–58MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shirvany Y, Hayati M, Moradian R (2009) Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of partial differential equation. Appl Soft Comput 9(1):20–29CrossRefGoogle Scholar
  21. 21.
    McFall KS (2013) Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities. J Frankl Inst 350(2):300–317MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yadav N, Yadav A, Kumar M, Kim JH (2015) An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch’s problem. Neural Comput Appl. doi: 10.1007/s00521-015-2046-1 Google Scholar
  23. 23.
    McFall KS, Mahan JR (2009) Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions. IEEE Trans Neural Netw 20(8):1221–1233CrossRefGoogle Scholar
  24. 24.
    Lagaris IE, Likas A, Papageorgiou DG (2000) Neural network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041–1049CrossRefGoogle Scholar
  25. 25.
    Fogler HS (1999) Elements of chemical reaction engineering, 3rd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  26. 26.
    Rawlings JB, Ekerdt JG (2002) Chemical reactor analysis and design fundamentals. Nob Hill Publishing, New JerseyGoogle Scholar
  27. 27.
    Krogh A, Vedelsby J (1995) Neural network ensembles, cross validation, and active learning. Adv Neural Inf Process Syst 7:231–238Google Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • Neha Yadav
    • 1
  • Kevin Stanley McFall
    • 2
  • Manoj Kumar
    • 3
  • Joong Hoon Kim
    • 1
  1. 1.School of Civil Environmental and Architectural EngineeringKorea UniversitySeoulSouth Korea
  2. 2.Department of Mechatronics EngineeringKennesaw State UniversityKennesawUSA
  3. 3.Department of MathematicsMotilal Nehru National Institute of TechnologyAllahabadIndia

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