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.
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.
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.
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.
R. Emden, Gaskugeln: Anwendungen der Mechanischen Warmetheorie Auf kosmologische und Meteorologische Probleme, BG Teubner, Berlin, 1907.
R. Aris, Introduction to the Analysis of Chemical Reactors, Englewood Cliffs, N. J. Prentice-Hall, 1965.
B. N. Metha and R. Aris, Mass transfer effects in coal combustion, J. Math. Anal.Appl. 36 (1971) 611–621.
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.
S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, 1967.
A. M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118 (2001) 287–310.
A. M. Wazwaz, The modifed decomposition method for analytic treatment of differential equations, Appl. Math. Comput. 173 (2006) 165–176.
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.
J. H. He, Variational approach to the Lane-Emden equation, Appl. Math.Comput. 143 (2003) 539–541.
J. I. Ramos, Series approach to the Lane-Emden equation and comparison with the Homotopy perturbation method, Chaos Soliton. Fract. 38 (2008) 400–408.
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.
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.
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.
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.
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.
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.
M. Bisheh-Niasar, A Computational Method for Solving the Lane-Emden Initial Value Problems, Comput. Methods Differ. Equ. 8 (2020) 673–684.
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.
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.
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.
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.
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.
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.
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.
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.
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.
G. Liu and Y. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, Netherlands, 2005.
M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, New York, 2004.
H. Wendland, Scattered data approximation, Cambridge University Press, New York, 2005.
R. Franke, Scattered data interpolation: Tests of some methods, Math. Comp. 48 (1982) 181–200.
S. Rippa, An algorithm for selecting a good parameter c in radial basis function interpolation, Advan. Comp. Math. 11 (1999) 193–210.
B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl.48 (2004) 853–867.
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.
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.
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.
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.
Hajiollow, A., & Zabihi, F. (2021). The effect of radial basis functions (RBFs) method in solving coupled Lane–Emden boundary value problems in Catalytic Diffusion Reactions. Iranian Journal of Mathematical Chemistry, 12(4), 239-261. doi: 10.22052/ijmc.2021.242232.1561
MLA
Ashraf Hajiollow; Fatemeh Zabihi. "The effect of radial basis functions (RBFs) method in solving coupled Lane–Emden boundary value problems in Catalytic Diffusion Reactions". Iranian Journal of Mathematical Chemistry, 12, 4, 2021, 239-261. doi: 10.22052/ijmc.2021.242232.1561
HARVARD
Hajiollow, A., Zabihi, F. (2021). 'The effect of radial basis functions (RBFs) method in solving coupled Lane–Emden boundary value problems in Catalytic Diffusion Reactions', Iranian Journal of Mathematical Chemistry, 12(4), pp. 239-261. doi: 10.22052/ijmc.2021.242232.1561
VANCOUVER
Hajiollow, A., Zabihi, F. The effect of radial basis functions (RBFs) method in solving coupled Lane–Emden boundary value problems in Catalytic Diffusion Reactions. Iranian Journal of Mathematical Chemistry, 2021; 12(4): 239-261. doi: 10.22052/ijmc.2021.242232.1561
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