Eigenfunction Expansions for Second-Order Boundary Value Problems with Separated Boundary Conditions

Document Type: Original Scientific Paper

Author

University of Kashan

Abstract

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem. Finally, by the sequence of orthogonal eigenfunctions, we provide the eigenfunction expansions for twice continuously differentiable functions.

Keywords

Main Subjects


1. R. Kh. Amirov, V. A. Yurko, On differential operators with singularity and discontinuity conditions inside an interval, Ukrainian Math. J. 53 (2001) 1751–1770.

2. K. Aydemir, Boundary value problems with eigenvalue-dependent boundary and transmission conditions, Bound. Value Probl. 2014, 2014:131.

3. K‎. ‎Aydemir‎, ‎O‎. ‎Sh‎. ‎Mukhtarov‎, ‎Spectrum‎ of one Sturm-Liouville type problem on‎ two disjoint intervals‎, Gen‎. Math‎. ‎Notes 21 (2014) 43-51‎.

 

4. K‎. ‎Aydemir‎, ‎O‎. ‎Sh‎. ‎Mukhtarov‎, ‎Second-orde‎r‎ differential operators with interior singularity‎,‎ Adv‎. ‎Difference Equ. 2015‎, ‎2015:26‎, ‎10 pp‎.

 

5. M‎. ‎Braun‎, ‎Differential Equations and their‎ Applications‎: ‎An Introduction to Applied Mathematics,‎ 3rd ed.‎, Springer-Verlag‎, ‎New York‎, ‎1983‎.

 

6. H‎. ‎Coskun‎, ‎Asymptotic approximations‎ of eigenvalues and eigenfunctions for‎ regular Sturm-Liouville problems‎, ‎ Rocky Mountain J‎. ‎Math. 36 (2006) 867-883‎.

 

7. G‎. ‎Freiling‎, ‎V‎. ‎Yurko‎, ‎On the determination‎ of differential equations with singularities‎ and turning points‎, Results Math. 41 (2002) 275-290‎.

 

8. G‎. ‎Freiling‎, ‎V‎. ‎Yurko‎, ‎On the solvability ‎of an inverse problem in the ‎central-symmetric case‎, ‎Appl‎. ‎Anal. 90 (2011) 1819-1828‎.

 

9. G‎. ‎Sh‎. ‎Guseinov‎, ‎Eigenfunction expansions‎ for a Sturm-Liouville problem on time scales‎‎,‎Int‎. ‎J‎. ‎Difference Equ.2 (2007) 93-104.

 

10. A‎. ‎P‎. ‎Khromov‎, ‎Expansion in eigenfunctions‎ of ordinary linear differential operators with‎ irregular decomposing boundary conditions‎,‎ Mat‎. ‎Sb‎. ‎(N.S.) 70 (1966) 310-329‎.

 

11. Q‎. ‎Kong‎, ‎A‎. ‎Zettl‎, ‎Eigenvalues ‎of regular Sturm-Liouville problems‎, ‎‎ J‎. ‎Differential Equations 131 (1996) 1-19‎.

 

12. R‎. ‎K‎. ‎Miller‎, ‎A‎. ‎N‎. ‎Michel‎,Ordinary Differential Equations‎,‎ Academic Press‎, ‎INC.‎, ‎New York‎, ‎1982‎.

 

13. S‎. ‎Mosazadeh‎, ‎The stability of the solution‎ of an inverse spectral problem with a singularity‎,‎ Bull‎. ‎Iranian Math‎. ‎Soc. 41 (2015) 1061-1070.

 

14. O‎. ‎Sh‎. ‎Mukhtarov‎, ‎H‎. ‎Olgar‎, ‎K‎. ‎Aydemir‎, Resolvent operator and spectrum of ‎ new type boundary value problems‎, ‎Filomat 29 (2015) 1671-1680.

 

15. A‎. ‎Neamaty‎, ‎S‎. ‎Mosazadeh‎, ‎On the‎  canonical solution of the Sturm-Liouville‎ problem with singularity and turning point of even order‎,‎ Canad‎. ‎Math‎. ‎Bull. 54 (2011) 506-518‎.

 

16. E‎. ‎Sen‎, ‎Asymptotic properties of ‎ eigenvalues and eigenfunctions of a ‎ Sturm-Liouville problems with discontinuous‎ weight function‎, ‎Miskolc Math‎. ‎Notes 15 (2014) 197-209‎.

 

17. J‎. ‎Smoller‎, ‎Shock Waves and ‎ Reaction-Diffusion Equations‎, ‎Springer-Verlag‎, ‎INC.‎, ‎New York‎, ‎1983‎.

 

18. V‎. ‎A‎. ‎Yurko‎, ‎The inverse spectral problem‎ for differential operators with nonseparated ‎ boundary conditions‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl. 250 (2000) 266-289.

 

19. V‎. ‎A‎. ‎Yurko‎, ‎An inverse spectral problem‎ for  non-selfadjoint Sturm-Liouville operators‎ with nonseparated boundary conditions‎, ‎Tamkang J‎. ‎Math. 43 (2012) 289-299‎.