Some properties of band matrix and its application to the numerical solution one-dimensional Bratu's problem

محورهای موضوعی : Linear and multilinear algebra; matrix theory

R. Jalilian ¹ , Y. Jalilian ² , H. Jalilian ³

1 - Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
2 - Department of Mathematics, Razi University Tagh Bostan, Kermanshah P.O. Box 6714967346 Iran
3 - School of Mathematics, Iran University of Science and Technology Narmak, Tehran 16844, Iran

تاریخ دریافت : 1391/10/12 تاریخ پذیرش : 1391/10/12 تاریخ انتشار : 1392/06/10

کلید واژه: Two-point boundary value problem, Non-polynomial spline, Convergence analysis, Bratu's problem,

چکیده مقاله :

A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary formulas are developed. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods.

منابع و مأخذ:

[1] G. Akram, and S. S. Siddiqi, End conditions for interpolatory septic splines, International Journal of Computer Mathematics, Vol. 82, No. 12 (2005), pp. 1525-1540.

[2] G. Akram, and S. S. Siddiqi, Solution of sixth order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 181( 2006), pp. 708-720.

[3] Y.A.S. Aregbesola, Numerical solution of Bratu problem using the method of weighted residual, Electron. J. South. Afr. Math. Sci. 3(1)(2003), pp. 1-7.

[4] J. P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 142(2003), pp. 189-200.

[5] A. Boutayeb, and E. H. Twizell, Numerical methods for the solution of special sixth-order boundary-value problems, Intern. J. Computer Math. 45(1992), pp. 207-223.

[6] A. Boutayeb, and E. H. Twizell, Finite-dierence methods for the solution of special eighth-order boundary-value problems, International Journal of Computer Mathematics, Volume 48(1993 ), pp. 63-75.

[7] R., Buckmire, Application of a Mickens nite-dierence scheme to the cylindrical Bratu-Gelfand problem, Numer. Methods Partial Dieren. Eqns 20(3)(2004), pp. 327-337.

[8] H. Caglar, N. Caglar, and M.  Ozer, Antonios Valaristos and Antonios N. Anagnostopoulos, B-spline method for solving Bratus problem, International Journal of Computer Mathematics, 87(2010), pp. 1885-1891.

[9] E. Deeba, S. A. Khuri, and S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys. 159(2000), pp. 125-138.

[10] D. A. Frank-Kamenetski, Diusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, NJ, 1955.

[11] P. Henrici, Discrete Variable Methods in Ordinary Dierential Equations, Wiley, New York, 1961.

[12] I.H.A.H. Hassan, and V. S. Erturk, Applying dierential transformation method to the one-dimensional planar Bratu problem, Int. J. Contemp. Math. Sci. 2(2007), pp. 1493-1504.

[13] J.H. He, Variational Approach to the Bratu's problem, Journal of Physics: Conference Series 96(2008), pp. 012-087.

[14] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20(10)(2006), pp. 1141-1199.

[15] R. Jalilian, Non-polynomial spline method for solving Bratus problem, Computer Physics Communications, 181(2010),pp. 1868-1872.

[16] R. Jalilian, and J. Rashidinia, Convergence analysis of nonic-spline solutions for special nonlinear sixth-order boundary value problems, Commun Nonlinear Sci Numer Simulat, 15(2010), pp. 3805-3813.

[17] J. Jacobsen, and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differen. Eqns. 184 (2002), pp. 283-298.

[18] S. A. Khuri, A new approach to Bratus problem, Appl. Math. Comput. 147(2004), pp. 131-136.

[19] S. Li, and S. J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem, Appl. Math. Comput. 169(2005), pp. 854-865.

[20] S. Liao, and Y. Tan, A general approach to obtain series solutions of nonlinear dierential equations, Stud. Appl. Math. 119(2007), pp. 297-354.

[21] J. S. McGough, Numerical continuation and the Gelfand problem, Appl. Math. Comput. 89(1998), pp. 225-239.

[22] A. S. Mounim, and B. M. de Dormale, From the tting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem, Numer. Methods Partial Dieren.Volume 22, Issue 4(2006), pp. 761-775.

[23] A. Mohsen, L.F. Sedeek, and S.A. Mohamed, New smoother to enhance multigrid-based methods for Bratu problem, Applied Mathematics and Computation 204(2008), pp. 325-339.

[24] j. Rashidinia, and R. Jalilian, Non-polynomial spline for solution of boundary-value problems in plate defection theory, International Journal of Computer Mathematics, 84(2007), pp. 1483-1494.

[25] J. Rashidinia, R. Jalilian, and R. Mohammadi, Non-polynomial spline methods for the solution of a system of obstacle problems, Appl. Math. Comput. 188(2007), pp. 1984-1990.

[26] M. A. Ramadan, I. F. Lashien, and W. K. Zahra, A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems, International Journal of Computer Mathematics Vol. 85, No. 5(2008) 759-770.

[27] M. Ramadan, I. Lashien, and W. Zahra, Quintic non-polynomial spline solutions for fourth order boundary value problem, Commun Nonlinear Sci Numer Simulat, 14(2009), pp. 1105-1114.

[28] S. S. Siddiqi, and G. Akram, Septic spline solutions of sixth-order boundary value problems, Journal of Computational and Applied Mathematics 215(2008), pp. 288-301.

[29] M. I. Syam, and A. Hamdan, An ecient method for solving Bratu equations, Appl. Math. Comput. 176(2006), pp. 704-713.

[30] E. H. Twizell, and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems with applications to Bnard layer eigenvalue problems, Proc. R. Soc. Lond. A, 431(1990), pp. 433-450.

[31] I. A. Tirmizi, and E. H. Twizell, Higher-Order Finite-Dierence Methods for Nonlinear Second-Order Two-Point Boundary-Value Problems, Applied Mathematics Letters 15(2002), pp. 897-90.

[32] R. A. Usmani, and S. A. Wasrt, Quintic spline solutions of boundary value problems, Comput. Math. with Appl. 6(1980), pp. 197-203.

[33] R. A. Usmani, and M. Sakai, A connection between quartic spline and Numerov solution of a boundary value problem, Int. J. Comput. Math. 26(1989), pp. 263-273.

[34] S. Ul Islam, I. A. Tirmizi, F. Haq, and S. K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun Nonlinear Sci Numer Simulat, 13(2008), pp. 1448-1460.

[35] M. Van Daele, G. Vanden berghe, and H. A. De Meyer, Smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines, J. Comput. Appl. Math. Vol. 51(1994), pp. 383-394.

[36] A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166(2005), pp. 652-663.