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An implicit series solution for a boundary value problem modelling a thermal explosion. (English) Zbl 1211.65101
Summary: An implicit series solution admitted by a boundary value problem modelled by a Lane-Emden equation of the second kind is obtained. The boundary value problem was derived by Frank-Kamenetskii to model the steady temperature in a vessel in which a thermal explosion is taking place. The Lane-Emden equation is reduced to an autonomous second-order ordinary differential equation by means of a coordinate transformation. The autonomous second-order ordinary differential equation is reduced to a first-order Abel equation. A power series solution of the first-order Abel equation is obtained. The power series solution of the Abel equation is transformed into an implicit series solution of the original Lane-Emden equation satisfying the boundary conditions of the original problem. We show that the implicit power series solution is valid for values of the dimensionless Frank-Kamenetskii parameter δ<0·02.
65L10Boundary value problems for ODE (numerical methods)
[1]Frank-Kamenetzkii, D. A.: Diffusion and heat transfer in chemical kinetics, (1969)
[2]Chandrasekhar, S.: An introduction to the study of stellar structure, (1939) · Zbl 0022.19207
[3]Bozkhov, Y.; Martins, A. C. G.: Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents, Nonlinear anal. 57, 773-793 (2004) · Zbl 1061.34030 · doi:10.1016/j.na.2004.03.016
[4]Harley, C.; Momoniat, E.: First integrals and bifurcations of a Lane–Emden equation of the second-kind, J. math. Anal. appl. 344, 757-764 (2008) · Zbl 1143.80005 · doi:10.1016/j.jmaa.2008.03.014
[5]Chambré, P. L.: On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions, J. chem. Phys. 20, 1795-1797 (1952)
[6]Harley, C.; Momoniat, E.: Steady state solutions for a thermal explosion in a cylindrical vessel, Modern phys. Lett. B 21, 831-842 (2007) · Zbl 1115.80005 · doi:10.1142/S0217984907013250
[7]Russell, R. D.; Shampine, L. F.: Numerical methods for singular boundary value problems, SIAM J. Numer. anal. 12, 13-36 (1975) · Zbl 0271.65051 · doi:10.1137/0712002
[8]Balakrishnan, E.; Swift, A.; Wake, G. C.: Critical values for some non-class a geometries in thermal ignition theory, Math. comput. Modelling 24, l-10 (1996) · Zbl 0880.35123 · doi:10.1016/0895-7177(96)00133-1
[9]Harley, C.; Momoniat, E.: Instability of invariant boundary conditions of a generalized Lane–Emden equation of the second-kind, Appl. math. Comput. 198, 621-633 (2008) · Zbl 1146.34031 · doi:10.1016/j.amc.2007.08.077
[10]Harley, C.; Momoniat, E.: Alternate derivation of the critical value of the Frank-kamenetskii parameter in cylindrical geometry, J. nonlinear math. Phys. 15, 69-76 (2008)
[11]Ramos, J. I.: Piecewise quasilinearization techniques for singular boundary-value problems, Comput. phys. Comm. 158, 12-25 (2004) · Zbl 1196.65122 · doi:10.1016/j.comphy.2003.11.003
[12]Binney, J.; Tremaine, S.: Galactic dynamics, (1987)
[13]Davis, H. T.: Introduction to nonlinear differential and integral equations, (1962) · Zbl 0106.28904
[14]Emden, R.: Gaskugeln–anwendungen der mechan. Warmtheorie, (1907)
[15]Kippenhahn, R.; Weigert, A.: Stellar structure and evolution, (1990)
[16]Richardson, O. U.: The emission of electricity from hot bodies, (1921)
[17]Shawagfeh, N. T.: Nonperturbative approximate solution for Lane–Emden equation, J. math. Phys. 34, 4364-4369 (1993) · Zbl 0780.34007 · doi:10.1063/1.530005
[18]Nouh, M. I.: Accelerated power series solution of polytropic and isothermal gas spheres, New astron. 9, 467-473 (2004)
[19]Mirza, B. M.: Approximate analytical solutions of the Lane–Emden equation for a self-gravitating isothermal gas sphere, Mon. not. R. astron. Soc. 395, 2288-2291 (2009)
[20]Wazwaz, A. M.: A new algorithm for solving differential equations of Lane–Emden type, Appl. math. Comput. 118, 287-310 (2001) · Zbl 1023.65067 · doi:10.1016/S0096-3003(99)00223-4
[21]Wazwaz, A. M.: A new method for solving singular initial value problems in the second-order ordinary differential equations, Appl. math. Comput. 128, 45-57 (2002) · Zbl 1030.34004 · doi:10.1016/S0096-3003(01)00021-2
[22]Wazwaz, A. M.: Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Appl. math. Comput. 161, 543-560 (2005) · Zbl 1061.65064 · doi:10.1016/j.amc.2003.12.048
[23]Wazwaz, A. M.: The modified decomposition method for analytic treatment of differential equations, Appl. math. Comput. 173, 165-176 (2006) · Zbl 1089.65112 · doi:10.1016/j.amc.2005.02.048
[24]Aslanov, A.; Abu-Alshaikhb, I.: Further developments to the decomposition method for solving singular initial-value problems, Math. comput. Modelling 48, 700-711 (2008) · Zbl 1156.65312 · doi:10.1016/j.mcm.2007.10.013
[25]Liao, S.: A new analytic algorithm of Lane–Emden type equations, Appl. math. Comput. 142, 1-16 (2003) · Zbl 1022.65078 · doi:10.1016/S0096-3003(02)00943-8
[26]Chowdhury, M. S. H.; Hashim, I.: Solutions of Emden–Fowler equations by homotopy–perturbation method, Nonlinear anal. RWA 10, 104-115 (2009) · Zbl 1154.34306 · doi:10.1016/j.nonrwa.2007.08.017
[27]Ramos, J. I.: Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method, Chaos solitons fractals 38, 400-408 (2008) · Zbl 1146.34300 · doi:10.1016/j.chaos.2006.11.018
[28]He, J. -H.: Variational approach to the Lane–Emden equation, Appl. math. Comput. 143, 539-541 (2003)
[29]Yousefi, S. A.: Legendre wavelets method for solving differential equations of Lane–Emden type, Appl. math. Comput. 181, 1417-1422 (2006) · Zbl 1105.65080 · doi:10.1016/j.amc.2006.02.031
[30]Dehghan, M.; Shakeri, F.: Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New astron. 13, 53-59 (2008)
[31]Yildirim, A.; Öziş, T.: Solutions of singular ivps of Lane–Emden type by the variational iteration method, Nonlinear anal. 70, 2480-2484 (2009) · Zbl 1162.34005 · doi:10.1016/j.na.2008.03.012
[32]Ertürk, V. S.: Differential transformation method for solving differential equations of Lane–Emden type, Math. comput. Appl. 12, 135-139 (2007) · Zbl 1175.34007
[33]Van Gorder, R. A.; Vajravelu, K.: Analytic and numerical solutions to the Lane–Emden equation, Phys. lett. A 372, 6060-6065 (2008) · Zbl 1223.85004 · doi:10.1016/j.physleta.2008.08.002
[34]Vanani, S. K.; Aminataei, A.: On the numerical solution of differential equations of Lane–Emden type, Comput. math. Appl. 59, 2815-2820 (2010) · Zbl 1193.65151 · doi:10.1016/j.camwa.2010.01.052
[35]Parand, K.; Shahini, M.; Dehghan, M.: Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type, J. comput. Phys. 228, 8830-8840 (2009) · Zbl 1177.65100 · doi:10.1016/j.jcp.2009.08.029
[36]Parand, K.; Dehghan, M.; Rezaei, A. R.; Ghaderi, S. M.: An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. phys. Comm. 181, 1096-1108 (2010) · Zbl 1216.65098 · doi:10.1016/j.cpc.2010.02.018
[37]Momoniat, E.; Harley, C.: Approximate implicit solution of a Lane–Emden equation, New astron. 11, 520-526 (2006)
[38]Khalique, C. M.; Mahomed, F. M.; Muatjetjeja, B.: Lagrangian formulation of a generalized Lane–Emden equation and double reduction, J. nonlinear math. Phys. 15, 152-161 (2008) · Zbl 1169.34033 · doi:10.2991/jnmp.2008.15.2.3
[39]Khalique, C. M.; Ntsime, P.: Exact solutions of the Lane–Emden-type equation, New astron. 13, 476-480 (2008)
[40]Hunter, C.: Series solutions for polytropes and the isothermal sphere, Mon. not. R. astron. Soc. 328, 839-847 (2001)
[41]Aslanov, A.: A generalization of Lane–Emden equation, Int. J. Comput. math. 85, 1709-1725 (2008) · Zbl 1154.65059 · doi:10.1080/00207160701558457
[42]Aslanov, A.: Approximate solutions of Emden–Fowler type equations, Int. J. Comput. math. 86, 807-826 (2009) · Zbl 1170.34011 · doi:10.1080/00207160701708235 · doi:http://www.informaworld.com/smpp/./content~db=all~content=a794908507
[43]Aslanov, A.: Determination of convergence intervals of the series solutions of Emden–Fowler equations using polytropes and isothermal spheres, Phys. lett. A 372, 3555-3561 (2008) · Zbl 1220.35084 · doi:10.1016/j.physleta.2008.02.019
[44]Dresner, L.: Phase-plane analysis of nonlinear, second-order, ordinary differential equations, J. math. Phys. 12, 1339-1348 (1971) · Zbl 0266.34034 · doi:10.1063/1.1665739