277
Views
23
CrossRef citations to date
0
Altmetric
Original Articles

Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter

&
Pages 951-961 | Received 06 Jul 2010, Accepted 18 Feb 2011, Published online: 19 Apr 2011

References

  • McLaughlin, JR, 1988. Inverse spectral theory using nodal points as data–a uniqueness result, J. Differ. Eqns. 73 (1988), pp. 354–362.
  • Hald, OH, and McLaughlin, JR, 1989. Solutions of inverse nodal problems, Inv. Probl. 5 (1989), pp. 307–347.
  • Buterin, SA, and Shieh, CT, 2009. Inverse nodal problem for differential pencils, Appl. Math. Lett. 22 (2009), pp. 1240–1247.
  • Chen, YT, Cheng, YH, Law, CK, and Tsay, J, 2002. L1 convergence of the reconstruction formula for the potential function, Proc. Am. Math. Soc. 130 (2002), pp. 2319–2324.
  • Cheng, YH, Law, CK, and Tsay, J, 2000. Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), pp. 145–155.
  • Currie, S, and Watson, BA, 2007. Inverse nodal problems for Sturm–Liouville equations on graphs, Inv. Probl. 23 (2007), pp. 2029–2040.
  • Koyunbakan, H, 2006. A new inverse problem for the diffusion operator, Appl. Math. Lett. 19 (2006), pp. 995–999.
  • Law, CK, Shen, CL, and Yang, CF, 1999. The inverse nodal problem on the smoothness of the potential function, Inv. Probl. 15 (1999), pp. 253–263, Errata 17 (2001) pp. 361–364.
  • Law, CK, and Tsay, J, 2001. On the well-posedness of the inverse nodal problem, Inv. Probl. 17 (2001), pp. 1493–1512.
  • Law, CK, and Yang, CF, 1998. Reconstructing the potential function and its derivatives using nodal data, Inv. Probl. 14 (2) (1998), pp. 299–312.
  • McCarthy, CM, and Rundell, W, 2003. Eigenparameter dependent inverse Sturm–Liouville problems, Numer. Funct. Anal. Optim. 24 (2003), pp. 85–105.
  • Shen, CL, 1988. On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math. Anal. 19 (1988), pp. 1419–1424.
  • Shen, CL, and Shieh, CT, 2000. An inverse nodal problem for vectorial Sturm–Liouville equation, Inv. Probl. 16 (2000), pp. 349–356.
  • Shieh, CT, and Yurko, VA, 2008. Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008), pp. 266–272.
  • Yang, CF, and Yang, XP, 2010. Inverse nodal problems for differential pencils on a star-shaped graph, Inv. Probl. 26 (2010), p. 085008, (15pp.).
  • Yang, XF, 1997. A solution of the inverse nodal problem, Inv. Probl. 13 (1997), pp. 203–213.
  • Yang, XF, 2001. A new inverse nodal problem, J. Differ. Eqns. 169 (2001), pp. 633–653.
  • Yurko, VA, 2008. Inverse nodal problems for Sturm–Liouville operators on star-type graphs, J. Inv. Ill-Posed Probl. 16 (2008), pp. 715–722.
  • Binding, PA, Browne, PJ, and Watson, BA, 2002. Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter: II, J. Comput. Appl. Math. 148 (2002), pp. 147–168.
  • Binding, PA, Browne, PJ, and Watson, BA, 2004. Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, J. Math. Anal. Appl. 291 (2004), pp. 246–261.
  • Browne, PJ, and Sleeman, BD, 1996. Inverse nodal problem for Sturm–Liouville equation with eigenparameter dependent boundary conditions, Inv. Probl. 12 (1996), pp. 377–381.
  • Chernozhukova, AY, and Freiling, G, 2009. A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter, Inv. Probl. Sci. Eng. 17 (2009), pp. 777–785.
  • Chugunova, MV, 2001. Inverse spectral problem for the Sturm–Liouville operator with eigenvalue parameter dependent boundary conditions, Oper. Theory: Adv. Appl. 123 (2001), pp. 187–194.
  • Freiling, G, and Yurko, VA, 2010. Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inv. Probl. 26 (2010), p. 055003, (17pp.).
  • Guliyev, NJ, 2005. Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary condition, Inv. Probl. 21 (2005), pp. 1315–1330.
  • R. Mennicken, M. Möller, Non-self-adjoint Boundary Value Problems, North-Holland Mathematic Studies, Vol. 192, North-Holland, Amsterdam, 2003.
  • Ramm, AG, 2005. Inverse Problems Mathematical and Analytical Techniques with Applications to Engineering. New York: Springer; 2005.
  • Rundell, W, and Sacks, PE, 1992. The reconstruction of Sturm–Liouville operators, Inv. Probl. 8 (1992), pp. 457–482.
  • Yamamoto, M, 1990. Inverse eigenvalue problem for a vibration of a string with viscous drag, J. Math. Anal. Appl. 152 (1990), pp. 20–34.
  • Collatz, L, 1963. Eigenwertaufgaben mit Technischen Anwendungen. Leipzig: Akad. Verlagsgesellschaft Geest & Portig; 1963.
  • Kraft, RE, and Wells, WR, 1977. Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flow-duct acoustics, J. Acoust. Soc. Am. 61 (1977), pp. 913–922.
  • Fulton, CT, 1977. Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinb. 77 (A) (1977), pp. 293–308.
  • Walter, J, 1973. Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133 (1973), pp. 301–312.
  • Freiling, G, and Yurko, VA, 2001. Inverse Sturm–Liouville Problems and their Applications. New York: NOVA Science Publishers; 2001.
  • B.M. Levitan, I.S. Sargsjan, Sturm–Liouville and Dirac Operators (Russian), Nauka, Moscow, 1988; English transl.: Kluwer, Dordrecht, 1991.
  • Marchenko, VA, 1977. Sturm–Liouville Operators and their Applications. Kiev: Naukova Dumka; 1977, English transl.: Birkhäser, 1986.
  • V.A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications, Gordon and Breach, Amsterdam, 2000.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.