JAMP  Vol.2 No.13 , December 2014
A Generalization of Ince’s Equation
Author(s) Ridha Moussa*
ABSTRACT
We investigate the Hill differential equation  where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.

Cite this paper
Moussa, R. (2014) A Generalization of Ince’s Equation. Journal of Applied Mathematics and Physics, 2, 1171-1182. doi: 10.4236/jamp.2014.213137.
References
[1]   Whittaker, E.T. (1915) On a Class of Differential Equations Whose Solutions Satisfy Integral Equations. Proceedings of the Edinburgh Mathematical Society, 33, 14-33.
http://dx.doi.org/10.1017/S0013091500002297

[2]   Ince, E.L. (1923) A Linear Differential Equation with Periodic Coefficients. Proceedings of the London Mathematical Society, 23, 800-842.

[3]   Ince, E.L. (1925) The Real Zeros of Solutions of a Linear Differential Equation with Periodic Coefficients. Proceedings of the London Mathematical Society, 25, 53-58.

[4]   Magnus, W. and Winkler, S. (1966) Hill’s Equation. John Wiley & Sons, New York.

[5]   Arscott, F.M. (1964) Periodic Differential Equations. Pergamon Press, New York.

[6]   Volkmer, H. (2003) Coexistence of Periodic Solutions of Ince’s Equation. Analysis, 23, 97-105.
http://dx.doi.org/10.1524/anly.2003.23.1.97

[7]   Recktenwald, G. and Rand, R. (2005) Coexistence Phenomenon in Autoparametric Excitation of Two Degree of Freedom Systems. International Journal of Non-Linear Mechanics, 40, 1160-1170.
http://dx.doi.org/10.1016/j.ijnonlinmec.2005.05.001

[8]   Hemerey, A.D. and Veselov, A.P. (2009) Whittaker-Hill Equation and Semifinite Gap Schrodinger Operators. 1-10.
arXiv:0906.1697v2

[9]   Eastham, M. (1973) The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh, London.

[10]   Volkmer, H. (2004) Four Remarks on Eigenvalues of Lamé’s Equation. Analysis and Applications, 2, 161-175.
http://dx.doi.org/10.1142/S0219530504000023

[11]   Coddington, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. Robert E. Krieger Publishing Company, Malarbar.

[12]   Kato, T. (1980) Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, Heidelberg, New York.

 
 
Top