×

Uniform stability of the inverse spectral problem for a convolution integro-differential operator. (English) Zbl 1508.65086

Summary: The operator of double differentiation perturbed by the composition of the differentiation operator and a convolution one on a finite interval with Dirichlet boundary conditions is considered. We obtain uniform stability of recovering the convolution kernel from the spectrum both in a weighted \(L_2\)-norm and in a weighted uniform norm. For this purpose, we successively prove uniform stability of each step of the algorithm for solving this inverse problem in both norms. The obtained results reveal some essential difference from the classical inverse Sturm-Liouville problem.

MSC:

65L09 Numerical solution of inverse problems involving ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
47G20 Integro-differential operators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] English transl., Birkhauser, 1986. · Zbl 0399.34022
[2] English transl., VNU Sci. Press, Utrecht, 1987. · Zbl 0575.34001
[3] Freiling, G.; Yurko, V., Inverse Sturm-Liouville Problems and Their Applications (2001), Huntington: Huntington NY, Nova Science Publishers · Zbl 1037.34005
[4] Yurko, V. A., Method of Spectral Mappings in the Inverse Problem Theory: Inverse and Ill-posed Problems Series (2002), VSP: VSP Utrecht · Zbl 1098.34008
[5] Ambarzumian, V., Über eine Frage der Eigenwerttheorie, Z. Phys, 53, 690-695 (1929) · JFM 55.0868.01
[6] Borg, G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78, 1-96 (1946) · Zbl 0063.00523
[7] Karaseva, T. M., On the inverse Sturm-Liouville problem for a non-Hermitian operator, Mat. Sbornik, 32, 74, 477-484 (1953) · Zbl 0050.08801
[8] Buterin, S.; Kuznetsova, M., On Borg’s method for non-selfadjoint Sturm-Liouville operators, Anal. Math. Phys., 9, 4, 2133-2150 (2019) · Zbl 1434.34025
[9] English transl. in Math. USSR-Sb. 26 (4) (1975) 493-554. · Zbl 0327.34021
[10] English transl. in Funk. Anal. Appl. 44 (4) (2010) 270-285. · Zbl 1271.34017
[11] 14pp. · Zbl 1327.34031
[12] Malamud, M. M., On some inverse problems, Boundary Value Problems of Mathematical Physics, Kiev, 116-124 (1979)
[13] Yurko, V. A., Inverse problem for integro-differential operators of the first order, Funct. Anal. Ul’janovsk, 144-151 (1984) · Zbl 0566.47031
[14] Eremin, M. S., An inverse problem for a second-order integro-differential equation with a singularity, Diff. Uravn., 24, 2, 350-351 (1988) · Zbl 0662.45006
[15] English transl. in Math. Notes 50 (5-6) (1991), 1188-1197. · Zbl 0744.45004
[16] Buterin, S. A., Recovering a convolution integro-differential operator from the spectrum, Matematika. Mekhanika, vol. 6, Saratov Univ., Saratov, 15-18 (2004)
[17] Buterin, S. A., On an inverse spectral problem for a convolution integro-differential operator, Results Math., 50, 3-4, 173-181 (2007) · Zbl 1135.45007
[18] English transl. in Math. Notes 81 (6) (2007) 767-777. · Zbl 1142.45006
[19] English transl. in Diff. Eqns. 46 (1) (2010) 150-154. · Zbl 1197.34017
[20] Kuryshova, Y.; Shieh, C. T., An inverse nodal problem for integro-differential operators, J. Inverse Ill-Posed Problems, 18, 4, 357-369 (2010) · Zbl 1279.34090
[21] Wang, Y.; Wei, G., The uniqueness for Sturm-Liouville problems with aftereffect, Acta Math Sci., 32A, 6, 1171-1178 (2012) · Zbl 1289.34033
[22] Yang, C. F., Trace formulae for matrix integro-differential operators, Zeitschrift für Naturforschung, 67a, 180-184 (2012)
[23] Yurko, V. A., An inverse spectral problem for integro-differential operators, Far East J. Math. Sci., 92, 2, 247-261 (2014) · Zbl 1328.47051
[24] Buterin, S. A.; Choque Rivero, A. E., On inverse problem for a convolution integro-differential operator with Robin boundary conditions, Appl. Math. Lett., 48, 150-155 (2015) · Zbl 1325.45011
[25] Buterin, S. A.; Sat, M., On the half inverse spectral problem for an integro-differential operator, Inverse Probl Sci Eng, 25, 10, 1508-1518 (2017) · Zbl 1390.45022
[26] Bondarenko, N.; Buterin, S., On recovering the Dirac operator with an integral delay from the spectrum, Results Math., 71, 3-4, 1521-1529 (2017) · Zbl 1408.34019
[27] Yurko, V. A., Inverse spectral problems for first order integro-differential operators, Boundary Value Probl., 2017 (2017), Art. No. 98, 7pp · Zbl 1381.47032
[28] Buterin, S. A., On inverse spectral problems for first-order integro-differential operators with discontinuities, Appl. Math. Lett., 78, 65-71 (2018) · Zbl 1381.45026
[29] Engl. transl. in J. Math. Sci. (to appear)
[30] Bondarenko, N. P., An inverse problem for an integro-differential operator on a star-shaped graph, Math. Meth. Appl. Sci., 41, 4, 1697-1702 (2018) · Zbl 1392.45014
[31] Buterin, S. A.; Vasiliev, S. V., On uniqueness of recovering the convolution integro-differential operator from the spectrum of its non-smooth one-dimensional perturbation, Boundary Value Probl., 2018 (2018), Art. No. 55, 12pp · Zbl 1499.34115
[32] Ignatyev, M., On an inverse spectral problem for the convolution integro-differential operator of fractional order, Results Math., 73, 34 (2018), 8pp · Zbl 1444.45008
[33] Ignatiev, M., On an inverse spectral problem for one integro-differential operator of fractional order, J. Inverse Ill-posed Probl., 27, 1, 17-23 (2019) · Zbl 1411.45005
[34] Zolotarev, V. A., Inverse spectral problem for the operators with non-local potential, Mathematische Nachrichten, 292, 3, 661-681 (2019) · Zbl 1506.47018
[35] Bondarenko, N. P., An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernel, J. Inverse Ill-Posed Probl., 27, 2, 151-157 (2019) · Zbl 1472.45007
[36] Bondarenko, N. P., An inverse problem for an integro-differential pencil with polynomial eigenparameter-dependence in the boundary condition, Anal. Math. Phys., 9, 4, 2227-2236 (2019) · Zbl 1440.45004
[37] Bondarenko, N. P., An inverse problem for an integro-differential equation with a convolution kernel dependent on the spectral parameter, Results Math., 74 (2019), Art. No. 148, 7pp · Zbl 1431.45010
[38] Bondarenko, N. P., An inverse problem for the second-order integro-differential pencil, Tamkang J. Math., 50, 3, 223-231 (2019) · Zbl 1440.45005
[39] Hu, Y.-T.; Bondarenko, N. P.; Shieh, C.-T.; Yang, C. F., Traces and inverse nodal problems for Dirac-type integro-differential operators on a graph, Appl. Math. Comput., 363, 124606 (2019) · Zbl 1433.34027
[40] Buterin, S. A., An inverse spectral problem for Sturm-Liouville-type integro-differential operators with Robin boundary conditions, Tamkang J. Math., 50, 3, 207-221 (2019) · Zbl 1440.45006
[41] Bondarenko, N.; Buterin, S., An inverse spectral problem for integro-differential Dirac operators with general convolution kernels, Appl. Anal., 99, 700-716 (2020) · Zbl 1436.34012
[42] Buterin, S. P., On a transformation operator approach in the inverse spectral theory of integral and integro-differential operators, (Kravchenko, V.; Sitnik, S., Transmutation Operators and Applications. Trends in Mathematics. Birkhäuser, Cham (2020)), 337-367 · Zbl 1467.45022
[43] Bondarenko, N.; Buterin, S., Numerical solution and stability of the inverse spectral problem for a convolution integro-differential operator, Commun. Nonlinear Sci. Numer. Simulat, 89, 105298 (2020), , 11pp · Zbl 1452.65416
[44] English transl. in Math. Notes 80 (5) (2006), 631-644. · Zbl 1134.47003
[45] Buterin, S.; Malyugina, M., On global solvability and uniform stability of one nonlinear integral equation, Results Math., 73 (2018), Art. No. 117, 19pp · Zbl 1401.45005
[46] Buterin, S. A.; Terekhin, P. A., On solvability of one nonlinear integral equation in the class of analytic functions, Appl. Math. Lett., 96, 27-32 (2019) · Zbl 1436.45004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.