**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30743

##### Application of Higher Order Splines for Boundary Value Problems

**Authors:**
Pankaj Kumar Srivastava

**Abstract:**

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

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

**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.