International Journal of Applied Science and Engineering
Published by Chaoyang University of Technology

K. N. S. Kasi Viswanadham* and Y. Showri Raju

Department of Mathematics National Institute of Technology, Warangal, India


 

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ABSTRACT


A finite element method involving collocation method with sextic B-splines as basis functions has been developed to solve eighth order boundary value problems. The sixth order, seventh order and eighth order derivatives for the dependent variable are approximated by the central differences of fifth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.


Keywords: Collocation method; sextic B-spline; basis function; eighth order boundary value problem; absolute error.


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REFERENCES


[1] Davies, A. R., Karageorghis, A., and Phillips, T. N. 1988. Spectral galerkin methods for the primary two point boundary-value problem in modelling viscoelastic flow. International Journal for Numerical Methods in Engineering, 26: 647-662.

[2] Karageorghis, A, Phillips, T. N., and Davies, A. R. 1988. Spectral collocation methods for the primary two point boundary-value problem in modelling viscoelastic flows. International Journal of Numerical Methods in Engineering, 26 : 805-813.

[3] Chandrasekhar, S. 1961. “Hydrodynamic and Hydromagnetic Stability’’. Clarendon Press, Oxford (Reprinted: Dover Books, New York, 1981).

[4] Bishop, R. E. D., Cannon, S. M., and Miao, S. 1989. On coupled bending and torsional vibration of uniform beams. Journal of Sound and Vibration, 131: 457-464.

[5] Agarwal, R. P. 1986. “Boundary Value Problems for Higher Order Differential Equations”. World Scientific, Singapore.

[6] Boutayeb, A. and Twizell, E. H. 1993. Finite-difference methods for the solution of eighth-order boundary-value problems. Applied Mathematics Letters, 48 : 63-75.

[7] Twizell, E. H., Boutayeb, A., and Djidjeli, K. 1994. Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability. Advances in Computational Mathematics, 2: 407-436.

[8] Twizell, E. H. and Siddiqi, S. S. 1996. Spline solution of linear eighth-order boundary value problems. Computer Methods in Applied Mechanics and Engineering, 131: 309-325.

[9] Inc, M. and Evans, D. J. 2004. An efficient approach to approximate solutions of eighth order boundary-value problems. International Journal of Computer Mathematics, 81: 685-692.

[10] Shahid, Siddiqi, S. and Ghazala Akram. 2006. Nonic spline solutions of eighth order boundary value problems. Journal of Computational and Applied Mathematics, 182: 829-845.

[11] Shahid, Siddiqi, S. and Ghazala Akram. 2007. Solutions of eighth order boundary value problems using non-polynomial spline technique. International Journal of Computer Mathematics, 84, 3 : 347-368.

[12] Scott, M. R. and Watts, H. A. 1977. Computional solution of linear two points bvp via orthonormalization. SIAM Journal on Numerical Analysis, 14 : 40-70.

[13] Scott, M. R. and Watts, H. A. 1976. “A Systematized Collection of Codes for Solving Two-Point Bvps, Numerical Methods for Differential Systems”, Academic Press.

[14] Layne, T., Watson, R., Melvin, R., and Scott. 1987. Solving spline-collocation approximations to nonlinear two-point boundary value problems by a homotopy method. Applied Mathematics and Computation, 24 : 333-357.

[15] Liu, G. R. and Wu, T. Y. 1973. Differential quadrature solutions of eighth order boundary value differential equations. Journal of Computational and Applied Mathematics, 145: 223-235.

[16] He, J. H. 1998. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 167, 1-2 : 57-68.

[17] He, J. H. 1999. Variational iteration method a kind of nonlinear analytical technique: some examples. International Journal of Non-Linear Mechanics, 34 : 699-708.

[18] He, J. H. 2000. Variational method for autonomous ordinary differential equations. Applied Mathematics and Computation, 114 : 115-123.

[19] He, J. H. 2001. Variational theory for linear magneto-elasticity. International Journal of Nonlinear Sciences and Numerical Simulation, 2 : 309-316.

[20] He, J. H. 2004. Variational principle for some nonlinear partial differential equations with variables coefficients. Chaos, Solitons & Fractals, 19: 847-851.

[21] Wazwaz, A. M. 2000. Approximate solutions to boundary value problems of higher order by the modified decomposition method. Computers & Mathematics with Applications, 40:  679-691.

[22] Wazwaz, A. M. 2000. The numerical solution of special eighth-order boundary value problems by the modified decomposition method. Neural, Parallel, and Scientific Computations, 8, 2 : 133-146.

[23] Bellman, R. E. and Kalaba, R. E. 1965. “Qusilinearization and Nonlinear Boundary Value Problems”. American Elsevier, New York.

[24] Reddy, J. N. 2005. “An Introduction to the Finite Element Method”. Tata Mc-GrawHill Publishing Company Limited, 3rd Edition, New Delhi.

[25] Prenter, P. M. 1989. “Splines and Variational Methods”. John-Wiley and Sons, New York.

[26] Carl de Boor. 1989. “A Practical Guide to Splines”. Springer-Verlag, Berlin.

[27] Schoenberg, I. J. 1966. “On Spline Functions”. MRC Report 625, University of Wisconsin.


ARTICLE INFORMATION


Received: 2012-12-27
Revised: 2013-08-06
Accepted: 2013-11-19
Publication Date: 2014-03-01


Cite this article:

Viswanadham, K.N.S.K., Raju, Y.S. 2014. Sextic B-Spline collocation method for eighth order boundary value problems. International Journal of Applied Science and Engineering, 12, 43–57. https://doi.org/10.6703/IJASE.2014.12(1).43

 


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