194
Views
6
CrossRef citations to date
0
Altmetric
Articles

Reconstruction for the spherically symmetric speed of sound from nodal data

, &
Pages 1032-1046 | Received 11 Sep 2011, Accepted 17 Jul 2012, Published online: 14 Aug 2012

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, Inverse Probl. 5 (1989), pp. 307–347.
  • Browne, PJ, and Sleeman, BD, 1996. Inverse nodal problem for Sturm–Liouville equation with eigenparameter dependent boundary conditions, Inverse Probl. 12 (1996), pp. 377–381.
  • Cheng, YH, Law, CK, and Tsay, J, 2000. Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), pp. 145–155.
  • 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.
  • Shen, CL, 1988. On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math. Anal. 19 (1988), pp. 1419–1424.
  • Yang, XF, 1997. A solution of the inverse nodal problem, Inverse Probl. 13 (1997), pp. 203–213.
  • Law, CK, Shen, CL, and Yang, CF, 1999. The inverse nodal problem on the smoothness of the potential function, Inverse Probl. 15 (1999), pp. 253–263.
  • Law, CK, and Tsay, J, 2001. On the well–posedness of the inverse nodal problem, Inverse Probl. 17 (2001), pp. 1493–1512.
  • Law, CK, and Yang, CF, 1998. Reconstructing the potential function and its derivatives using nodal data, Inverse Probl. 14 (1998), pp. 299–312.
  • Panakhov, ES, Koyunbakan, H, and Ic, U, 2010. Reconstruction formula for the potential function of problem with eigenparameter boundary condition, Inverse Probl. Sci. Eng. 18 (2010), pp. 173–180.
  • Yang, CF, and Yang, XP, 2011. Inverse nodal problem for the Sturm–Liouville equation with polynomially dependent on the eigenparameter, Inverse Probl. Sci. Eng. 19 (2011), pp. 951–961.
  • Buterin, SA, and Shieh, CT, 2009. Inverse nodal problem for differential pencils, Appl. Math. Lett. 22 (2009), pp. 1240–1247.
  • Currie, S, and Watson, BA, 2007. Inverse nodal problems for Sturm–Liouville equations on graphs, Inverse Probl. 23 (2007), pp. 2029–2040.
  • Shen, CL, and Shieh, CT, 2000. An inverse nodal problem for vectorial Sturm–Liouville equation, Inverse 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, 2010. Reconstruction of the diffusion operator from nodal data, Z. Naturforsch. 65a (1) (2010), pp. 100–106.
  • Yang, CF, and Huang, ZY, 2010. Reconstruction of the Dirac Operator from Nodal Data, Int. Eqns Operator Theory 66 (2010), pp. 539–551.
  • Yurko, VA, 2008. Inverse nodal problems for Sturm–Liouville operators on star–type graphs, J. Inverse Ill–Posed Probl. 16 (2008), pp. 715–722.
  • Marchenko, VA, 1977. Sturm–Liouville Operators and their Applications. Kiev: Naukova Dumka; 1977, English transl. Basel, Birkhäser, 1986.
  • Chernozhukova, AY, and Freiling, G, 2009. A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter, Inverse Probl. Sci. Eng. 17 (2009), pp. 777–785.
  • Freiling, G, and Yurko, VA, 2010. Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Probl. 26 (6) (2010), p. 17, 055003.
  • Colton, D, Kirsch, A, and Päivärinta, A, 1989. Far–field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal. 20 (1989), pp. 1472–1483.
  • Colton, D, and Monk, P, 1988. The inverse scattering problem for time–harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math. 41 (1988), pp. 97–125.
  • Hartman, P, and Wilcox, C, 1961. On solutions of the Helmholtz equation in exterior domains, Math. Z. 75 (1961), pp. 228–255.
  • Rynne, BP, and Sleeman, BD, 1991. The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal. 22 (1991), pp. 1755–1762.
  • Mclaughlin, JR, and Polyakov, PL, 1994. On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differ. Eqns 107 (1994), pp. 351–382.
  • Gesztesy, F, and Simon, B, 2000. Inverse spectral analysis with partial information on the potential II: The case of discrete spectrum, Trans. Am. Math. Soc. 352 (2000), pp. 2765–2787.
  • Hochstadt, H, and Lieberman, B, 1978. An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), pp. 676–680.
  • Levitan, BM, and Sargsjan, IS, 1990. Sturm–Liouville and Dirac Operators. Dordrecht: Kluwer Academic; 1990.
  • Yurko, VA, 2002. Method of Spectral Mappings in the Inverse Problem Theory, Inverse Ill–posed Probl. Series. Utrecht: VSP; 2002.

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.