Geodesic
Inverse Boundary-Value Problem for an Integro-Differential Boussinesq-type Equation with Degenerate Kernel
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 129-140.

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We discuss questions on the unique solvability of inverse boundary-value source problems for a certain nonlinear integro-differential equation of Boussinesq type with degenerate kernel. We develop the method of degenerate kernels for the inverse boundary-value problem for a fourth-order integro-differential partial differential equation. Using the Banach fixed-point theorem, we prove the uniquely solvability of the problem and establish a criterion of stability of solutions with respect to recovery functions.
Keywords: inverse boundary-value problem, integro-differential Boussinesq-type equation, degenerate kernel, unique solvability. 
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     author = {T. K. Yuldashev},
     title = {Inverse {Boundary-Value} {Problem} for an {Integro-Differential} {Boussinesq-type} {Equation} with {Degenerate} {Kernel}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {129--140},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2018_149_a14/}
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T. K. Yuldashev. Inverse Boundary-Value Problem for an Integro-Differential Boussinesq-type Equation with Degenerate Kernel. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 129-140. http://geodesic.mathdoc.fr/item/INTO_2018_149_a14/

[1] Abzalimov R. R., Salyakhova E. V., “Raznostno-analiticheskii metod vychisleniya sobstvennykh znachenii dlya uravnenii chetvertogo poryadka s razdelennymi kraevymi usloviyami”, Izv. vuzov. Ser. mat., 11 (2008), 3–11 | MR | Zbl

[2] Algazin S. D., Kiiko I. A., Flatter plastin i obolochek, Nauka, M., 2006

[3] Akhtyamov A. M., Ayupova A. R., “O reshenii zadachi diagnostirovaniya defektov v vide maloi polosti v sterzhne”, Zh. Srednevolzh. mat. o-va, 12:3 (2010), 37–42

[4] Buryachenko E. A., “O razmernosti yadra zadachi Dirikhle dlya uravnenii chetvertogo poryadka”, Differ. uravn., 51:4 (2015), 472–480 | DOI | MR | Zbl

[5] Denisov A. M., Vvedenie v teoriyu obratnykh zadach, MGU, M., 1994 | MR

[6] Zamyshlyaeva A. A., “Matematicheskie modeli sobolevskogo tipa vysokogo poryadka”, Vestn. Yuzhno-Ural. gos. un-ta. Ser. mat. model. programm., 7:2 (2014), 5–28 | Zbl

[7] Karimov Sh. T., “Ob odnom metode resheniya zadachi Koshi dlya odnomernogo polivolnovogo uravneniya s singulyarnym operatorom Besselya”, Izv. vuzov. Ser. mat., 8 (2017), 27–41 | Zbl

[8] Koshanov B. D., Soldatov A. P., “Kraevaya zadacha s normalnymi proizvodnymi dlya ellipticheskogo uravneniya vysokogo poryadka na ploskosti”, Differ. uravn., 52:12 (2016), 1666–1681 | DOI | MR | Zbl

[9] Lavrentev M. M., Savelev L. Ya., Lineinye operatory i nekorrektnye zadachi, Nauka, M., 1991 | MR

[10] Mamedov I. G., “Fundamentalnoe reshenie nachalno-kraevoi zadachi dlya psevdoparabolicheskogo uravneniya chetvertogo poryadka s negladkimi koeffitsientami”, Vladikavkaz. mat. zh., 12:1 (2010), 17–32 | MR | Zbl

[11] Megraliev Ya. T., “Ob odnoi obratnoi kraevoi zadache dlya ellipticheskogo uravneniya vtorogo poryadka s integralnym usloviem pervogo roda”, Tr. in-ta mat. mekh. UrO RAN, 19:1 (2013), 226–235 | MR

[12] Pokhozhaev S. I., “O razreshimosti kvazilineinykh ellipticheskikh uravnenii proizvolnogo poryadka”, Mat. sb., 117:2 (1982), 251–265 | MR | Zbl

[13] Romanov V. G., Obratnye zadachi matematicheskoi fiziki, Nauka, M., 1984

[14] Skrypnik I. V., Nelineinye ellipticheskie uravneniya vysshego poryadka, Naukova dumka, Kiev, 1973

[15] Smirnov M. M., Modelnye uravneniya smeshannogo tipa chetvertogo poryadka, LGU, L., 1972

[16] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980

[17] Turbin M. V., “Issledovanie nachalno-kraevoi zadachi dlya modeli dvizheniya zhidkosti Gershel—Balkli”, Vestn. Voronezh. gos. un-ta. Ser. fiz. mat., 2 (2013), 246–257

[18] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977

[19] Shabrov S. A., “Ob otsenkakh funktsii vliyaniya odnoi matematicheskoi modeli chetvertogo poryadka”, Vestn. Voronezh. gos. un-ta. Ser. fiz. mat., 2 (2015), 168–179 | Zbl

[20] Yuldashev T. K., “Smeshannaya zadacha dlya nelineinogo differentsialnogo uravneniya chetvertogo poryadka s malym parametrom pri parabolicheskom operatore”, Zh. vychisl. mat. mat. fiz., 51:9 (2011), 1703–1711 | MR | Zbl

[21] Yuldashev T. K., “O smeshannoi zadache dlya nelineinogo uravneniya v chastnykh proizvodnykh chetvertogo poryadka s otrazhayuschim otkloneniem”, Vest. Yuzhno-Ural. gos. un-ta. Ser. Mat. Mekh. Fiz., 10 (227) (2011), 40–48 | Zbl

[22] Yuldashev T. K., “Smeshannaya zadacha dlya nelineinogo integro-differentsialnogo uravneniya s parabolicheskim operatorom vysokoi stepeni”, Zh. vychisl. mat. mat. fiz., 52:1 (2012), 112–123 | MR | Zbl

[23] Yuldashev T. K., “Obratnaya zadacha dlya nelineinogo integro-differentsialnogo uravneniya s giperbolicheskim operatorom vysokoi stepeni”, Vestn. Yuzhno-Ural. gos. un-ta. Ser. mat. mekh. fiz., 5:1 (2013), 69–75 | Zbl

[24] Yuldashev T. K., “Obratnaya zadacha dlya odnogo nelineinogo integro-differentsialnogo uravneniya tretego poryadka”, Vest. Samar. gos. un-ta. Ser. estestvennonauch., 1 (2013), 58–66 | Zbl

[25] Yuldashev T. K., “Ob obratnoi zadache dlya nelineinykh integro-differentsialnykh uravnenii vysshego poryadka”, Vestn. Voronezh. gos. un-ta. Ser. Fiz. Mat., 1 (2014), 153–163

[26] Yuldashev T. K., “Ob odnom integro-differentsialnom uravnenii Fredgolma v chastnykh proizvodnykh tretego poryadka”, Izv. vuzov. Ser. Mat., 9 (2015), 74–79 | Zbl

[27] Yuldashev T. K., “Smeshannaya zadacha dlya psevdoparabolicheskogo integro-differentsialnogo uravneniya s vyrozhdennym yadrom”, Differ. uravn., 53:1 (2017), 101–110 | DOI | MR | Zbl

[28] Yuldasheva A. V., “Ob odnoi zadache dlya kvazilineinogo uravneniya chetnogo poryadka”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obzory, 140 (2017), 43–49 | MR

[29] Benney D. J., Luke J. C., “Interactions of permanent waves of finite amplitude”, J. Math. Phys., 43 (1964), 309–313 | DOI | MR | Zbl

[30] Karimov Sh. T., “Multidimensional generalized Erdélyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficients”, Fract. Calc. Appl. Anal., 18:4 (2015), 845–861 | DOI | MR | Zbl