An efficient numerical solution of time--fractional batch reactor system

  • Saad Yusur Alwahhah Directorate of Education, Dhi Qar, Iraq
Keywords: Fractional differential equation, Volterra integral equation, Predictor–Corrector approach, Runge-Kutta method, Chemical process

Abstract

Recently, an approximate solution of the time fractional chemical engineering equations by means of the homotopy perturbation method (HPM) has been presented by Khan et al. \cite{khan2010approximate}. As a disadvantage of the HPM, to have reasonable solution at large values of $t$ we should truncate \text{HPM} series with more terms, while this task is so complicated and even takes too time to complete. In this paper, we have successively applied the predictor--corrector approach on fractional chemical systems to obtain an accurate numerical solution. The numerical results are compared with obtained results by HPM. This comparison shows that predictor--corrector approach is more accurate than the HPM. As advantages of predictor-corrector over the HPM, the method reduces the computational difficulties and is easy to implement.

Downloads

Download data is not yet available.

References

[1] N. A. Khan, A. Ara, and A. Mahmood, “Approximate solution of timefractional chemical engineering equations: a comparative study,” International Journal of Chemical Reactor Engineering, vol. 8, no. 1, 2010.
[2] K. S. Miller and B. Ross, “An introduction to the fractional calculus and
fractional differential equations,” 1993.
[3] I. Podlubny, “Fractional-order systems and fractional-order controllers,”
Institute of Experimental Physics, Slovak Academy of Sciences, Kosice,
vol. 12, no. 3, pp. 1–18, 1994.
[4] Y. Luchko and R. Gorenflo, “The initial value problem for some fractional differential equations with the caputo derivatives,” 1998.
[5] K. Diethelm, “An algorithm for the numerical solution of differential
equations of fractional order,” Electron. Trans. Numer. Anal, vol. 5,
no. 1, pp. 1–6, 1997.
[6] S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional integrals and
derivatives: theory and applications,” 1993.
[7] R. Lewandowski and B. Chorążyczewski, “Identification of the parameters of the kelvin–voigt and the maxwell fractional models, used to modeling of viscoelastic dampers,” Computers & structures, vol. 88, no. 1-2,
pp. 1–17, 2010.
[8] A. Carpinteri and F. Mainardi, Fractals and fractional calculus in continuum mechanics, vol. 378. Springer, 2014.
[9] L. Debnath, “Fractional integral and fractional differential equations in
fluid mechanics......... 119,” 2003.
[10] R. L. Bagley and R. Calico, “Fractional order state equations for the control of viscoelasticallydamped structures,” Journal of Guidance, Control,
and Dynamics, vol. 14, no. 2, pp. 304–311, 1991.
[11] K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,”
Nonlinear Dynamics, vol. 29, no. 1-4, pp. 3–22, 2002.
[12] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential
equations. I, Nonstiff problems. Springer, 1987.
[13] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential
Equations I: Nonstiff Problems. Springer, 1993.
[14] K. Diethelm, The analysis of fractional differential equations: An
application-oriented exposition using differential operators of Caputo
type. Springer, 2010.
[15] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265,
no. 2, pp. 229–248, 2002.
[16] K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a
fractional adams method,” Numerical algorithms, vol. 36, no. 1, pp. 31–
52, 2004.
[17] S. C. Chapra and R. P. Canale, Numerical methods for engineers, vol. 2.
McGraw-Hill New York, 2012.
[18] J. R. Dormand and P. J. Prince, “A family of embedded runge-kutta
formulae,” Journal of computational and applied mathematics, vol. 6,
no. 1, pp. 19–26, 1980.
[19] O. Abdulaziz, I. Hashim, and S. Momani, “Application of homotopyperturbation method to fractional ivps,” Journal of Computational and
Applied Mathematics, vol. 216, no. 2, pp. 574–584, 2008.
[20] I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method
for fractional ivps,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674–684, 2009.
20
[21] N. A. Khan, M. Jamil, A. Ara, and S. Das, “Explicit solution of timefractional batch reactor system,” International Journal of Chemical Reactor Engineering, vol. 9, no. 1, 2011.
[22] K. Diethelm, “An extension of the well-posedness concept for fractional
differential equations of caputo?s type,” Applicable Analysis, vol. 93,
no. 10, pp. 2126–2135, 2014.
Published
2019-02-26