‎The effect of radial basis functions (RBFs) method in solving coupled Lane–Emden boundary value problems in‎ ‎Catalytic Diffusion Reactions

Document Type : Research Paper


Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Kashan‎, ‎Kashan‎, ‎87317-53153‎, ‎Iran.


‎‎In this paper‎, ‎the radial basis functions (RBFs) method is applied to solve the coupled Lane–Emden boundary value problems arising in catalytic diffusion reactions‎. ‎First‎, ‎we multiply the equations by x to overcome the difficulties of the singularity at the origin‎. ‎Then‎, ‎the Kansa collocation method based on radial basis functions is used to approximate the unknown functions‎. ‎By this technique‎, ‎the problem with boundary conditions is reduced to a system of algebraic equations‎. ‎We solve this system and compare the maximal residual error with the results previously‎, ‎which show the presented method is efficient and produces very accurate and rapidly convergent numerical results in considerably low computational effort and easy implementation‎.



    1. K‎. ‎Parand‎, ‎Z‎. ‎Roozbahani and F‎. ‎B‎. ‎Babolghani‎, ‎Solving nonlinear Lane-Emden type equations with unsupervised combined artifcial neural networks, Int. J Ind. Ind. ‎‎Math.‎ ‎5 (2013) 355–366‎.
    2. R‎. ‎Rach‎, ‎J‎. ‎S‎. ‎Duan and A‎. ‎M‎. ‎Wazwaz‎, ‎Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method‎, ‎J. Math. Chem.52 (2014) 255–267‎.
    3. H‎. ‎J‎. ‎Lane‎, ‎On the theoretical temperature of the Sun‎, ‎under the hypothesis of a gaseous mass maintaining its volume by its internal heat‎, ‎and depending on the laws of gases as known to terrestrial experiment‎, Am. J. Sci.50 (1870) 57–74‎.
    4. R‎. ‎Emden‎, ‎Gaskugeln‎: Anwendungen der Mechanischen Warmetheorie Auf kosmologische und Meteorologische Probleme, BG Teubner‎, ‎Berlin, ‎1907‎.
    5. R‎. ‎Aris,‎ Introduction to the Analysis of Chemical Reactors‎, ‎Englewood Cliffs‎, N‎. ‎J‎. ‎Prentice-Hall‎, ‎1965‎.‎
    6. B‎. ‎N‎. ‎Metha‎ and ‎R‎. ‎Aris‎, ‎Mass transfer effects in coal combustion, J. Math. Anal. Appl.‎ 36 (1971) 611–621‎.‎
    7. A‎. ‎M‎. ‎Wazwaz‎, ‎Solving the non-isothermal reaction–diffusion model equations in a spherical catalyst by the variational iteration method‎, ‎Chem. Phys. Lett.‎ ‎679 (2017) 132–136‎.
    8. S‎. ‎Chandrasekhar‎, Introduction to the Study of Stellar Structure‎, ‎Dover‎, ‎New York‎, ‎1967‎.
    9. A‎. ‎M‎. ‎Wazwaz‎, ‎A new algorithm for solving differential equations of Lane-Emden type‎, Appl. Math. Comput.‎ ‎118 (2001) 287–310‎.
    10. A‎. ‎M‎. ‎Wazwaz‎, ‎The modifed decomposition method for analytic treatment of differential equations‎, Appl. Math. Comput.‎ ‎173 (2006) 165–176‎.
    11. C‎. ‎M‎. ‎Bender‎, ‎K‎. ‎A‎. ‎Milton‎, ‎S‎. ‎S‎. ‎Pinsky and L‎. ‎M‎. ‎Simmons‎, ‎A new perturbative approach to nonlinear problems‎, J. Math. Phys.30 (1989) 1447–1455‎.
    12. J‎. ‎H‎. ‎He‎, ‎Variational approach to the Lane-Emden equation, Appl. Math. Comput.‎ ‎143 (2003) 539–541‎.
    13. J‎. ‎I‎. ‎Ramos‎, ‎Series approach to the Lane-Emden equation and comparison with the Homotopy perturbation method‎, ‎Chaos Soliton. Fract.‎ ‎38 (2008) 400–408‎.
    14. M‎. ‎Dehghan‎, ‎J‎. ‎Manafian and A‎. ‎Saadatmandi‎, ‎Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Meth. PDE‎, ‎26 (2010) 448–479‎.
    15. O‎. ‎P‎. ‎Singh‎, ‎R‎. ‎K‎. ‎Pandey and V‎. ‎K‎. ‎Singh‎, ‎An analytic algorithm of Lane-Emden type equations arising in astrophysics using modifed homotopy analysis method‎, ‎Comput. Phys. Commun. ‎180 (2009) 1116–1124‎.
    16. K‎. ‎Parand‎, ‎M‎. ‎Dehghan‎, ‎A‎. ‎R‎. ‎Rezaei and S‎. ‎M‎. ‎Ghaderi‎, ‎An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. ‎181 (2010) 1096–1108‎.
    17. K‎. ‎Parand‎, ‎M‎. ‎Shahini and M‎. ‎Dehghan‎, ‎Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type‎, ‎J. Comput. Phys.‎ ‎228 (2009) 8830–8840‎.
    18. A‎. ‎Yíldírím and ‎T‎. öziş‎, ‎Solutions of singular IVPs of Lane-Emden type by the variational iteration method‎, ‎Nonlinear Anal‎. ‎Theor.70 (2009) 2480–2484‎.
    19. H‎. ‎R‎. ‎Marzban‎, ‎H‎. ‎R‎. ‎Tabrizidooz and M‎. ‎Razzaghi‎, ‎Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations‎, ‎Phys. Lett. A.‎ ‎37 (2008) 5883–5886‎.
    20. M‎. ‎Bisheh-Niasar‎, ‎A Computational Method for Solving the Lane-Emden Initial Value Problems‎, ‎Comput‎. ‎Methods Differ. Equ.8 (2020) 673–684‎.
    21. D‎. ‎Flockerzi‎ and ‎K‎. ‎Sundmacher‎, ‎On coupled Lane-Emden equations arising in dusty fluid models‎, ‎J. Phys. Conf. Ser.268 (2011) 012006‎.
    22. B‎. ‎Muatjetjeja‎ and ‎C‎. ‎M‎. ‎Khalique‎, ‎Noether‎, ‎partial Noether operators and first integrals for the coupled Lane–Emden system‎, ‎Math. Comput. Appl.‎ ‎15 (2010) 325–333‎.
    23. R‎. ‎Singh‎ and ‎A‎. ‎M‎. ‎Wazwaz‎, ‎An efficient algorithm for solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions‎: ‎the homotopy analysis method‎, ‎Math‎. ‎Comp‎. ‎Chem.‎ ‎81 (2019) 785–800‎.‎
    24. T‎. ‎C‎. ‎Hao‎, ‎F‎. ‎Z‎. ‎Cong and Y‎. ‎F‎. ‎Shang‎, ‎An efficient method for solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions and error estimate‎, J. Math. Chem. ‎56 (2018) 2691–2706‎.
    25. A‎. ‎Saadatmandi‎ and ‎S‎. ‎Fayyaz‎, ‎Chebyshev finite difference method for solving a mathematical model arising in wastewater treatment plants‎, ‎Comput. Methods Differ. Equ.‎ ‎6‎, ‎(2018) 448–455‎.
    26. F‎. ‎Zabihi‎, ‎Chebyshev finite difference method for Steady-State concentrations of carbon dioxide absorbed into phenyl glycidyl ether‎, ‎MATCH Commun. Math. Comput. Chem.84 (2020) 131–140‎.
    27. A‎. ‎Saadatmandi‎ and ‎S‎. ‎Fayyaz‎, ‎Numerical study of oxygen and carbon substrate concentrations in excess sludge production using sinc-collocation method, MATCH Commun‎. ‎Math. Comput. Chem.‎ 80 (2018) 355–368‎.‎
    28. E‎. ‎J‎. ‎Kansa‎, ‎Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates‎, ‎Comput. Math. Appl.‎ ‎19 (1990) 127–145‎.
    29. F‎. ‎Zabihi‎ and ‎M‎. ‎Saffarian‎, ‎A meshless method using radial basis functions for the numerical solution of two-dimensional ZK–BBM equation, Appl. Comput. Math. 3 (2017) 4001–4013‎.
    30. F‎. ‎Zabihi and ‎M‎. ‎Saffarian‎, ‎A not-a-knot meshless method with radial basis functions for numerical solutions of Gilson–Pickering equation‎, ‎Eng‎. ‎Comput.34 (2018) 37–44‎.
    31. G‎. ‎Liu‎ and ‎Y‎. ‎Gu‎, ‎An Introduction to Meshfree Methods and Their Programming, Springer‎, ‎Netherlands‎, ‎2005‎.
    32. M‎. ‎D‎. ‎Buhmann‎, Radial Basis Functions: Theory and Implementations‎, ‎Cambridge University Press‎, ‎New York‎, ‎2004‎.
    33. H‎. ‎Wendland‎, ‎Scattered data approximation‎, ‎Cambridge University Press‎, ‎New York‎, ‎2005‎.
    34. R‎. ‎L‎. ‎Hardy‎, ‎Geodetic application of multiquadric analysis, AVN. Allg. Vermess. Nachr.79 (1972) 389–406‎.
    35. R‎. ‎Franke‎, ‎Scattered data interpolation‎: ‎Tests of some methods‎, ‎Math. Comp. ‎48 (1982) 181–200‎.
    36. S‎. ‎Rippa‎, ‎An algorithm for selecting a good parameter c in radial basis function interpolation‎, Advan. Comp. Math.‎ ‎11 (1999) 193–210‎.
    37. B‎. ‎Fornberg and ‎G‎. ‎Wright‎, ‎Stable computation of multiquadric interpolants for all values of the shape parameter‎, ‎Comput‎. ‎Math‎. ‎Appl. ‎48 (2004) 853–867‎.
    38. J‎. ‎A‎. ‎Rad‎, ‎J‎. ‎Hook‎, ‎E‎. ‎Larsson and L‎. ‎Sydow‎, ‎Forward deterministic pricing of options using Gaussian radial basis functions‎, ‎J. Comput. Sci.‎ ‎24 (2018) 209–217‎.
    39. A. J‎. ‎Khattak‎, ‎S‎. ‎I‎. ‎A‎. ‎Tirmizi and S‎. ‎U‎. ‎Islam‎, ‎Application of meshfree collocation method to a class of nonlinear partial differential equations, Eng. Anal. Bound. Elem.‎ ‎33 (2009) 661–667‎.
    40. A‎. ‎Hajiollow‎, ‎Y‎. ‎Lotfi‎, ‎K‎. ‎Parand‎, ‎A‎. ‎H‎. ‎Rasanan‎, ‎K‎. ‎Rashedi and J. A. Rad, Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment‎: ‎the (quasi)linearization idea combined with radial basis functions (RBFs) approximation‎, ‎Eng. Comput.‎ ‎(2020) 1–15‎.
    41. F‎. ‎Zabihi‎ and ‎M‎. ‎Saffarian‎, ‎A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equations, Eur. Phys. J. Plus.‎ 131 (2016) 243‎.