Journal of Chemical Engineering And Bioanalytical Chemistry

ISSN: 2575-5641


A cubic spline collocation method for integrating a class of chemical reactor equations


N. Barje, A. El hajaji , A. Serghini, K. Hilal ,E.B. Mermri


Abdelhafid Serghini, A cubic spline collocation method for integrating a class of chemical reactor equations(2018)Journal of Chemical Engineering And Bioanalytical Chemistry 2(1)


In this paper, the cubic spline collocation method is implemented to find the numerical solution of the nonlinear partial differential equations PDEs of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a Crank-Nicolson, and a cubic spline collocation method on meshes, respectively. The method is shown to be unconditionally stable and second order accurate with respect to both the variables. Numerical results are presented and compared with other collocation methods given in the literature.


  1. N. Barje, M.E. Achhab, V. Wertz, Observer Design for a class of Exothermal Pug-Flow Tubular Reactors, Int. Journal of Applied Mathematical Research, 2 (2013), pp. 273-282.

  2. P. D. Christo?des, P. Daoutidis, Robust control of hyperbolic PDE systems. Chemical Engineering Science, 53 (1998), pp. 85-105. 87571-9

    View Article           

  3. P.D. Christo?des, Nonlinear and robust control of the PDE systems: Methods and application to transport-reactor processs. Boston: Birkh?auser, (2001).

    View Article           

  4. C.Clavero, J.C. Jorge, F,Lisbona, Uniformly convergent schemes for singular perturbation problems combining alternating directions and exponential ?tting techniques, in: J.J.H. Miller (Ed.), Applications of Advanced Computational Methods for Boundary and Interior Layers, Boole, Dublin, (1993) pp.33-52.

  5. C.Clavero, J.C. Jorge, F,Lisbona, A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, Journal of Computational and Applied Mathematics 154 (2003) pp. 415-429. 00861-0

    View Article           

  6. R.A. DeVore, G.G. Lorentz, Constructive approximation, SpringerVerlag, Berlin, (1993).

  7. E.A., Garnica, J.P.G., Sandoval, Carlos Gonzalez-Figueredo, A robust monitoring tool for distributed parameter plug ?ow reactos, Computers and chemical engineering 35 (2011) pp. 510-518.

    View Article           

  8. M. K. Kadalbajoo, L. P. Tripathi, A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Mathematical and Computer Modelling 55 (2012) pp. 14831505.

    View Article           

  9. O.A.Ladyzenskaja,V.A,Solonnikov,N.N.Ural'ceva,LinearandQuasilinear Equations of Parabolic Type, In: Amer. Math. Soc. Transl., Vol. 23 (1968) Providence, RI. H. H. Lou, J., Chandrasekaran, and R.A. Smith, Large-scale dynamic simulation for security assessment of an ethylene oxide manufacturing process. Computers and Chemical Engineering, 30 (2006) pp. 11021118.

  10. E.Mermri,A.Serghini,A.ElhajajiandK.Hilal,ACubicSplineMethod for Solving a Unilateral Obstacle Problem, American Journal of Computational Mathematics, Vol. 2 No. 3 (2012) pp. 217-222.

    View Article           

  11. W. Ray, Some recent applications of distributed parameter systems theory - a survey, Automatica, vol. 14, no. 3 (1978) pp. 2816287. 90092-4

    View Article           

  12. R.G.,Rice,andD.D.,Do, Appliedmathematicsandmodelingforchemical engineers, New York: John Wiley and Sons Inc., (1995). [16] P. Sablonni?A?re, Univariate spline quasi-interpolants and applications to numerical analysis, Rend. Sem. Mat. Univ. Pol. Torino 63 (2005) pp. 211-222.

  13. F.Sandelin, P.,Oinas, T.,Salmi, J., Paloniemi,andH., Haario, Dynamic modelling of catalytic liquid-phase reactions in ?xed beds-kinetics and catalystdeactivationintherecoveryofanthraquinones,ChemicalEngineering Science, 61 (2006) pp.4528-4539.

    View Article           

  14. J.Villadsen,andM.L.Michelsen,Solutionofdifferentialequationmodels by polynomial approximation. Englewood Cliffs, NJ: Prentice-Hall, (1987).

  15. W.,Wu,Nonlinearcontrolofadiabatictubularreactors: Acomputerassisteddesign,JournaloftheChineseInstituteofChemicalEngineers,32 (2001), pp. 373-382.

  16. A. El hajaji, N. Barje, A. Serghini, K. Hilal, E.B. Mermri, A spline collocation method for integrating a class of chemical reactor equations, International Journal of Nonlinear Analysis and Applications(IJNAA) Volume 8, Issue 1, Spring and Summer 2017, Page 69-80.