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 JAMP  Vol.4 No.2 , February 2016
Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions
Abstract: In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.
Cite this paper: Kalyani, P. and Lemma, M. (2016) Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions. Journal of Applied Mathematics and Physics, 4, 249-261. doi: 10.4236/jamp.2016.42032.
References

[1]   Bickely, W.G. (1968) Piecewise Cubic Interpolation and Two-Point Boundary Value Problem. Computer Journal, 11, 202-208.

[2]   Schoenberg, J. (1946) Contributions to the Problems of Approximation of Equidistant Data by Analytic Functions, Quart. Applied Mathematics, 4, 45-99.

[3]   Maclaren, D.H. (1958) Formula for Fitting a Spline Curve through a Set of Points, Boeing. Applied Mathematics.

[4]   Rubin, S.G. and Khosla, P.K. (1976) Higher Order Numerical Solutions Using Cubic Splines, AIAAJ, 14, 851-858.
http://dx.doi.org/10.2514/3.61427

[5]   Sastry, S.S. (1976) Finite-Difference Approximations Toone Dimensional Parabolic Equations Using Cubic Spline Technique. Journal of Computational and Applied Mathematics, 2, 23-26.
http://dx.doi.org/10.1016/0771-050X(76)90035-8

[6]   Schoenberg, J. (1958) Spline Functions, Convex Curves and Mechanical Quadrature. Bulletin of the American Mathematical Society, 64, 352-357.
http://dx.doi.org/10.1090/S0002-9904-1958-10227-X

[7]   Ahlberg, J.H. and Nilson, E.N. (1963) Convergence Properties of the Spline Fit. SIAM Journal, 11, 95-104.
http://dx.doi.org/10.1137/0111007

[8]   Ahlberg, J.H., Nilson, E.N. and Walsh, J.L. (1967) The Theory of Splines and Their Application. Academic Press Inc., New York.

[9]   De Boor, C. (2001) A Practical Guide to Splines. Springer-Verlag, New York.

[10]   Prenter, P.M. (1975) Splines and Variational Methods. Wiley, New York.

[11]   Schumaker, L.L. (1981) Spline Functions: Basic Theory. John Wiley and Son, New York.

[12]   Shikin, E.V. and Plis, A.I. (1995) Handbook on Splines for the User. CRC Press.

[13]   Spath, H. (1995) One Dimensional Spline Interpolation Algorithm. A.K. Peters.

[14]   Chawla, M.M. (1978) A Fourth Order Tridiagonal Finite Difference Method for General Two Point Boundary Value Problems with Mixed Boundary Conditions. Journal of the Institute of Mathematics and Its Applications, 21, 83-93.
http://dx.doi.org/10.1093/imamat/21.1.83

[15]   Karageorghis, A., Phillips, T.N. and Davies, A.R. (1988) Spectral Collocation Methods for the Primary Two-Point Boundary-Value Problem in Modeling Viscoelastic Flows. International Journal for Numerical Methods in Engineering, 26, 805-813.
http://dx.doi.org/10.1002/nme.1620260404

[16]   Kasi Viswanadham, K.N.S. and Murali Krishna, P. (2010) Quintic B-Splines Galerkin Method for Fifth Order Boundary Value Problems. ARPN Journal of Engineering and Applied Sciences, 5.

[17]   Kumar, M. and Srivastava, P. K. (2009) Computational Techniques for Solving Differentia Equations by Cubic, Quintic, and Sextic Spline. Computational Methods in Engg—Mechanical Engineering, 10, 108-115.

[18]   Grandine, T.A. (2005) The Extensive Use of Splines at Boeing. SIAM News, 38.

[19]   Mihretu, N.L. and Kalyani, P. (2015) Eighth Degree Spline Solution for Seventh Order Boundary Value Problem. Journal of Multidisplinary Engineering Science and Technology, 2, 1185-1191.

 
 
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