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

K. N. S. Kasi Viswanadham* and Sreenivasulu Ballem

Department of Mathematics, National Institute of Technology, Warangal, India


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ABSTRACT


In this paper, we present a finite element method involving Galerkin method with septic B-splines as basis functions to solve a general tenth order boundary value problem. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the given set of boundary conditions are prescribed. The proposed method was applied to solve several examples of tenth order linear and nonlinear boundary value problems. The solution of a nonlinear boundary value problem has been obtained as the limit of a sequence of solution of linear boundary value problems generated by quasilinearization technique. The obtained numerical results are compared with the exact solutions available in the literature.


Keywords: Absolute error; basis function; Galerkin method; septic B-spline; tenth order boundary value problem.


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REFERENCES


[1] Chandrasekhar, S. 1981. “Hydrodynamics and Hydromagnetic Stability”. New York: Dover.

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

[3] Twizell, E. H. and Boutayeb, A. 1994. Numerical methods for eighth, tenth and twelfth order eigenvalue problems arising in thermal instability. Advances in Computational Mathematics, 2, 4: 407- 436.

[4] Siddiqi, S. S. and Twizell, E. H. 1998. Spline solution of linear tenth order boundary value problems. International Journal of Computer Mathematics, 68, 3-4: 345-362.

[5] Siddiqi, S. S. and Akram, G. 2007. Solutions of tenth order boundary value problems using the non-polynomial spline technique. Applied Mathematics and Computation, 185, 1: 115- 127.

[6] Akram, G. and Siddiqi, S. S. 2006. Solutions of tenth order boundary value problems using the eleventh degree spline. Applied Mathematics and Computation, 182, 1: 829- 845.

[7] Scott, M. R. and Watts, H. A. 1977. Computational solution of linear two points boundary value problems via orthonormalization. SIAM Journal on Numerical Analysis, 14, 1: 40-70.

[8] Watson, L. M. and Scott, M. R. 1987. Solving spline Collocation approximations to nonlinear two point boundary value problems by a homotopy method. Applied Mathematics and Computation, 24, 4: 333-357.

[9] Rashidinia, J., Jalilian, R. and Farajeyan, K. 2011. Non polynomial spline solutions for special linear tenth order boundary value problems. World Journal of Modelling and Simulation, 7, 1: 40-51.

[10] Dijidejeli, K. and Twizell, E. H. 1993. Numerical methods for special nonlinear boundary value problems of order 2m.  Journal of Computational and Applied Mathematics, 47, 1: 35- 45.

[11] Lamnii, A., Mraoui, H., Sbibih D., Tijini A. and Zidna A. 2008. Spline solution of some linear boundary value problems. Applied Mathematics E-Notes, 8: 171-178.

[12] Ramadan, M. A., Lashien, I. F. and Zahra, W. K. 2008. High order accuracy nonpolynomial spline solutions for 2μ th order two point boundary value problems. Applied Mathematics and Computation, 204, 2: 920- 927.

[13] Erturk, V. S. and Momani, S. 2007. A reliable algorithm for solving tenth order boundary value problems. Numerical Algorithms, 44, 2: 147-158.

[14] Wazwaz, A.-M. 2000. The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth order and twelfth order. International Journal of Nonlinear Sciences and Numerical Simulation, 1, 1: 17-24.

[15] Farajeyan, K. and Maleki, N. R. 2012. Numerical solution of tenth order boundary value problems in off step points. Journal of Basic and Applied Scientific Research, 2, 6: 6235-6244.

[16] Noor, M. A., Al-Said, E., and Mohyud-Din, S. T. 2012. A reliable algorithm for solving tenth order boundary value problems. Applied Mathematics and Information Sciences, 6, 1: 103-107.

[17] Geng, F. and Li, X. 2009. Variational iteration method for solving tenth order boundary value problems. Mathematical Sciences, 3, 2: 161- 172.

[18] Mirmoradi, H., Mazaheripour, H., Ghanbarpour, S., and Barari, A. 2009. Homotopy perturbation method for solving twelfth order boundary value problems. International Journal of Research and Reviews in Applied Sciences, 1, 2: 163- 173.

[19]  Mohyud-Din, S. T. and Yildirim, A. 2010. Solutions of Tenth and Ninth and Ninth-order Boundary Value Problems by Modified Variational Iteration Method. Applications and Applied Mathematics, 5, 1: 11-25.

[20] Siddiqi, S. S. and Akram, G., and Zaheer, S. 2009. Solutions of tenth order boundary value problems using Variational iteration technique. European Journal of Scientific Research, 30, 3: 326- 347.

[21] Viswanadham, K. N. S. K. and Raju, Y. S. 2012. Quintic B-spline Collocation method for tenth boundary value problems. International Journal of Computer Applications, 51, 15: 7-13.

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

[23] Bers, L. John, F. and Schecheter, M. 1964. “Partial Differential equations”. John Wiley Inter Science, New York.

[24] Lions, J. L. and Magenes, E. 1972. “Non-homogeneous Boundary Value Problem and Applications”. Springer-Verlag, Berlin.

[25] Mitchel, A. R. and Wait, R. 1977. “The Finite Element Method in Partial Differential Equations”. John Wiley and Sons, London.

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

[27] de Boor, C. 2001. “A Practical Guide to Splines”. Springer-Verlag.

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


ARTICLE INFORMATION


Received: 2014-10-11
Revised: 2015-06-12
Accepted: 2015-10-27
Available Online: 2015-09-01


Cite this article:

Viswanadham, K.N.S.K., Ballem, S. 2015. Numerical solution of tenth order boundary value problems by galerkin method with septic B-splines. International Journal of Applied Science and Engineering, 13, 247–260. https://doi.org/10.6703/IJASE.2015.13(3).247


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