Journal of the Korean Society for Industrial and Applied Mathematics

  • 제26권4호
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  • Pages.185-223
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  • 2022
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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ANALYTICAL AND NUMERICAL SOLUTIONS OF A CLASS OF GENERALISED LANE-EMDEN EQUATIONS

  • 투고 : 2022.08.22
  • 심사 : 2022.12.07
  • 발행 : 2022.12.25
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⟨ 이전 논문 다음 논문 ⟩

초록

The classical equation of Jonathan Homer Lane and Robert Emden, a nonlinear second-order ordinary differential equation, models the isothermal spherical clouded gases under the influence of the mutual attractive interaction between the gases' molecules. In this paper, the Adomian decomposition method (ADM) is presented to obtain highly accurate and reliable analytical solutions of a class of generalised Lane-Emden equations with strong nonlinearities. The nonlinear term f(y(x)) of the proposed problem is given by the integer powers of a continuous real-valued function h(y(x)), that is, f(y(x)) = hm(y(x)), for integer m ≥ 0, real x > 0. In the end, numerical comparisons are presented between the analytical results obtained using the ADM and numerical solutions using the eighth-order nested second derivative two-step Runge-Kutta method (NSDTSRKM) to illustrate the reliability, accuracy, effectiveness and convenience of the proposed methods. The special cases h(y) = sin y(x), cos y(x); h(y) = sinh y(x), cosh y(x) are considered explicitly using both methods. Interestingly, in each of these methods, a unified result is presented for an integer power of any continuous real-valued function - compared with the case by case computations for the nonlinear functions f(y). The results presented in this paper are a generalisation of several published results. Several examples are given to illustrate the proposed methods. Tables of expansion coefficients of the series solutions of some special Lane-Emden type equations are presented. Comparisons of the two results indicate that both methods are reliably and accurately efficient in solving a class of singular strongly nonlinear ordinary differential equations.

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참고문헌

  1. C. Mohan, A.R. Al-Bayaty, Power series solutions of the Lane-Emden equation, Astro. Space Sci., 73 (1980), 227-239. https://doi.org/10.1007/BF00642378
  2. A.M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput., 118 (2001), 287-310. https://doi.org/10.1016/S0096-3003(99)00223-4
  3. A. Saadatmandi, A. Ghasemi-Nasrabady, A. Eftekhari, Numerical study of singular fractional Lane-Emden type equations arising in astrophysics, J. Astrophys. Astr. 40 (2019), 1-12. https://doi.org/10.1007/s12036-018-9570-1
  4. R. Saadeh, A. Burqan, A. El-Ajou, Reliable solutions of fractional Lane-Emden equations via Laplace transform and residual error function, Alenxandria Eng. J., 61 (2022), 10551-10562. https://doi.org/10.1016/j.aej.2022.04.004
  5. 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. https://doi.org/10.1016/j.cplett.2017.04.077
  6. K. Boubaker, R. A. V. Gorder, Application of the BPES to Lane-Emden equations governing polytropic and isothermal gas spheres, New Astron., 17 (2012) 565-569. https://doi.org/10.1016/j.newast.2012.02.003
  7. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Courier Corporation, Dover, New York, 1962.
  8. O.U. Richardson, The Emission of Electricity from Hot Bodies, Longman, Green and Co., London, New York, 1921.
  9. M. C, Khalique, F. M. Mahomed, B. Muatjetjeja, Lagrangian formulation of a generalized Lane-Emden equation and double reduction, J. Nonl. Math. Phys.,15 (2008), 152-161. https://doi.org/10.2991/jnmp.2008.15.2.3
  10. H. Madduri, P. Roul, T.C. Hao, F.Z. Cong,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. https://doi.org/10.1007/s10910-018-0912-7
  11. H. Madduri, P. Roul, A fast-converging iterative scheme for solving a system of Lane-Emden equations arising in catalytic diffusion reactions, J. Math. Chem., 57 (2019), 570-582. https://doi.org/10.1007/s10910-018-0964-8
  12. P. Roul, A new mixed MADM-Collocation approach for solving a class of Lane-Emden singular boundary value problems, J. Math. Chem., 57 (2019), 945-969. https://doi.org/10.1007/s10910-018-00995-x
  13. A.K. Verma, S. Kayenat, On the convergence of Mickens' type nonstandard finite difference schemes on Lane-Emden type equations, J. Math. Chem., 56 (2018), 1667-1706. https://doi.org/10.1007/s10910-018-0880-y
  14. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, 1967.
  15. J.H. He, F. Y. Ji, Taylor series solution for Lane-Emden equation, Journal of Mathematical Chemistry, 57 (2019), 1932-1934 https://doi.org/10.1007/s10910-019-01048-7
  16. J.I Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons & Fractals, 38 (2008), 400-408. https://doi.org/10.1016/j.chaos.2006.11.018
  17. S.K Vanani, A. Aminataei, On the numerical solutions of differential equations of Lane-Emden type, Comput. Math. Appl., 59 (2010), 2815-2820.
  18. M. S. H. Chowdhury, I. Hashim, Solutions of a class of singular second-order IVPs by homotopyperturbation method, Phys. Lett. A, 365 (2007), 439-447. https://doi.org/10.1016/j.physleta.2007.02.002
  19. O. P. Singh, R. K. Pandey, V. K. Singh, An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified Homotopy analysis method, Comput. Phys. Commun., 180 (2009), 1116-1124. https://doi.org/10.1016/j.cpc.2009.01.012
  20. A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonl. Anal., 70, (2009), 2480-1484.
  21. S.A. Yousefi, Legendre wavelets method for solving differential equations of Lane-Emden type, Appl. Math. Comput., 181 (2006), 1417-1422. https://doi.org/10.1016/j.amc.2006.02.031
  22. E. A. -B. Abdel-Salam, M. I. Nouh, E. A. Elkholy, Analytical solution to the conformable fractional LaneEmden type equations arising in astrophysics, Scientific African, 8 (2020), e00386.
  23. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501-544. https://doi.org/10.1016/0022-247x(88)90170-9
  24. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994.
  25. G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of Lane-Emden equation, Foundations of Phys. Lett., 8 (1995), 161-181. https://doi.org/10.1007/BF02187585
  26. U. Saeed, Haar Adomian method for the solution of fractional nonlinear Lane-Emden type equations arising in astrophysics, Taiwanese J. Math., 21 (2017), 1175-1192, https://doi.org/10.11650/tjm/7969
  27. M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron., 13 (2008), 53-59. https://doi.org/10.1016/j.newast.2007.06.012
  28. J.H. He, Variational approach to the Lane-Emden equation, Appl. Math. Comput., 143 (2003), 539-541. https://doi.org/10.1016/S0096-3003(02)00382-X
  29. A. Aslanov, A generalization of the Lane-Emden equation, Int. J. Comput. Math., 85 (2008), 661-663. https://doi.org/10.1080/00207160701558457
  30. P. Mach, All solutions of the n = 5 Lane-Emden equation, J. Math. Phys., 53 (2012), 062503.
  31. M. S. Hashemi, A. Akgul, M. Inc, I. S. Mustafa, D. Baleanu, Solving the Lane-Emden equation within a reproducing kernel method and group preserving scheme, Mathematics, 5 (2017), 1-13. https://doi.org/10.3390/math5010001
  32. A.M. Malik, O.H. Mohammed, Two efficient methods for solving fractional Lane-Emden equations with conformable fractional derivative, J. Egyptian Math. Soc., 28 (2020)
  33. P. K. Sahu, B. Mallick, Approximate solution of fractional order Lane-Emden type differential equation by orthonormal Bernoulli's polynomials, Int. J. Appl. Comput. Math, 5 (2019)
  34. Z. Sabir, M. A. Z. Raja, M. Umar, M. Shoaib, Neuro-swarm intelligent computing to solve the second-order singular functional differential model, Eur. Phys. J. Plus, 135 (2020)
  35. W. Adel, Z. Sabir, Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model, Eur. Phys. J. Plus, 135 (2020)
  36. R. Gupta, S. Kumar, Numerical simulation of variable-order fractional differential equation of nonlinear Lane-Emden type appearing in astrophysics, Int. J. Nonlinear Sci. Num. Simul., 23 (2022)
  37. M. A. Abdelkawy, Z. Sabir, J. L. G. Guirao, T. Saeed, Numerical investigations of a new singular secondorder nonlinear coupled functional Lane-Emden model, Open Phys.,18 (2020), 770-778. https://doi.org/10.1515/phys-2020-0185
  38. K. Tablennehas, Z. Dahmani, M. M. Belhamiti, A. Abdelnebi, M. Z. Sarikaya, On a fractional problem of Lane-Emden type: Ulam type stabilities and numerical behaviors, Adv. Differ. Equat.,2021 (2021)
  39. H. F. Ahmed, M. B. Melad, A new numerical strategy for solving nonlinear singular Emden-Fowler delay differential models with variable order, Math. Sci., (2022)
  40. R. O. Awonusika, Analytical solution of a class of fractional Lane-Emden equation: a power series method, Int. J. Appl. Comput. Math,8 (2022)
  41. R. O. Awonusika, O. A. Mogbojuri, Approximate analytical solution of fractional Lane-Emden equation by Mittag-Leffler function method, J. Nig. Soc. Phys. Sci., 4 (2022), 265-280.
  42. M.S. Mechee, N. Senu, Numerical study of fractional differential equations of Lane-Emden type by method of collocation, Appl. Math., 3 (2012), 851-856. https://doi.org/10.4236/am.2012.38126
  43. M. I. Nouh, E. A.-B. Abdel-Salam, Approximate Solution to the Fractional Lane-Emden Type Equations, Iran J Sci Technol Trans Sci, 42 (2017), 2199-2206.
  44. A. Akgul, M. Inc, E. Karatas, D. Baleanu, Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Adv. Differ. Equat., 2015 (2015)
  45. B. Caruntu, C. Bota, M. Lapadat, M. S. Pasca, Polynomial least squares method for fractional Lane-Emden equations, Symmetry, 11 (2019)
  46. J. Davila, L. Dupaigne, J. Wei, On the fractional Lane-Emden equation, Trans. Am. Math. Soc.,369 (2017), 6087-6104. https://doi.org/10.1090/tran/6872
  47. A. K. Nasab, Z. P. Atabakan, A. I. Ismail, R. W. Ibrahim, A numerical method for solving singular fractional Lane-Emden type equations, J. King Saud University-Science,30 (2018)
  48. A. H. Bhrawy, A. S. Alofi, A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 62-70. https://doi.org/10.1016/j.cnsns.2011.04.025
  49. P. O. Olatunji, Second derivative multistep methods with nested hybrid evaluation, M.Sc. Thesis, Department of Mathematics, University of Benin, Nigeria, 2017.
  50. P. O. Olatunji, M. N. O. Ikhile, Variable order nested hybrid multistep methods for stiff ODEs, J. Math. Comput. Sci., 10 (2020), 78-94.
  51. A. Figueroa, Z. Jackiewicz, R. Lohner, Explicit two-step Runge-Kutta methods for computational fluid dynamics solvers, Int. J. Numer. Methods Fluids, 93 (2020), 429-444.
  52. A. Figueroa, Z. Jackiewicz, R. Lohner, Efficient two-step Runge-Kutta methods for fluid dynamics simulations, Appl. Numer. Math., 159 (2021), 1-20. https://doi.org/10.1016/j.apnum.2020.08.013
  53. S. E. Ogunfeyitimi and M. N. O. Ikhile, Second derivative generalized extended backward differentiation formulas for stiff problems, J. Korean Soc. Ind. Appl. Math., 23 (2019) 179-202. https://doi.org/10.12941/jksiam.2019.23.179
  54. S. E. Ogunfeyitimi and M. N. O. Ikhile, Multiblock boundary value methods for ordinary differential and differential algebraic equations, J. Korean Soc. Ind. Appl. Math., 24 (2020), 243-291. https://doi.org/10.12941/JKSIAM.2020.24.243
  55. J. C. Butcher, Numerical methods for solving ordinary differential equations, Wiley, Chichester, 2016.
  56. P. O. Olatunji, Nested general linear methods for stiff differential equations and differential algebraic equations, PhD Thesis, Department of Mathematics, University of Benin, Nigeria, 2021.
  57. P. O. Olatunji, M. N. O. Ikhile, Strongly regular general linear methods, J. Sci. Comput., 82 (2020), 1-25. https://doi.org/10.1007/s10915-019-01102-1
  58. P. O. Olatunji, M. N. O. Ikhile, FSAL mono-implicit Nordsieck general linear methods with inherent Runge - Kutta stability for DAEs, J. Korean Soc. Ind. Appl. Math., 25 (2021), 262-295.
  59. P. O. Olatunji, M. N. O. Ikhile, R. I. Okuonghae, Nested second derivative two-step Runge-Kutta methods, Int. J. Appl. Comput. Math., 7 (2021), 1-39. https://doi.org/10.1007/s40819-020-00933-z
  60. K. Parand, M. Dehghan, A. R. Rezaei, 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. https://doi.org/10.1016/j.cpc.2010.02.018
  61. A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111 (2000), 53-69. https://doi.org/10.1016/S0096-3003(99)00063-6
  62. K. Abbaoui, Y. Cherruault, Convergence of Adomian's method applied to differential equations, Comput. Math. Appl., 102 (1999), 77-86. https://doi.org/10.1016/S0096-3003(98)10024-3
  63. R. Rach, A convenient computational form for the Adomian polynomials, J. Math. Anal. Appl., 102 (1984), 415-419. https://doi.org/10.1016/0022-247X(84)90181-1
  64. V. Seng, K. Abbaoui, Y. Cherruault, Adomian's polynomials for nonlinear operators, Mathl. Comput. Modelling, 24 (1996), 59-65.
  65. A.M. Wazwaz, The decomposition method for approximate solution of the Goursat problem, Appl. Math. Comput., 69 (1995), 299-311. https://doi.org/10.1016/0096-3003(94)00137-S
  66. A.M. Wazwaz, A reliable modification of Adomian's decomposition method, Appl. Math. Comput., 102 (1999), 77-86. https://doi.org/10.1016/S0096-3003(98)10024-3