Application of Higher Order Splines for Boundary Value Problems
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32794
Application of Higher Order Splines for Boundary Value Problems

Authors: Pankaj Kumar Srivastava

Abstract:

Bringing forth a survey on recent higher order spline techniques for solving boundary value problems in ordinary differential equations. Here we have discussed the summary of the articles since 2000 till date based on higher order splines like Septic, Octic, Nonic, Tenth, Eleventh, Twelfth and Thirteenth Degree splines. Comparisons of methods with own critical comments as remarks have been included.

Keywords: Septic spline, Octic spline, Nonic spline, Tenth, Eleventh, Twelfth and Thirteenth Degree spline, parametric and non-parametric splines, thermal instability, astrophysics.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1107165

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References:


[1] M. Kumar and P. K. Srivastava, Computational Techniques for Solving Differential Equations by Quadratic, Quartic and Octic Spline, Advances in Engineering Software, 39 (2008) 646-653.
[2] M. Kumar and P. K. Srivastava, Computational Techniques for Solving Differential Equations by Cubic, Quintic and Sextic Spline, International Journal for Computational Methods in Engineering Science & Mechanics, 10(1) (2009) 108-115.
[3] P. K. Srivastava and M. Kumar, Numerical Treatment of Nonlinear Third Order Boundary Value Problem, Applied Mathematics, 2 (2011) 959-964.
[4] P. K. Srivastava, M. Kumar and R. N. Mohapatra, Quintic Nonpolynomial Spline Method for the Solution of a Special Second- Order Boundary-value Problem with engineering application, Computers & Mathematics with Applications, 62 (4) (2011) 1707-1714.
[5] G. Akram, S. S. Siddiqi, Solution of sixth order boundary value problems using non-polynomial spline technique, Applied Mathematics and Computation 181 (2006) 708–720.
[6] M. El-Gamel, J.R. Cannon, J. Latour, A.I. Zayed, Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. Comput. 73 (247) (2003) 1325–1343.
[7] S. S. Siddiqi, E.H. Twizell, Spline solutions of linear sixth-order boundary value problems, Int. J. Comput. Math. 60 (1996) 295–304.
[8] Siraj-ul-Islam, I. A. Tirmizi, Fazal-i-Haq, M. Azam Khan, Nonpolynomial splines approach to the solution of sixth-order boundaryvalue problems, Applied Mathematics and Computation 195 (2008) 270–284.
[9] M. A. Ramadan, I. F. Lashien, W. K. Zahara, 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, 85 (2008) 759-770.
[10] M. A. Ramadan, I. F. Lashien, W. K. Zahara, Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems, Applied Mathematics and computation, 184 (2007) 476-484.
[11] M. A. Ramadan, I. F. Lashien, W. K. Zahara, Quintic nonpolynomial spline solutions for fourth order two-point boundary value problems Communications in Nonlinear Science and Numerical Simulation, 14(4) (2007) 1105-1114.
[12] J. Toomre, J.R. Zahn, J. Latour, E. A. Spiegel, Stellar convection theory II: single-mode study of the second convection zone in A-type stars, Astrophysics J. 207 (1976) 545–563.
[13] G. A. Glatzmaier, Numerical simulations of stellar convective dynamics III: at the base of the convection zone, Geophysics Astrophysics Fluid Dynamics 31 (1985) 137–150.
[14] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford 1961 (Reprinted: Dover Books, New York, 1981).
[15] R. P. Agarwal, Boundary-Value Problems for Higher-Order Differential Equations, World Scientific, Singapore, 1986.
[16] P. Baldwin, A localized instability in a Bernard layer, Appl. Anal. 24 (1987) 117–156.
[17] P. Baldwin, Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global-phase integral methods, Phil. Trans. R. Soc. Lond. A 322 (1987) 281–305.
[18] M. M. Chawala, C. P Katti, Finite difference methods for two-point boundary-value problems involving higher-order differential equations, BIT 19 (1979) 27–33.
[19] E. H. Twizell, A second order convergent method for sixth-order boundary-value problems, in: R.P. Agarwal, Y.M. Chow, S.J.Wilson (Eds.), Numerical Mathematics, Birkhhauser Verlag, Basel, 1988, pp. 495–506 (Singapore).
[20] E. H. Twizell, Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Benard layer eigenvalue problems, Proc. R. Soc. Lond. A. 431 (1990) 433–450.
[21] A. M. Wazwaz, The numerical solution of sixth-order boundary-value problems by modified decomposition method, Appl. Math. Comput. 118 (2001) 311–325.
[22] Shahid S. Siddiqui, E. H. Twizell, Spline solutions of linear eighth-order boundary-value problems, Comput. Methods Appl. Mech. Engg. 131 (1996) 309-325.
[23] Karwan H., F. Jwamer and Aryan Ali. M., Second Order Initial Value Problem and its Eight Degree Spline Solution, World Applied Sciences Journal 17 (12) (2012) 1694-1712.
[24] Jwamer, K. H.-F., Approximation Solution of Second Order Initial Value Problem by Spline Function of Degree Seven. International Journal of Contemporary Mathematical. Sciences, Bulgaria, 5 (46) (2010) 2293 – 2309.
[25] Ghazala Akram, Shahid S. Siddiqi, Nonic spline solutions of eighth order boundary value problems, Applied Mathematics and Computation 182 (2006) 829–845.
[26] M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth-order boundary-value problems, Int. J. Comput. Math. 81 (6) (2004) 685–692.
[27] A. Boutayeb, E. H. Twizell, Finite-difference methods for the solution of eighth-order boundary-value problems, Int. J. Comput. Math. 48 (1993) 63–75.
[28] E. H. Twizell, A. Boutayeb, K. Djidjeli, Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability, Adv. Comput. Math. 2 (1994) 407–436.
[29] S. S. Siddiqi and G. Akram, End Conditions for Interpolatory Nonic Splines, Southeast Asian Bulletin of Mathematics 34 (2010) 469-488.
[30] S. S. Siddiqi and E.H. Twizell, Spline solutions of linear tenth-order boundary-value problems, Intern. J. Computer Math. 68 (1998) 345-362.
[31] S. S. Siddiqi, G. Akram, Solutions of tenth-order boundary value problems using eleventh degree spline, Applied Mathematics and Computation 185 (2007) 115–127.
[32] S. S. Siddiqi, G. Akram, Solution of 10th-order boundary value problems using non-polynomial spline technique, Applied Mathematics and Computation 190 (2007) 641–651.
[33] G. Akram, S. S. Siddiqi, Solution of eighth order boundary value problems using non polynomial spline technique, International Journal of Computer Mathematics, 84(3) ( 2007) 347-368.
[34] J. Rashidinia, R. Jalilian, K. Farajeyan, Non polynomial spline solutions for special linear tenth-order boundary value problems, World Journal of Modelling and Simulation 7(1) (2011) 40-51.
[35] S. S. Siddiqi and E. H. Twizell, Spline solutions of linear twelfth-order boundary-value problems, Journal of Computational and Applied Mathematics78 (1997) 371- 390.
[36] S. S. Siddiqi, Ghazala Akram, Solutions of twelfth order boundary value problems using thirteen degree spline, Applied Mathematics and Computation 182 (2006) 1443–1453.
[37] S. S. Siddiqi, G. Akram, Solutions of 12th order boundary value problems using non-polynomial spline technique, Applied Mathematics and Computation 199 (2008) 559–571.
[38] G. Akram, S. S. Siddiqi, Solution of eighth order boundary value problems using non polynomial spline technique, Int. J. Comput. Math. 84 (3) (2007) 347–368.
[39] P. K. Srivastava, M. Kumar, and R. N. Mohapatra: Solution of Fourth Order Boundary Value Problems by Numerical Algorithms Based on Nonpolynomial Quintic Splines, Journal of Numerical Mathematics and Stochastics, 4 (1) 13-25, 2012.
[40] P. K. Srivastava and M. Kumar, Numerical Algorithm Based on Quintic Nonpolynomial Spline for Solving Third-Order Boundary value Problems Associated with Draining and Coating Flow, Chinese Annals of Mathematics, Series B, 33B(6), 2012, 831–840
[41] P. Singh, P. K. Srivastava, R. K. Patne, S. D. Joshi, and K. Saha, Nonpolynomial Spline Based Empirical Mode Decomposition in the proceedings of “International Conference on Signal Processing and Communications” (ICSC-2013), 480-577, 978-1-4799-1607-8/13.
[42] P. K. Srivastava, Study of Differential Equations With Their Polynomial And Nonpolynomial Spline Based Approximation, published in “Acta Tehnica Corviniensis – Bulletin of Engineering” 7(3) (2014) 139-150.
[43] M. R. Scott, H. A. Watts, Computational solution of linear two-point BVP via orthonormailzation, SIAM J. Numer. Anal. 14 (1977) 40–70.
[44] M.R. Scott, H.A. Watts, A systematized collection of codes for solving two-point BVPs, Numerical Methods for Differential Systems, Academic Press, 1976.
[45] L. Watson, M.R. Scott, Solving spline-collocation approximations to nonlinear two-point boundary value problems by a homotopy method, Appl. Math. Comput. 24 (1987) 333–357.