Numerical solution of system of Emden-Fowler type equations by Bernstein collocation method
Tóm tắt
The system of Emden-Fowler equations occurs frequently in physical and natural sciences. These system of equations are very difficult to handle due to their singularity and the nonlinearity. We provide a numerically robust technique for approximating the solution of the system of Emden-Fowler equations. This algorithm is based on the Bernstein polynomial accompanied by the collocation method. First, we transform the system of Emden-Fowler equations into their integral form. Then we implement the Bernstein polynomial approximation and the collocation technique to acquire a nonlinear system of equations. We apply the Newton-Raphson method to analyze the resulting nonlinear system of equations. We also discuss the error analysis of this method. We provide the numerical results of the
$$L_{\infty }$$
error, the
$$L_{2}$$
-norm error, and the residual error of some numerical supporting problems to examine the accuracy of the current technique. The advantage of the present method is exhibited by comparing the numerical results obtained by the proposed technique and other known techniques.
Tài liệu tham khảo
S. Chandrasekhar, An introduction to the study of stellar structure. Ciel et Terre 55, 412 (1939)
D. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. J. Theoret. Biol. 71(2), 255–263 (1978)
R.A. Van Gorder, Exact first integrals for a Lane-Emden equation of the second kind modeling a thermal explosion in a rectangular slab. New Astron. 16(8), 492–497 (2011)
K. Reger, R. Van Gorder, Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere. Appl. Math. Mech. 34(12), 1439–1452 (2013)
M. Chawla, C. Katti, Finite difference methods and their convergence for a class of singular two point boundary value problems. Numerische Mathematik 39(3), 341–350 (1982)
R. Pandey, On the convergence of a finite difference method for a class of singular two point boundary value problems. Int. J. Comput. Math. 42, 237–241 (1992)
S. Iyengar, P. Jain, Spline finite difference methods for singular two point boundary value problems. Numerische Mathematik 50(3), 363–376 (1986)
M. Kumar, A three-point finite difference method for a class of singular two-point boundary value problems. J. Comput. Appl. Math. 145(1), 89–97 (2002)
J. Rashidinia, Z. Mahmoodi, M. Ghasemi, Parametric spline method for a class of singular two-point boundary value problems. Appl. Math. Comput. 188(1), 58–63 (2007)
A.R. Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189(2), 2017–2022 (2007)
A. Taghavi, S. Pearce, A solution to the Lane-Emden equation in the theory of stellar structure utilizing the Tau method. Math. Methods Appl. Sci. 36(10), 1240–1247 (2013)
M. Lakestani, M. Dehghan, Four techniques based on the B-spline expansion and the collocation approach for the numerical solution of the Lane-Emden equation. Math. Methods Appl. Sci. 36(16), 2243–2253 (2013)
R. Singh, J. Kumar, G. Nelakanti, Numerical solution of singular boundary value problems using Green’s function and improved decomposition method. J. Appl. Math. Comput. 43(1–2), 409–425 (2013)
R. Singh, J. Kumar, An efficient numerical technique for the solution of nonlinear singular boundary value problems. Comput. Phys. Commun. 185(4), 1282–1289 (2014)
R. Singh, J. Kumar, G. Nelakanti, Approximate series solution of singular boundary value problems with derivative dependence using Green’s function technique. Comput. Appl. Math. 33(2), 451–467 (2014)
R. Singh, J. Kumar, The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems. J. Appl. Math. Comput. 44(1–2), 397–416 (2014)
R. Mohammadzadeh, M. Lakestani, M. Dehghan, Collocation method for the numerical solutions of Lane-Emden type equations using cubic Hermite spline functions. Math. Methods Appl. Sci. 37(9), 1303–1717 (2014)
F. Zhou, X. Xu, Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets. Adv. Differ. Equ. 2016(1), 17 (2016)
M. Turkyilmazoglu, Solution of initial and boundary value problems by an effective accurate method. Int. J. Comput. Methods 14(06), 1750069 (2017)
R. Singh, N. Das, J. Kumar, The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions. Europ. Phys. J.Plus 132(6), 251 (2017)
A.K. Verma, S. Kayenat, On the convergence of Mickens’ type nonstandard finite difference schemes on Lane-Emden type equations. J. Math. Chem. 56(6), 1667–1706 (2018)
R. Singh, Optimal homotopy analysis method for the non-isothermal reaction-diffusion model equations in a spherical catalyst. J. Math. Chem. 56(9), 2579–2590 (2018)
R. Singh, Analytic solution of singular Emden-Fowler-type equations by Green’s function and homotopy analysis method. Europ. Phys. J. Plus 134(11), 583 (2019)
R. Singh, A modified homotopy perturbation method for nonlinear singular Lane-Emden equations arising in various physical models. Int. J. Appl. Comput. Math. 5(3), 64 (2019)
R. Singh, H. Garg, V. Guleria, Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. J. Comput. Appl. Math. 346, 150–161 (2019)
R. Singh, J. Shahni, H. Garg, A. Garg, Haar wavelet collocation approach for Lane-Emden equations arising in mathematical physics and astrophysics. Europ. Phys. J. Plus 134(11), 548 (2019)
A.K. Verma, D. Tiwari, Higher resolution methods based on quasilinearization and Haar wavelets on Lane-Emden equations, International Journal of Wavelets. Multiresolution Inf. Process. 17(03), 1950005 (2019)
R. Singh, V. Guleria, M. Singh, Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations. Math. Comput. Simul. 174, 123–133 (2020)
M. Chapwanya, R. Dozva, G. Muchatibaya, A nonstandard finite difference technique for singular Lane-Emden type equations. Eng. Comput. 36(5), 1566–1578 (2019)
M. Umesh, Kumar, Numerical solution of singular boundary value problems using advanced Adomian decomposition method. Engineering with Computers 1–11 (2020)
A.M. Wazwaz, R. Rach, J.S. Duan, A study on the systems of the Volterra integral forms of the Lane-Emden equations by the adomian decomposition method. Math. Methods Appl. Sci. 37(1), 10–19 (2014)
L.J. Xie, C.L. Zhou, S. Xu, Solving the systems of equations of Lane-Emden type by differential transform method coupled with adomian polynomials. Mathematics 7(4), 377 (2019)
R. Singh, Analytical approach for computation of exact and analytic approximate solutions to the system of Lane-Emden-Fowler type equations arising in astrophysics. Europ. Phys. J. Plus 133(8), 320 (2018)
Y. Öztürk, An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering. Nonlinear Eng. 8(1), 429–437 (2019)
J.H. He, F.Y. Ji, Taylor series solution for Lane-Emden equation. J. Math. Chem. 57(8), 1932–1934 (2019)
J. Shahni, R. Singh, An efficient numerical technique for Lane-Emden-Fowler boundary value problems: Bernstein collocation method. Europ. Phys. J. Plus 135(06), 1–21 (2020)
D. Flockerzi, K. Sundmacher, On coupled Lane-Emden equations arising in dusty fluid models. J. Phys. 268(1), 012006 (2011)
S. Muthukumar, M. Veeramuni, R. Lakshmanan, Analytical expression of concentration of substrate and oxygen in excess sludge production using Adomian decomposition method. Indian J. Appl. Res. 4, 387–391 (2014)
R. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52(1), 255–267 (2014)
J.S. Duan, R. Rach, A.M. Wazwaz, Oxygen and carbon substrate concentrations in microbial floc particles by the Adomian decomposition method. MATCH Commun. Math. Comput. Chem. 73, 785–796 (2015)
A.M. Wazwaz, R. Rach, J.S. Duan, Variational iteration method for solving oxygen and carbon substrate concentrations in microbial floc particles. MATCH Commun. Math. Comput. Chem. 76, 511–523 (2016)
R. Ma, Multiple nonnegative solutions of second-order systems of boundary value problems. Nonlinear Anal.: Theory, Methods Appl. 42(6), 1003–1010 (2000)
J.S. Duan, R. Rach, A.M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method. J. Math. Chem. 53(4), 1054–1067 (2015)
M. Dehghan, A. Saadatmandi, The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method. Math. Comput. Model. 46(11–12), 1434–1441 (2007)
M. Dehghan, M. Lakestani, Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions. Int. J. Comput. Math. 85(9), 1455–1461 (2008)
F. Geng, M. Cui, Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems. J. Comput. Appl. Math. 235(8), 2405–2411 (2011)
R. Singha, A.M. Wazwazb, An efficient algorithm for solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions: The homotopy analysis method. MATCH Commun. Math. Comput. Chem. 81(3), 785–800 (2019)
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(9), 2691–2706 (2018)
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(2), 570–582 (2019)
R. Singh, Solving coupled Lane-Emden equations by Green’s function and decomposition technique. Int. J. Appl. Comput. Math. 6, 80 (2020)
A.K. Verma, N. Kumar, D. Tiwari, Haar wavelets collocation method for a system of nonlinear singular differential equations. Eng. Comput. 37(9), 1–40 (2020)
D.D. Bhatta, M.I. Bhatti, Numerical solution of KdV equation using modified Bernstein polynomials. Appl. Math. Comput. 174(2), 1255–1268 (2006)
M.I. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205(1), 272–280 (2007)
B.N. Mandal, S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput. 190(2), 1707–1716 (2007)
K. Maleknejad, E. Hashemizadeh, R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Commun. Nonlinear Sci. Numeric. Simul. 16(2), 647–655 (2011)
S.S. Ray, S. Singh, Numerical solution of nonlinear stochastic Itô-Volterra integral equation driven by fractional Brownian motion, Engineering Computations
Ş Yüzbaşı, A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics. Math. Methods Appl. Sci. 34(18), 2218–2230 (2011)
O.R. Isik, M. Sezer, Bernstein series solution of a class of Lane-Emden type equations. Math. Probl. Eng. 2013, 423797 (2013)
S.N. Bernstein, Démo istration du th'eorème de Weierstrass fondée sur le calcul des probabilités, Communications de la Soci'et'e math'ematique de Kharkow 13(1), 1–2 (1912)
M.J.D. Powell, Approximation Theory and Methods (Cambridge University Press, Cambridge, 1981).
G. Lorentz, R. DeVore, Constructive Approximation, Polynomials and Splines Approximation (Springer, Berlin, 1993).