An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems

Mohamed El-Gamel, Mona Sameeh

Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt.

DOI: 10.4236/am.2012.38137                

Abstract

In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.

Keywords

Chebychev Polynomial; Fourth-Order; Sturm-Liouville; Eigenvalue

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. Greenberg and M. Marletta, “Algorithm 775: The Code SLEUTH for Solving Fourth-Order Sturm-Liouville Problems,” ACM Transactions on Mathematical Software, Vol. 23, No. 4, 1997, pp. 453-493. doi:10.1145/279232.279231
[2]	P. Bailey, M. Gordon and L. Shampine, “Automatic Solution of the Sturm-Liouville Problem,” ACM Transactions on Mathematical Software, Vol. 4, No. 3, 1978, pp. 193-208. doi:10.1145/355791.355792
[3]	P. Bailey, W. Everitt and A. Zeal, “Computing Eigenvalues of Singular Sturm-Liouville Problems,” Results in Mathematics, Vol. 20, Birkhauser, Basel, 199l.
[4]	C. Fulton and S. Pruess, “Mathematical Software for SturmLiouville Problems,” INSF Final Report for Grants DMS88-13113 and DMS88-00839, Computational Mathematics Division, 1991.
[5]	B. Chanane, “Eigenvalues of Fourth-Order SturmLiouville Problems Using Fliess Series,” Journal of Computational and Applied Mathematics, Vol. 96, No. 2, 1998, pp. 91-97. doi:10.1016/S0377-0427(98)00086-7
[6]	B. Chanane, “Fliess Series Approach to the Computation of the Eigenvalues of Fourth-Order Sturm-Liouville Problems,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 459-463. doi:10.1016/S0893-9659(01)00159-8
[7]	M. Chawla, “A New Fourth-Order Finite Difference Method for Computing Eigenvalues of Fourth-Order Linear Boundary-Value Problems,” IMA Journal of Numerical Analysis, Vol. 3, No. 3, 1983, pp. 291-293. doi:10.1093/imanum/3.3.291
[8]	R. Usmani and M. Sakai, “Two New Finite Difference Method for Computing Eigenvalues of a Fourth-Order Two-Point Boundary-Value Problems,” International Journal of Mathematics and Mathematical Sciences, Vol. 10, No. 3, 1983, pp. 525-530. doi:10.1155/S0161171287000620
[9]	E. Twizell and S. Matar, “Numerical Methods for Computing the Eigenvalues of Linear Fourth-Order Boundary-Value Problems,” Journal of Computational and Applied Mathematics, Vol. 40, No. 1, 1992, pp. 115-125. doi:10.1016/0377-0427(92)90046-Z
[10]	B. Attili and D. Lesnic, “An Efficient Method for Computing Eigenelements of Sturm-Liouville Fourth-Order Boundary Value Problems,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 1247-1254. doi:10.1016/j.amc.2006.05.011
[11]	M. Syam and H. Siyyam, “An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems,” Chaos, Solitons & Fractals, Vol. 39, No. 2, 2009, pp. 659-665. doi:10.1016/j.chaos.2007.01.105
[12]	B. Chanane, “Accurate Solutions of Fourth Order Sturm-Liouville Problems,” Journal of Computational and Applied Mathematics, Vol. 234, No. 10, 2010, pp. 3064-3071. doi:10.1016/j.cam.2010.04.023
[13]	B. Chanane, “Sturm-Liouville Problems with Parameter Dependent Potential and Boundary Conditions,” Journal of Computational and Applied Mathematics, Vol. 212, No. 2, 2008, pp. 282-290. doi:10.1016/j.cam.2006.12.006
[14]	S. Abbasbandy and A. Shirzadi, “A New Application of the Homotopy Analysis Method: Solving the Sturm-Liouville Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 1, 2011, pp. 112-126. doi:10.1016/j.cnsns.2010.04.004
[15]	Z. Shi and Y. Cao, “Application of Haar Wavelet Method to Eigenvalue Problems of High Order Differential Equations,” Applied Mathematical Modelling, Vol. 36, No. 9, 2012, pp. 4020-4026. doi:10.1016/j.apm.2011.11.024
[16]	U. Ycel and K. Boubaker, “Differential Quadrature Method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for Efficient Computation of the Eigenvalues of Fourth-Order Sturm-Liouville Problems,” Applied Mathematical Modelling, Vol. 36, No. 1, 2012, pp. 158-167. doi:10.1016/j.apm.2011.05.030
[17]	A. Aky?z and M. Sezer, “A Chebyshev Collocation Method for the Solution Linear Integro Differential Equations,” J. Comput. Math, Vol. 72, 1999, pp. 491-507.
[18]	H. ?erdk-Yaslan and A. Aky?z-Daciolu, “Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra Integro-Differential Equations,” Journal of Science and Arts, Vol. 6, 2006, pp. 89-101.
[19]	M. Sezer and M. Kaynak, “Chebyshev Polynomial Solutions of Linear Differential Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 27, No. 4, 1996, pp. 607-611. doi:10.1080/0020739960270414
[20]	T. Rivlin, “An Introduction to the Approximation of Functions,” Dover Publications, Inc., New York, 1969.
[21]	M. Sezer and M. Kaynak, “Chebyshev Polynomial Solutions of Linear Differential Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 27, No. 4, 1996, pp. 607-618. doi:10.1080/0020739960270414