The Effect of the Singular Potential Functions for Eigenvalues of Sturm-Lıouville Operators

Yıl 2016, Cilt: 4 Sayı: 1, 15 - 23, 15.04.2016

Bülent Yılmaz

https://doi.org/10.36753/mathenot.421349

Öz

In this article we obtain asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the

non-self adjoint Sturm–Liouville operators with Dirichlet boundary conditions, when the potential is a

summable function. Then using these we compute the main part of the eigenvalues in special cases. The

eigenvalues obtained by the asymptotic method and the eigenvalues obtained by the finite difference

method followed by a numerical correction, are compared.

Anahtar Kelimeler

Asymptotic formulas, Sturm–Liouville operators

Kaynakça

• [1] B. Yilmaz and O. A. Veliev, “Asymptotic formulas for Dirichlet boundary value problems,” Studia Scientiarum Mathematicarum Hungarica, vol. 42, no. 2, pp. 153–171, 2005.
• [2] O. A. Veliev and M. Toppamuk Duman, “The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential,” Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 76–90, 2002.
• [3] G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Transactions of the AmericanMathematical Society, vol. 9, no. 4, pp. 373–395, 1908.
• [4] N. Dunford and J. T. Schwartz, Linear Operators. Part III, Spectral Operators,Wiley-Interscience, New York, NY, USA, 1988.
• [5] W. N. Everitt, J. Gunson, and A. Zettl, “Some comments on Sturm-Liouville eigenvalue problems with interior singularities,”Journal of Applied Mathematics and Physics, vol. 38, no. 6, pp. 813–838, 1987.
• [6] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkh¨auser, Basel, Switzerland, 1986.
• [7] M. A.Naimark, Linear Differential Operators, GeorgeG. Harrap and Company, 4th edition, 1967.
• [8] B. N. Parlett,The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, USA, 1980.
• [9] J. D. Tamarkin, “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions,” Mathematische Zeitschrift, vol. 27, no. 1, pp. 1–54, 1928.
• [10] E. C. Titchmarsh, Eigenfunction Expansions, vol. I, Oxford University Press, 1962.
• [11] R. L. Burden, Numerical Analysis, Brooks Cole, Pacific Grove
• [12] A. L. Andrew, “Correction of finite difference eigenvalues of periodic Sturm-Liouville problems,” Australian Mathematical Society Journal Series B, vol. 30, no. 4, pp. 460–469, 1989.
• [13] J.W. Paine, F. R. de Hoog, and R. S. Anderssen, “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems,” Computing, vol. 26, no. 2, pp. 123– 139, 1981.
• [14] R. S. Anderssen and F. R. de Hoog, “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions,” BIT, vol. 24, no. 4, pp. 401–412, 1984.
• [15] A. L. Andrew and J. W. Paine, “Correction of finite element estimates for Sturm-Liouville eigenvalues,” Numerische Mathematik, vol. 50, no. 2, pp. 205–215, 1986.
• [16] C.-K. Chen and S.-H. Ho, “Application of differential transformation to eigenvalue problems,” Applied Mathematics and Computation, vol. 79, no. 2-3, pp. 173–188, 1996.
• [17] P. Ghelardoni, “Approximations of Sturm-Liouville eigenvalues using boundary value methods,” Applied Numerical Mathematics, vol. 23, no. 3, pp. 311–325, 1997.
• [18] P. Ghelardoni and G. Gheri, “Improved shooting technique for numerical computations of eigenvalues in Sturm-Liouville problems,” Nonlinear Analysis, vol. 47, pp. 885–896, 2001.
• [19] J. D. Pryce, Numerical Solution of Sturm-Liouville Problems, Clarendon Press, Oxford, UK, 1993.
• [20] B.Yilmaz, Guldem Yildiz„ O. A. Veliev, "Asymptotic and Numerical Methods in Estimating Eigenvalues", Mathematical Problems in Engineering , 2013.