Skip to main content
SpringerLink
Log in

Computational Method for Reaction Diffusion-Model Arising in a Spherical Catalyst

  • Full length article
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider Lane–Emden problems which have many applications in sciences. Mainly we focus on two special cases of Lane–Emden boundary value problems which models reaction–diffusion equations in a spherical catalyst and spherical biocatalyst. Here we propose a method to obtain approximate solution of these models. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. Numerical results are shown using tables and figures. Accuracy of the computational method is shown by comparing numerical results by analytical methods.

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

Similar content being viewed by others

References

  1. Davis, H.T.: Introduction to Nonlinear Differential and Integral Equations. Dover, New York (1962)

    Google Scholar 

  2. Lane, J.H.: On theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its internal heat and depending on the laws of gases known to terrestrial experiment. Am. J. Sci. Arts Ser. 50(2), 57–74 (1870)

    Article  Google Scholar 

  3. Van Gorder, R.A.: Exact first integrals for a Lane–Emden equation of the second kind modeling a thermal explosion in a rectangular slab. New Astron. 16(8), 492–497 (2011)

    Article  Google Scholar 

  4. Singh, H.: An efficient computational method for the approximate solution of nonlinear Lane–Emden type equations arising in astrophysics. Astrophys. Space Sci. 363(4), 363–371 (2018)

    Article  MathSciNet  Google Scholar 

  5. Rach, R., Duan, J.S., Wazwaz, A.M.: On the solution of non-isothermal reaction–diffusion model equations in a spherical catalyst by the modified Adomian method. Chem. Eng. Commun. 202(8), 1081–1088 (2015)

    Article  Google Scholar 

  6. Singh, H., Srivastava, H.M., Kumar, D.: A reliable algorithm for the approximate solution of the nonlinear Lane–Emden type equations arising in astrophysics. Numer. Methods Partial Differ. Equ. 34(5), 1524–1555 (2018)

    Article  MathSciNet  Google Scholar 

  7. Chandrasekhar, S.: Introduction to Study of Stellar Structure. Dover, New York (1967)

    Google Scholar 

  8. Horedt, G.P.: Polytropes: Applications in Astrophysics and Related Fields. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  9. Duan, J.-S., Rach, R., Wazwaz, A.M.: Steady-state concentrations of carbon dioxide absorbed into phenylglycidyl ether solutions by the Adomian decomposition method. J. Math. Chem. 53(4), 1054–1067 (2015)

    Article  MathSciNet  Google Scholar 

  10. Wazwaz, A.M.: The variational iteration method for solving new fourth-order Emden–Fowler type equations. Chem. Eng. Commun. 202(11), 1425–1437 (2015)

    Article  Google Scholar 

  11. Wazwaz, A.M.: Solving systems of fourth-order Emden–Fowler type equations by the variational iteration method. Chem. Eng. Commun. 203(8), 1081–1092 (2016)

    Article  Google Scholar 

  12. Wazwaz, A.M.: Solving the non-isothermal reaction–diffusion model equations in a spherical catalyst by the variational iteration method. Chem. Phys. Lett. 679, 132–136 (2017)

    Article  Google Scholar 

  13. Rach, R., Duan, J.-S., Wazwaz, A.M.: Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52(1), 255–267 (2014)

    Article  MathSciNet  Google Scholar 

  14. Saadatmandi, A., Nafar, N., Toufighi, S.P.: Numerical study on the reaction cum diffusion process in a spherical biocatalyst. Iran. J. Math. Chem. 5(1), 47–61 (2014)

    MATH  Google Scholar 

  15. Sevukaperumal, S., Rajendran, L.: Analytical solution of the concentration of species using modified Adomian decomposition method. Int. J. Math. Arch. 4(6), 107–117 (2013)

    Google Scholar 

  16. Danish, M., Kumar, S., Kumar, S.: OHAM solution of a singular BVP of reaction cum diffusion in a biocatalyst. Int. J. Appl. Math. 41(3), 223–227 (2011)

    MathSciNet  Google Scholar 

  17. Singh, R.: Optimal homotopy analysis method for the non-isothermal reaction–diffusion model equations in a spherical catalyst. J. Math. Chem. 56(9), 2579–2590 (2018)

    Article  MathSciNet  Google Scholar 

  18. Li, X., Chen, X.D., Chen, N.: A third order approximate solution of the reaction diffusion process in an immobilized biocatalyst particle. Biochem. Eng. J. 17, 65–69 (2004)

    Article  Google Scholar 

  19. Singh, H.: A new numerical algorithm for fractional model of Bloch equation in nuclear magnetic resonance. Alex. Eng. J. 55, 2863–2869 (2016)

    Article  Google Scholar 

  20. Singh, H.: Operational matrix approach for approximate solution of fractional model of Bloch equation. J. King Saud Univ. Sci. 29(2), 235–240 (2017)

    Article  Google Scholar 

  21. Petráš, I.: Modeling and numerical analysis of fractional-order Bloch equations. Comput. Math. Appl. 61, 341–356 (2011)

    Article  MathSciNet  Google Scholar 

  22. Singh, H., Sahoo, M.R., Singh, O.P.: Numerical method based on Galerkin approximation for the fractional advection–dispersion equation. Int. J. Appl. Comput. Math. 3(3), 2171–2187 (2016)

    Article  MathSciNet  Google Scholar 

  23. Wu, J.L.: A wavelet operational method for solving fractional partial differential equations numerically. Appl. Math. Comput. 214, 31–40 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Singh, H., Srivastava, H.M., Kumar, D.: A reliable numerical algorithm for the fractional vibration equation. Chaos Solitons Fractals 103, 131–138 (2017)

    Article  MathSciNet  Google Scholar 

  25. Tohidi, E., Bhrawy, A.H., Erfani, K.: A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model. 37, 4283–4294 (2013)

    Article  MathSciNet  Google Scholar 

  26. Singh, H., Srivastava, H.M.: Numerical simulation for fractional-order Bloch equation arising in nuclear magnetic resonance by using the Jacobi polynomials. Appl. Sci. 10(8), 2850 (2020)

    Article  Google Scholar 

  27. Singh, C.S., Singh, H., Singh, V.K., Singh, O.P.: Fractional order operational matrix methods for fractional singular integro-differential equation. Appl. Math. Modell. 40, 10705–10718 (2016)

    Article  MathSciNet  Google Scholar 

  28. Singh, H., Srivastava, H.M.: Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients. Phys. A 523, 1130–1149 (2019)

    Article  MathSciNet  Google Scholar 

  29. Singh, C.S., Singh, H., Singh, S., Kumar, D.: An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics. Phys. A 525, 0–8 (2019)

    Article  MathSciNet  Google Scholar 

  30. Singh, H., Pandey, R.K., Baleanu, D.: Stable numerical approach for fractional delay differential equations. Few-Body Syst. 58, 156 (2017)

    Article  Google Scholar 

  31. Singh, H., Singh, C.S.: Stable numerical solutions of fractional partial differential equations using Legendre scaling functions operational matrix. Ain Shams Eng. J. 9, 717–725 (2018)

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Harendra Singh and Abdul-Majid Wazwaz have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Post Graduate College Ghazipur, Ghazipur, U. P., 233001, India

    Harendra Singh

  2. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul-Majid Wazwaz

Authors
  1. Harendra Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdul-Majid Wazwaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harendra Singh.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

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

Singh, H., Wazwaz, AM. Computational Method for Reaction Diffusion-Model Arising in a Spherical Catalyst. Int. J. Appl. Comput. Math 7, 65 (2021). https://doi.org/10.1007/s40819-021-00993-9

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-00993-9

Keywords

Search

Navigation