Asymptotic distributions of Neumann problem for Sturm-Liouville equation

Document Type : Research Paper

Authors

University of Bonab, Bonab, Iran

Abstract

In this paper we apply the Homotopy perturbation method to derive the higher-order asymptotic distribution of the eigenvalues and eigenfunctions associated with the linear real second order equation of Sturm-liouville type on $[0,pi]$ with Neumann conditions $(y'(0)=y'(pi)=0)$ where $q$ is a real-valued Sign-indefinite number of $C^{1}[0,pi]$ and $lambda$ is a real parameter.

Keywords


[1] F. V. Atkinson, A. B. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted Sturm-Liouville problems, J. Reine angev. Math. 375 (1987), 380-393.
[2] M. Duman, Asymptotics for the Sturm-Liouville problem by homotopy perturbation method, Applied Mathematics and Computation, 216 (2010), 492-496.
[3] J. H. He, Homotopy perturbation technique, computer Methods in Applied mathematics and Engineering, 178 (1999), 257-262.
[4] J. H. He, Homotopy perturbation method: a new nonlinear technique, Applied Mathematics and Computation, 135 (2003), 73-79.
[5] H. Hochstsdt, Differential equations, Dover, New york, 1957.
[6] E. L. Ince, Ordinary differential equations, Dover, New york,1956.
[7] B. M. Levitan, G. Gasymov, Determination of a differential equation by its spectra, Russian Math. Surveys. 19 (1964), 1-63.
[8] B. M. Levitan, G. Gasymov, Introduction to spectral theory, Translation of Math. Amer. Math. Society (1975).
[9] A. B. Mingarelli, A survey of the regular weighted Sturm-Liouville problem, The non-definite case, P. Fuquan, X. Shutie(Eds.), Applied Differential Equations, World scientific, Singapoure and Philadelphia(1986), 109-137.
[10] Z. M. Odibat, A new modification of the homotopy perturbation method for linear and nonlinear operators, Applied Mathematics and Computation, 189 (2007), 746-753.
[11] J. I. Ramos, Piecewise homotopy methods for nonlinear ordinary differential equations, Appled Mathematics and Computation, 198 (2008), 92-116.
[12] F. G. Tricomi, Differential equations, Hofner, New york, 1961.
[13] E. yusufoglu, A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations, Mathematical and Computer Modeling, 47 (2008), 1099-1107.