Bernstein polynomials method for numerical solutions of integro-differential form of the singular Emden-Fowler initial value problems

Volume 17, Issue 1, pp 66-75 http://dx.doi.org/10.22436/jmcs.017.01.06

Publication Date: March 15, 2017 Submission Date: February 03, 2016

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Authors

Abdelkrim Bencheikh - Department of Mathematics, University Kasdi-Merbah, Ouargla, 30000, Algeria. Lakhdar Chiter - Department of Mathematics, Ferhat-Abbas University, Setif, 19000, Algeria. Hocine Abbassi - Department of Mathematics, University Kasdi-Merbah, Ouargla, 30000, Algeria.

Abstract

In this paper, Bernstein polynomial method applied to the solutions of generalized Emden-Fowler equations as singular initial value problems is presented. Firstly, the singular differential equations are converted to Volterra integro-differential equations and then solved by the Bernstein polynomials method. The properties of Bernstein polynomials via Gauss-Legendre rule are used to reduce the integral equations to a system of algebraic equations which can be solved numerically. Some illustrative examples are discussed to demonstrate the validity and applicability of the present method.

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Keywords

• Bernstein polynomials
• Volterra integro-differential equations
• Emden-Fowler equation
• Gaussian integrations.

MSC

•  05C12
•  05C15
•  05C76

References

• [1] M. I. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein polynomial basis, J. Comput. Appl. Math., 205 (2007), 272–280.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York (1967)
  • MathSciNet


• [3] M. S. H. Chowdhury, I. Hashim, Solutions of Emden-Fowler equations by homotopy-perturbation method, Nonlinear Anal. Real World Appl., 10 (2009), 104–115.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] A. Constantinides, Applied numerical methods with personal computers, McGraw-Hill, New York (1987)
  • Google Scholar


• [5] H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York (1962)
  • MathSciNet


• [6] E. H. Doha, A. H. Bhrawy, M. A. Saker, Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations, Appl. Math. Lett., 24 (2011), 559–565.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] E. H. Doha, A. H. Bhrawy, M. A. Saker, On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equations, Bound. Value Probl., 2011 (2011 ), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] S. Karimi Vanani, A. Aminataei, On the numerical solution of differential equations of Lane-Emden type, Comput. Math. Appl., 59 (2010), 2815–2820.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] A. Kazemi Nasab, A. Kılıçman, Z. P. Atabakan, W. J. Leong, A numerical approach for solving singular nonlinear LaneEmden type equations arising in astrophysics , New Astron., 34 (2015), 178–186.
  • View Article
  • Google Scholar


• [10] T. Luo, Z.-P. Xin, H.-H. Zeng, On nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem, Adv. Math., 291 (2016), 90–182.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] K. Maleknejad, B. Basirat, E. Hashemizadeh, A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations, Math. Comput. Modelling, 55 (2012), 1363–1372.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S. Mall, S. Chakraverty, Chebyshev neural network based model for solving Lane-Emden type equations, Appl. Math. Comput., 247 (2014), 100–114.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] B. N. Mandal, S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190 (2007), 1707–1716.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] A. D. Polyanin, V. F. Zaitsev, Handbook of exact solutions for ordinary differential equations, CRC Press, Boca Raton, FL, (1995)
  • MathSciNet


• [15] O. W. Richardson, The emission of electricity from hot bodies, Longmans, Green and Company (1921)


• [16] T. J. Rivlin, An introduction to the approximation of functions, Waltham, Mass.-TorontoBlaisdell Publishing Co. Ginn and Co., , Ont.-London (1969)
  • MathSciNet


• [17] X.-F. Shang, P. Wu, X.-P. Shao, An efficient method for solving Emden-Fowler equations, J. Franklin Inst., 346 (2009), 889–897.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] Z. Šmarda, Y. Khan, An efficient computational approach to solving singular initial value problems for Lane-Emden type equations , J. Comput. Appl. Math., 290 (2015), 65–73.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] A. M.Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Appl. Math. Comput., 128 (2002), 45–57.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden-Fowler equation, Appl. Math. Comput., 161 (2005), 543–560.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] A. M. Wazwaz, Analytical solution for the time-dependent Emden-Fowler type of equations by Adomian decomposition method, Appl. Math. Comput., 166 (2005), 638–651.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. A. Yousefi, Legendre scaling function for solving generalized Emden-Fowler equations, Int. J. Inf. Syst. Sci., 3 (2007), 243–250.
  • MathSciNet
  • Google Scholar
  • MATH


• [23] S. A. Yousefi, M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Internat. J. Systems Sci., 41 (2010), 709–716.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH