Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials
Abstract
:1. Introduction
2. Fundamental Relations
3. Method of Solution
4. Algorithm
5. Examples
6. Conclusions
Funding
Conflicts of Interest
References
- Fowler, R.H. Further studies of Emden’s and similar differential equations. Q. J. Math. 1931, 2, 259–288. [Google Scholar] [CrossRef]
- Meerson, E.; Megged, E.; Tajima, T. On the quasi-hydrostatic flows of radiatively cooling self-gravitating gas clouds. Astrophys. J. 1996, 457, 321. [Google Scholar] [CrossRef]
- Chandrasekhar, S. Introduction to Study of Stellar Structure. Available online: https://www.amazon.com/Introduction-Study-Stellar-Structure-Astronomy/dp/0486604136 (accessed on 22 August 2018).
- Davis, H.T. Introduction to Nonlinear Differential and Integral Equations. J. Lond. Math. Soc. 1962, 16, 556. [Google Scholar] [CrossRef]
- Flockerzi, D.; Sundmacher, K. On coupled Lane-Emden equations arising in dusty fluid models. J. Phys. Conf. Ser. 2011, 268, 012006. [Google Scholar] [CrossRef] [Green Version]
- Parand, K.; Pirkhedri, K. Sinc-collocation method for solving astrophysics equations. New Astron. 2010, 15, 533–537. [Google Scholar] [CrossRef]
- Dehghan, M.; Shakeri, F. Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astron. 2008, 13, 53–59. [Google Scholar] [CrossRef]
- Parand, K.; Dehghan, M.; Rezaei, A.R.; Ghaderi, S. An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method. Comput. Phys. Commun. 2010, 181, 1096–1108. [Google Scholar] [CrossRef] [Green Version]
- Singh, O.P.; Pandey, R.K.; Singh, V.K. An analytic algorithm of Lane–Emden type equations arising in astrophysics using modified Homotopy analysis method. Comput. Phys. Commun. 2009, 180, 1116–1124. [Google Scholar] [CrossRef]
- Hasan, Y.Q.; Zhu, L.M. Solving singular boundary value problems of higher-order ordinary differential equations by modified Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 2592–2596. [Google Scholar] [CrossRef]
- Pandey, R.K.; Kumar, N.; Bhardwaj, A.; Dutta, G. Solution of Lane-Emden type equations using Legendre operational matrix of differentiation. Appl. Math. Comput. 2012, 218, 7629–7637. [Google Scholar] [CrossRef]
- Pandey, R.K.; Kumar, N. Solution of Lane-Emden type equations using Berstein operational matrix of differentiation. New Astron. 2012, 17, 303–308. [Google Scholar] [CrossRef]
- Agarwal, R.P.; O’Regan, D. Second order initial value problems of Lane-Emden type. Appl. Math. Lett. 2007, 20, 1198–1205. [Google Scholar] [CrossRef]
- Varani, S.K.; Aminataei, A. On the numerical solution of differential equations of Lane-Emden type. Comput. Math. Appl. 2010, 59, 2815–2820. [Google Scholar]
- Aslanov, A. A generalization of the Lane–Emden equation. Int. J. Comput. Math. 2008, 85, 1709–1725. [Google Scholar] [CrossRef]
- Wazwaz, A.M. A new algorithm for solving differential equations of Lane-Emden type. Appl. Math. Comput. 2001, 118, 287–310. [Google Scholar] [CrossRef]
- Wazwaz, A.M.; Rach, R.; Duan, J.S. A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method. Math. Method Appl. Sci. 2013, 37, 10–19. [Google Scholar] [CrossRef]
- Marin, M. An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 2016, 51, 1127–1133. [Google Scholar] [CrossRef]
- Marin, M. Some estimates on vibrations in Thermoelasticity of dipolar bodies. J. Vib. Control 2010, 16, 33–47. [Google Scholar] [CrossRef]
- Marin, M. A temporally evolutionary equation in elasticity of micropolar bodies with voids. U.P.B. Sci. Bull. Ser. A-Appl. Math. Phys. 1998, 60, 3–12. [Google Scholar]
- Gülsu, M.; Öztürk, Y.; Sezer, M. A new collocation method for solution of mixed linear integro-differential-difference equations. Appl. Math. Comput. 2010, 216, 2183–2198. [Google Scholar] [CrossRef] [Green Version]
- Gülsu, M.; Öztürk, Y.; Sezer, M. On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials. Appl. Math. Comput. 2011, 217, 4827–4833. [Google Scholar] [CrossRef]
- Daşçıoğlu, A.; Yaslan, H. The solution of high-order nonlinear ordinary differential equations by Chebyshev polynomials. Appl. Math. Comput. 2011, 217, 5658–5666. [Google Scholar]
- Öztürk, Y.; Anapalı, A.; Gülsu, M. A numerical scheme for continuous population models for single and interacting species. J. Balıkesir Univ. Inst. Sci. Technol. 2017, 19, 12–28. [Google Scholar] [CrossRef]
- Herrero, H.; Mancho, A.M. Numerical modeling in Chebyshev collocation methods applied to stability analysis of convection problems. Appl. Numer. Math. 2000, 33, 161–165. [Google Scholar] [CrossRef]
- Gürbüz, B.; Sezer, M. Modified Laguerre collocation method for solving 1-dimensional parabolic convection-diffusion problems. Math. Method. Appl. Sci. Spec. Issue Pap. 2017. [Google Scholar] [CrossRef]
- Jang, W.; Chen, Z.; Zhang, C. Chebshev collocation method for solving singular integral equation with cosecant kernel. Int. J. Comput. Math. 2012, 89, 975–982. [Google Scholar] [CrossRef]
- Boyd, J.P. Chebyshev and Fourier Spectral Methods. Available online: https://books.google.com.hk/books?hl=en&lr=&id=i9UoAwAAQBAJ&oi=fnd&pg=PP1&dq=Chebyshev+and+fourier+spectral+methods&ots=mCuqRE7G7u&sig=PdpyHdX0wV0ZzUCt1znEvdtFXAs&redir_esc=y&hl=zh-CN&sourceid=cndr#v=onepage&q=Chebyshev%20and%20fourier%20spectral%20methods&f=false (accessed on 22 August 2018).
- Mason, J.C.; Handscomb, D.C. Chebyshev Polynomials. Available online: https://www.crcpress.com/Chebyshev-Polynomials/Mason-Handscomb/p/book/9780849303555 (accessed on 22 August 2018).
t | Exact Solution | N = 5 | Ne = 5 | N = 6 | Ne = 6 | N = 8 | Ne = 8 |
---|---|---|---|---|---|---|---|
0.0 | 1.000000 | 0.999999 | 0.800 × 10−8 | 0.999999 | 0.500 × 10−9 | 1 | 0 |
0.2 | 1.040810 | 1.040834 | 0.238 × 10−4 | 1.040810 | 0.135 × 10−6 | 1.040810 | 0.102 × 10−6 |
0.4 | 1.173510 | 1.173384 | 0.126 × 10−3 | 1.173517 | 0.690 × 10−5 | 1.173510 | 0.261 × 10−6 |
0.6 | 1.433329 | 1.433539 | 0.209 × 10−3 | 1.433298 | 0.305 × 10−4 | 1.433329 | 0.471 × 10-6 |
0.8 | 1.896480 | 1.895792 | 0.688 × 10−2 | 1.896583 | 0.102 × 10−3 | 1.896481 | 0.909 × 10−6 |
1.0 | 2.718281 | 2.686791 | 0.314 × 10−1 | 2.712165 | 0.611 × 10−3 | 2.718083 | 0.197 × 10−3 |
t | Exact Solution | N = 5 | Ne = 5 | N = 6 | Ne = 6 | N = 8 | Ne = 8 |
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.2 | −0.0064 | −0.006400 | 0.436 × 10−7 | −0.006400 | 0.189 × 10−9 | −0.00640 | 0.122 × 10−9 |
0.4 | −0.0384 | −0.038399 | 0.343 × 10−6 | −0.038400 | 0.358 × 10−7 | −0.03840 | 0.399 × 10−9 |
0.6 | −0.0864 | −0.086399 | 0.770 × 10−5 | −0.086399 | 0.102 × 10−6 | −0.08640 | 0.138 × 10−8 |
0.8 | −0.1024 | −0.102400 | 0.620 × 10−5 | −0.102400 | 0.259 × 10−6 | −0.10240 | 0.455 × 10−8 |
1.0 | 0.0000 | −0.419×10−4 | 0.419 × 10−4 | −0.722×10−5 | 0.722 × 10−5 | 0.168 × 10−6 | 0.168 × 10−6 |
Present Method | ||||
---|---|---|---|---|
- | ||||
N = 5 | 0.615773 × 10−2 | 101474 × 10−4 | 10−2 | 10−6 |
N = 6 | 0.103500 × 10−2 | 0.148370 × 10−5 | 10−3 | 10−8 |
N = 8 | 0.263404 × 10−4 | 0.256832 × 10−7 | 10−5 | 10−10 |
t | Exact Solution | N = 4 | Ne = 4 | N = 5 | Ne = 5 | N = 6 | Ne = 6 |
---|---|---|---|---|---|---|---|
0.0 | 1.00000 | 1.000000 | 0 | 1.00000 | 0 | 1.00000 | 0 |
0.2 | 1.019803 | 1.020423 | 0.509 × 10−3 | 1.019815 | 0.565 × 10−4 | 1.019803 | 0.756 × 10−5 |
0.4 | 1.077032 | 1.077135 | 0.628 × 10−3 | 1.077064 | 0.216 × 10−4 | 1.077032 | 0.865 × 10−5 |
0.6 | 1.166190 | 1.166724 | 0.277 × 10−3 | 1.166185 | 0.557 × 10−5 | 1.166190 | 0.456 × 10−5 |
0.8 | 1.280624 | 1.280663 | 0.272 × 10−3 | 1.280699 | 0.738 × 10−4 | 1.280624 | 0.771 × 10−5 |
1.0 | 1.414213 | 1.414742 | 0.644 × 10−3 | 1.414217 | 0.746 × 10−4 | 1.414213 | 0.656 × 10−5 |
t | Exact Solution | N = 4 | Ne = 4 | N = 5 | Ne = 5 | N = 6 | Ne = 6 |
---|---|---|---|---|---|---|---|
0.0 | 1.000000 | 1.000000 | 0 | 1.000000 | 0 | 1.000000 | 0 |
0.2 | 0.980580 | 0.980103 | 0.103 × 10−3 | 0.980570 | 0.165 × 10−4 | 0.980580 | 0.659 × 10−5 |
0.4 | 0.928476 | 0.928687 | 0.227 × 10−3 | 0.928466 | 0.151 × 10−4 | 0.928476 | 0.765 × 10−5 |
0.6 | 0.857492 | 0.858961 | 0.100 × 10−3 | 0.857482 | 0.166 × 10−4 | 0.857492 | 0.963 × 10−5 |
0.8 | 0.780868 | 0.781652 | 0.692 × 10−3 | 0.780898 | 0.167 × 10−4 | 0.780868 | 0.865 × 10−5 |
1.0 | 0.707106 | 0.706123 | 0.262 × 10−3 | 0.707146 | 0.608 × 10−4 | 0.707106 | 0.653 × 10−6 |
© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
ÖZTÜRK, Y. Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials. Mathematics 2018, 6, 181. https://doi.org/10.3390/math6100181
ÖZTÜRK Y. Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials. Mathematics. 2018; 6(10):181. https://doi.org/10.3390/math6100181
Chicago/Turabian StyleÖZTÜRK, Yalçın. 2018. "Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials" Mathematics 6, no. 10: 181. https://doi.org/10.3390/math6100181
APA StyleÖZTÜRK, Y. (2018). Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials. Mathematics, 6(10), 181. https://doi.org/10.3390/math6100181