Abstract
An inverse spectral problem for the convolution integro-differential operator of fractional order \(\alpha >2\) is studied. We show that specification of one spectrum determines such operator uniquely independently of particular value of \(\alpha \). The convolution kernel can be recovered by solving a certain nonlinear equation.
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References
Buterin, S.A.: The inverse problem of recovering the volterra convolution operator from the incomplete spectrum of its rank-one perturbation. Inverse Probl. 22, 2223–2236 (2006)
Buterin, S.: On an inverse spectral problem for a convolution integro-differential operator. Results Math. 50(3–4), 173–181 (2007)
Eremin, M.S.: Inverse problem for second-order integro-differential equation with a singularity. Differ. Uravn. 24(2), 350–351 (1988)
Ignat’ev, MYu.: On the similarity between volterra operators and transformation operators for integro-differential operators of fractional order. Math. Notes 73(2), 192–201 (2003)
Kuryshova, YuV: The inverse spectral problem for integro-differential operators. Mat. Zametki 81(6), 855–866 (2007)
Malamud M.M.: On Some Inverse Problems, in Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Kiev, 1979, pp. 116–124
Malamud, M.M.: Similar volterra operators and related aspects of the theory of fractional differential equations. Tr. Mosk. Mat. Obs. 55, 73–148 (1993)
Popov, AYu., Sedletskii, A.M.: Distribution of roots of Mittag-Leffler functions. J. Math. Sci. 190(2), 209–409 (2013)
Yurko, V.A.: An inverse problem for integral operators. Math. Notes 37, 378–385 (1985)
Yurko, V.A.: Inverse problem for integro-differential operators. Mat. Zametki 50(5), 134–144 (1991)
Acknowledgements
This work was supported by the Russian Science Foundation (Project No. 17-11-01193).
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Ignatyev, M. On an Inverse Spectral Problem for the Convolution Integro-Differential Operator of Fractional Order. Results Math 73, 34 (2018). https://doi.org/10.1007/s00025-018-0800-2
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DOI: https://doi.org/10.1007/s00025-018-0800-2