The partial inverse nodal problem for differential pencils on a finite interval
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Abstract
In this paper, the partial inverse nodal problem for differential pencils with real-valued coefficients on a finite interval \([0,1]\) was studied. The authors showed that the coefficients \((q_{0}(x),q_{1}(x),h,H_0)\) of the differential pencil \(L_0\) can be uniquely determined by partial nodal data on the right(or, left) arbitrary subinterval \([a,b]\) of \([0,1].\) Finally, an example was given to verify the validity of the reconstruction algorithm for this inverse nodal problem.
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Wang, Y. P. ., Hu, Y., & Shieh, C.-T. (2019). The partial inverse nodal problem for differential pencils on a finite interval. Tamkang Journal of Mathematics, 50(3), 307–319. https://doi.org/10.5556/j.tkjm.50.2019.3359
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References
[1] N. Bondarenko, A partial inverse problem for the differential pencil on a star-shaped graph, Results in Mathematics 72(4) (2017), 1933-1942.
[2] N. Bondarenko, Recovery of the matrix quadratic differential pencil from the spectral data, Journal of Inverse and Ill-posed Problems 24(3)(2016), 245-263.
[3] P.J. Browne and B.D. Sleeman, Inverse nodal problem for Sturm-Liouville equation with eigenparameter dependent boundary conditions, Inverse Problems 12(1996), 377-381.
[4] S. Buterin, Inverse nodal problem for differential pencils of second order(Russian), Spect.
Evol. Problems 18(2008), 46-51.
[5] S. Buterin and C.-T. Shieh, Inverse nodal problem for differential pencils, Appl. Math.
Lett. 22 (2009), 1240-1247.
[6] S. Buterin and C.-T. Shieh, Incomplete inverse spectral and nodal problems for differential
pencils, Results Math. 62(2012), 167-179.
[7] S. Buterin and V. Yurko, Inverse spectral problem for pencils of differential operators on
a finite interval, Vestnik Bashkir. Univ. 4(2006), 8-12.
[8] S. Buterin, On half inverse problem for differential pencils with the spectral parameter
in boundary conditions, Tamkang J. Math. 42(2011), 355-364.
[9] S. Buterin and V. Yurko, Inverse problems for second-order differential pencils with dirichlet boundary conditions, J. Inverse Ill-Posed Probl. 20(5-6)(2012), 855-881.
[10] X.F. Chen, Y.H. Cheng and C.K. Law, Reconstructing potentials from zeros of one eigenfunction, Trans. Amer. Math. Soc. 363(2011), 4831-4851.
[11] Y.H. Cheng, C.K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248(2000), 145-155.
[12] M.G. Gasymov and G.Sh. Guseinov, Determination of the diffusion operator by spectral data(in Russian), Doklady Acad. Nauk AzSSR 37(1981), 19-23.
[13] I.M. Guseinov and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sbornik: Mathematics 198(11) (2007), 1579-1598.
[14] F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential II: The case of discrete spectrum, Trans. Amer. Math. Soc. 352(2000), 2765- 2787.
[15] G. Freiling and V.A. Yurko, Recovering nonselfadjoint differential pencils with nonsepa- rated boundary conditions, Applicable Analysis, 94(8)(2015), 1649-1661..
[16] Y. Guo and G. Wei, Inverse problem for diferential pencils with incompletely spectral information, Taiwanese Journal of mathematics 19(2015): 927-942.
[17] Y. Guo and G. Wei, Determination of differential pencils from dense nodal subset in an interior subinterval, Israel Journal of Mathematics 206(1)(2015), 213-231.
[18] Y. Guo and G. Wei, Inverse problems: Dense nodal subset on an interior subinterval, J. Differential Equations 255(2013), 2002-2017.
[19] M. Horvath, On the inverse spectral theory of Schro ̈dinger and Dirac operators, Trans. Amer. Math. Soc. 353(2001), 4155-4171.
[20] R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm- Liouville equations, Inverse Problems 28(2012): 085008(21 pp)
[21] M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys. 28(1972), 177-220.
[22] B.Ja. Levin, Distribution of zeros of entire functions (in Russian), GITTL, Moscow 1956.
[23] N. Levinson, Gap and density theorems, AMS Coll. Publ. 1940, New York.
[24] J.R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result,
J. Differential Equations 73(1988), 354-362.
[25] N. Pronska, Reconstruction of energy-dependent Sturm-Liouville operators from two
spectra, Integral Equations and Operator Theory 76(2013): 403-419.
[26] C.L. Shen, On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math.
Anal. 19(1988), 1419-1424.
[27] C.-T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontin-
uous boundary value problems, J. Math. Anal. Appl. 347(2008), 266-272.
[28] Y.P. Wang, Inverse problems for discontinuous Sturm-Liouville operators with mixed
spectral data, Inverse Probl. Sci. Eng., 23(2015), 1180-1198.
[29] Y.P. Wang and V. Yurko, On the inverse nodal problems for discontinuous Sturm- Liouville operators, J. Differential Equations 260(2016), 4086-4109.
[30] Y.P. Wang, K.Y. Lien and C.-T. Shieh, Inverse problems for the boundary value problem with the interior nodal subsets, Applicable Analysis 96(7)(2017), 1229-1239.
[31] Y.P. Wang, The inverse spectral problem for differential pencils by mixed spectral data, Applied Mathematics and Computation 338(2018), 544-551.
[32] Y.P. Wang and V. Yurko, Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter, Math. Meth. Appl. Sci. 40(2017), 3190- 3196.
[33] C.F. Yang, Reconstruction of the diffusion operator from nodal data, Z. Natureforsch 65a.1(2010), 100-106.
[34] C.F. Yang, Solution to open problems of Yang concerning inverse nodal problems, Isr. J. Math. 204(2014), 283-298.
[35] C.F. Yang, Inverse nodal problems for differential pencils on a star-shaped graph, Inverse Problems 26(2010), 085008.
[36] X.F. Yang, A new inverse nodal problem, J. Differential Equations 169(2001), 633-653.
[37] V. Yurko, Inverse problem for quasi-periodic differential pencils with jump conditions
inside the interval, Complex Anal. Oper. Theory, 10(2016), 1203-1212.
[38] V. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, VSP, Utrecht:
Inverse and Ill-posed Problems Ser. 2002.
[2] N. Bondarenko, Recovery of the matrix quadratic differential pencil from the spectral data, Journal of Inverse and Ill-posed Problems 24(3)(2016), 245-263.
[3] P.J. Browne and B.D. Sleeman, Inverse nodal problem for Sturm-Liouville equation with eigenparameter dependent boundary conditions, Inverse Problems 12(1996), 377-381.
[4] S. Buterin, Inverse nodal problem for differential pencils of second order(Russian), Spect.
Evol. Problems 18(2008), 46-51.
[5] S. Buterin and C.-T. Shieh, Inverse nodal problem for differential pencils, Appl. Math.
Lett. 22 (2009), 1240-1247.
[6] S. Buterin and C.-T. Shieh, Incomplete inverse spectral and nodal problems for differential
pencils, Results Math. 62(2012), 167-179.
[7] S. Buterin and V. Yurko, Inverse spectral problem for pencils of differential operators on
a finite interval, Vestnik Bashkir. Univ. 4(2006), 8-12.
[8] S. Buterin, On half inverse problem for differential pencils with the spectral parameter
in boundary conditions, Tamkang J. Math. 42(2011), 355-364.
[9] S. Buterin and V. Yurko, Inverse problems for second-order differential pencils with dirichlet boundary conditions, J. Inverse Ill-Posed Probl. 20(5-6)(2012), 855-881.
[10] X.F. Chen, Y.H. Cheng and C.K. Law, Reconstructing potentials from zeros of one eigenfunction, Trans. Amer. Math. Soc. 363(2011), 4831-4851.
[11] Y.H. Cheng, C.K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248(2000), 145-155.
[12] M.G. Gasymov and G.Sh. Guseinov, Determination of the diffusion operator by spectral data(in Russian), Doklady Acad. Nauk AzSSR 37(1981), 19-23.
[13] I.M. Guseinov and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sbornik: Mathematics 198(11) (2007), 1579-1598.
[14] F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential II: The case of discrete spectrum, Trans. Amer. Math. Soc. 352(2000), 2765- 2787.
[15] G. Freiling and V.A. Yurko, Recovering nonselfadjoint differential pencils with nonsepa- rated boundary conditions, Applicable Analysis, 94(8)(2015), 1649-1661..
[16] Y. Guo and G. Wei, Inverse problem for diferential pencils with incompletely spectral information, Taiwanese Journal of mathematics 19(2015): 927-942.
[17] Y. Guo and G. Wei, Determination of differential pencils from dense nodal subset in an interior subinterval, Israel Journal of Mathematics 206(1)(2015), 213-231.
[18] Y. Guo and G. Wei, Inverse problems: Dense nodal subset on an interior subinterval, J. Differential Equations 255(2013), 2002-2017.
[19] M. Horvath, On the inverse spectral theory of Schro ̈dinger and Dirac operators, Trans. Amer. Math. Soc. 353(2001), 4155-4171.
[20] R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm- Liouville equations, Inverse Problems 28(2012): 085008(21 pp)
[21] M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys. 28(1972), 177-220.
[22] B.Ja. Levin, Distribution of zeros of entire functions (in Russian), GITTL, Moscow 1956.
[23] N. Levinson, Gap and density theorems, AMS Coll. Publ. 1940, New York.
[24] J.R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result,
J. Differential Equations 73(1988), 354-362.
[25] N. Pronska, Reconstruction of energy-dependent Sturm-Liouville operators from two
spectra, Integral Equations and Operator Theory 76(2013): 403-419.
[26] C.L. Shen, On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math.
Anal. 19(1988), 1419-1424.
[27] C.-T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontin-
uous boundary value problems, J. Math. Anal. Appl. 347(2008), 266-272.
[28] Y.P. Wang, Inverse problems for discontinuous Sturm-Liouville operators with mixed
spectral data, Inverse Probl. Sci. Eng., 23(2015), 1180-1198.
[29] Y.P. Wang and V. Yurko, On the inverse nodal problems for discontinuous Sturm- Liouville operators, J. Differential Equations 260(2016), 4086-4109.
[30] Y.P. Wang, K.Y. Lien and C.-T. Shieh, Inverse problems for the boundary value problem with the interior nodal subsets, Applicable Analysis 96(7)(2017), 1229-1239.
[31] Y.P. Wang, The inverse spectral problem for differential pencils by mixed spectral data, Applied Mathematics and Computation 338(2018), 544-551.
[32] Y.P. Wang and V. Yurko, Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter, Math. Meth. Appl. Sci. 40(2017), 3190- 3196.
[33] C.F. Yang, Reconstruction of the diffusion operator from nodal data, Z. Natureforsch 65a.1(2010), 100-106.
[34] C.F. Yang, Solution to open problems of Yang concerning inverse nodal problems, Isr. J. Math. 204(2014), 283-298.
[35] C.F. Yang, Inverse nodal problems for differential pencils on a star-shaped graph, Inverse Problems 26(2010), 085008.
[36] X.F. Yang, A new inverse nodal problem, J. Differential Equations 169(2001), 633-653.
[37] V. Yurko, Inverse problem for quasi-periodic differential pencils with jump conditions
inside the interval, Complex Anal. Oper. Theory, 10(2016), 1203-1212.
[38] V. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, VSP, Utrecht:
Inverse and Ill-posed Problems Ser. 2002.