Gegenbauer Parameter Effect on Gegenbauer Wavelet Solutions of Lane-Emden Equations

Yıl 2024, Cilt: 29 Sayı: 1, 144 - 156, 30.04.2024

Demet Özdek

https://doi.org/10.53433/yyufbed.1330540

Öz

In this study, we aim to solve Lane-Emden equations numerically by the Gegenbauer wavelet method. This method is mainly based on orthonormal Gegenbauer polynomials and takes advantage of orthonormality which reduces the computational cost. As a further advantage, Gegenbauer polynomials are associated with a real parameter allowing them to be defined as Legendre polynomials or Chebyshev polynomials for some values. Although this provides an opportunity to be able to analyze the problem under consideration from a wide point of view, the effect of the Gegenbauer parameter on the solution of Lane-Emden equations has not been studied so far. This study demonstrates the robustness of the Gegenbauer wavelet method on three problems of Lane-Emden equations considering different values of this parameter.

Anahtar Kelimeler

Boundary value problems, Gegenbauer wavelets, Initial value problems, Lane-Emden equations

Lane-Emden Denklemlerinin Gegenbauer Dalgacık Çözümleri Üzerinde Gegenbauer Parametresinin Etkisi

Öz

Bu çalışmada Lane-Emden denklemlerini Gegenbauer dalgacık yöntemi ile sayısal olarak çözmeyi amaçlıyoruz. Bu yöntem temel olarak ortonormal Gegenbauer polinomlarına dayanır ve hesaplama maliyetini azaltan ortonormallik avantajını kullanır. Diğer bir avantaj olarak, Gegenbauer polinomları, bazı değerleri için Legendre polinomları veya Chebyshev polinomları olarak tanımlanabilmelerini sağlayan reel bir parametre ile ilişkilendirilir. Bu durum, ele alınan problemi geniş bir bakış açısıyla analiz edebilmek için bir fırsat sağlasa da Gegenbauer parametresinin Lane-Emden denklemlerinin çözümü üzerindeki etkisi şimdiye kadar çalışılmamıştır. Bu çalışma, bu parametrenin farklı değerlerini dikkate alarak Gegenbauer dalgacık yönteminin Lane-Emden denklemlerinin üç problemi üzerindeki doğruluğunu göstermektedir.

Anahtar Kelimeler

Başlangıç değer problemleri, Gegenbauer dalgacıkları, Lane-Emden denklemleri, Sınır değer problemleri

Destekleyen Kurum

İzmir Ekonomi Üniversitesi

Kaynakça

• Adibi, H., & Rismani, A. M. (2010). On using a modified Legendre-spectral method for solving singular IVPs of Lane–Emden type. Computers and Mathematics with Applications, 60, 2126-2130. doi:10.1016/j.camwa.2010.07.056
• Ahmed, H. M. (2023). Numerical solutions for singular Lane-Emden equations using shifted Chebyshev polynomials of first kind. Contemporary Mathematics, 4(1), 132-149. doi:10.37256/cm.4120232254
• Arfken, G. B., & Weber, H. J. (2005). Mathematical Methods for Physicists (6th Ed.). London: Elsevier Academic Press.
• Chambre, P. L. (1952). On the solution of the Poisson-Boltzmann equation with application to the theory of thermal explosions. The Journal of Chemical Physics, 20, 1795-1797. doi:10.1063/1.1700291
• Chandrasekhar, S. (1967). Introduction to the Study of Stellar Structure. New York: Dover.
• Çağlar, H., Çağlar, N., & Özer, M. (2009). B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos, Soliton & Fractals, 39, 1232-1237. doi:10.1016/j.chaos.2007.06.007
• Davis, H.T. (1962). Introduction to Nonlinear Differential and Integral Equations. New York: Dover,
• Doha, E. H, Abd-Elhameed, W. M., & Youssri, Y. H. (2013). Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type. New Astronomy, 23-24, 113-117. doi:10.1016/j.newast.2013.03.002
• Emden, R. (1907). Gaskugeln: Anwendungen der Mechanischen Warmetheorie auf Kosmologische und Meteorologische Probleme. Berlin: Teubner.
• Gümgüm, S. (2020). Taylor wavelet solution of linear and nonlinear Lane-Emden equations. Applied Numerical Mathematics, 158, 44-53. doi:10.1016/j.apnum.2020.07.019
• Gürbüz, B., & Sezer, M. (2014). Laguerre polynomial approach for solving Lane-Emden type functional differential equations. Applied Mathematics and Computation, 242, 255-264.
• doi:10.1016/j.amc.2014.05.058
• İdiz, F., Tanoğlu, G., & Aghazadeh, N. (2023). A numerical method based on Legendre wavelet and quasilinearization technique for fractional Lane-Emden type equations. Numerical Algorithms, 95, 181-206. doi:10.1007/s11075-023-01568-z
• Khalique, C. M., & Ntsime, P. (2008). Exact solutions of the Lane-Emden-type equations. New Astronomy, 13(7), 476-480. doi:10.1016/j.newast.2008.01.002
• Kim, D. S., Kim, T., & Rim, S. H. (2012). Some identities involving Gegenbauer polynomials. Advances in Difference Equations, 2012, 219. doi:10.1186/1687-1847-2012-219
• Krivec, R., & Mandelzweig, V. B. (2008). Quasilinearization approach to computations with singular potentials. Computer Physics Communications, 179(12), 865-867. doi:10.1016/j.cpc.2008.07.006
• Kumar, N., Pandey, R. K., & Cattani, C. (2011). Solution of the Lane-Emden Equation Using the Bernstein Operational Matrix of Integration. ISRN Astronomy and Astrophysics, 2011, 351747. doi:10.5402/2011/351747
• Kumar, S., Pandey, P., & Das, S. (2019). Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations. Computational and Applied Mathematics, 162, 1-22. doi:10.1007/s40314-019-0952-z
• Lane, J. H. (1870). On the 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. The American Journal of Science and Arts, 50, 57-74. doi:10.2475/ajs.s2-50.148.57
• Liao, S. J. (2003). A new analytic algorithm of Lane-Emden type equations. Applied Mathematics and Computation, 142(1), 1-16. doi:10.1016/S0096-3003(02)00943-8
• Lima, P. M., & Morgado, L. (2010). Numerical modeling of oxygen diffusion in cells with Michaelis–Menten uptake kinetics. Journal of Mathematical Chemistry, 48, 145-158. doi:10.1007/s10910-009-9646-x.
• Mandelzweig, V. B., & Tabakin, F. (2001). Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Computer Physics Communications, 141(2), 268-281. doi:10.1016/S0010-4655(01)00415-5
• Mall, S., & Chakraverty, S. (2015). Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev Neural Network method. Neurocomputing, 149, 975-982. doi:10.1016/j.neucom.2014.07.036
• Mohsenyzadeh, M., Maleknejad, K., & Ezzati, R. (2015). A numerical approach for the solution of a class of singular boundary value problems arising in physiology. Advances in Difference Equations, 231.
• Öztürk, Y. (2018). Solution for the system of Lane–Emden type equations using Chebyshev Polynomials. Mathematics, 6, 181. doi:10.3390/math6100181
• Öztürk, Y., & Gülsu, M. (2014). An operational matrix method for solving Lane-Emden equations arising in astrophysics. Mathematical Methods in the Applied Sciences, 37(15), 2227-2235. doi:10.1002/mma.2969
• Pandey, R. K., & Kumar, N. (2012). Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astronomy, 17(3), 303-308. doi: 10.1016/j.newast.2011.09.005.
• Parand, K., Dehghan, M., Rezaei, A. R., & Ghaderi, S. M. (2010). An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Computer Physics Communications, 181, 1096-1108. doi: 10.1016/j.cpc.2010.02.018
• Reimer, M. (2003). Gegenbauer Polynomials. In Multivariate Polynomial Approximation, (pp. 19-38). Birkhäuser Verlag: Springer Basel.
• Shawagfeh, N.T. (1993). Nonperturbative approximate solution for Lane-Emden equation. Journal of Mathematical Physics, 34, 4364-4369. doi:10.1063/1.530005
• Shiralashetti, S. C., & Kumbinarasaiah, S. (2017). Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations. Applied Mathematics and Computation, 315, 591-602. doi:10.1016/j.amc.2017.07.071
• Singh, R., & Kumar, J. (2014). An efficient numerical technique for the solution of nonlinear singular boundary value problems. Computer Physics Communications, 185, 1282-1289. doi:10.1016/j.cpc.2014.01.002
• Wazwaz, A. M. (2001). A new algorithm for solving differential equations of Lane-Emden type. Applied Mathematics and Computation, 118, 287-310. doi:10.1016/S0096-3003(99)00223-4
• Wazwaz, A. M. (2011). The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Communications in Nonlinear Science and Numerical Simulation, 16, 3881-3886. doi:10.1016/j.cnsns.2011.02.026
• Usman, M., Hamid, M., Zubair, T., Haq, R., & Wang, W. (2019). Operational-matrix-based algorithm for differential equations of fractional order with Dirichlet boundary conditions. The European Physical Journal Plus, 134, 279-294. doi: 10.1140/epjp/i2019-12653-7
• Van Gorder, R. A., & Vajravelu, K. (2008). Analytic and numerical solutions to the Lane-Emden Equations. Physics Letters, 372(39), 6060-6065. doi:10.1016/j.physleta.2008.08.002
• Yildirim, A., & Öziş, T. (2009). Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Analysis, Theory, Methods & Applications, 70(6), 2480-2484. doi:10.1016/j.na.2008.03.012
• Yousefi, A.S. (2006). Legendre wavelets method for solving differential equations of Lane–Emden type. Applied Mathematics and Computation, 181, 1417-1422. doi:10.1016/j.amc.2006.02.031
• Yüzbaşı, Ş. (2011). A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics. Mathematical Methods in the Applied Sciences, 34(18), 2218-2230. doi:10.1002/mma.1519
• Yüzbaşı, Ş., & Sezer, M. (2013). An improved Bessel collocation method with a residual error function to solve a class of Lane-Emden differential equations, Mathematical Computer Modelling, 57, 1298-1311. doi:10.1016/j.mcm.2012.10.032