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2013 | 11 | 11 | 2044-2051
Tytuł artykułu

Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

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EN
Abstrakty
EN
We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.
Twórcy
Bibliografia
  • [1] Andrew A.L., Computing Sturm-Liouville potentials from two spectra, Inverse Problems, 2006, 22(6), 2069–2081 http://dx.doi.org/10.1088/0266-5611/22/6/010
  • [2] Andrew A.L., Finite difference methods for half inverse Sturm-Liouville problems, Appl. Math. Comput., 2011, 218(2), 445–457 http://dx.doi.org/10.1016/j.amc.2011.05.085
  • [3] Chu M.T., Golub G.H., Inverse Eigenvalue Problems, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005 http://dx.doi.org/10.1093/acprof:oso/9780198566649.001.0001
  • [4] Freiling G., Yurko V., Inverse Sturm-Liouville Problems and their Applications, Nova Science, Huntington, 2001
  • [5] Freiling G., Yurko V., Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point, Inverse Problems, 2002, 18(3), 757–773 http://dx.doi.org/10.1088/0266-5611/18/3/316
  • [6] Gel’fand I.M., Levitan B.M., On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl., 1955, 1, 253–305
  • [7] Hald O.H., Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 1984, 37(5), 539–577 http://dx.doi.org/10.1002/cpa.3160370502
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  • [9] Ignatiev M., Yurko V., Numerical methods for solving inverse Sturm-Liouville problems, Results Math., 2008, 52(1–2), 63–74 http://dx.doi.org/10.1007/s00025-007-0276-y
  • [10] Krueger R.J., Inverse problems for nonabsorbing media with discontinuous material properties, J. Math. Phys., 1982, 23(3), 396–404 http://dx.doi.org/10.1063/1.525358
  • [11] Levitan B.M., Sargsjan I.S., Sturm-Liouville and Dirac Operators, Math. Appl. (Soviet Series), 59, Kluwer, Dordrecht, 1991 http://dx.doi.org/10.1007/978-94-011-3748-5
  • [12] Marchenko V.A., Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel, 1986 http://dx.doi.org/10.1007/978-3-0348-5485-6
  • [13] Plum M., Eigenvalue problems for differential equations, In: Wavelets, Multilevel Methods and Elliptic PDEs, Leicester, 1996, Numer. Math. Sci. Comput., Oxford University Press, New York, 1997, 39–83
  • [14] Pöschel J., Trubowitz E., Inverse Spectral Theory, Pure Appl. Math., 130, Academic Press, Boston, 1987
  • [15] Pryce J.D., Numerical Solution of Sturm-Liouville Problems, Monogr. Numer. Anal., Oxford University Press, New York, 1993
  • [16] Rafler M., Böckmann C., Reconstructive method for inverse Sturm-Liouville problems with discontinuous potentials, Inverse Problems, 2007, 23(3), 933–946 http://dx.doi.org/10.1088/0266-5611/23/3/006
  • [17] Rundell W., Sacks P.E., Reconstruction techniques for classical inverse Sturm-Liouville problems, Math. Comp., 1992, 58(197), 161–183 http://dx.doi.org/10.1090/S0025-5718-1992-1106979-0
  • [18] Sacks P.E., An iterative method for the inverse Dirichlet problem, Inverse Problems, 1988, 4(4), 1055–1069 http://dx.doi.org/10.1088/0266-5611/4/4/009
  • [19] Shepel’sky D.G., The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions, Adv. Soviet Math., 1994, 19, 209–232
  • [20] Vinokurov V.A., Sadovnichiı V.A., Asymptotics of arbitrary order of the eigenvalues and eigenfunctions of the Sturm-Liouville boundary value problem in an interval with a summable potential, Izv. Math., 2000, 64(4), 695–754 http://dx.doi.org/10.1070/IM2000v064n04ABEH000295
  • [21] Yurko V., Integral transforms connected with discontinuous boundary value problems, Integral Transform. Spec. Funct., 2000, 10(2), 141–164 http://dx.doi.org/10.1080/10652460008819282
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-013-0301-1
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