##
**A coefficient determination in nonlocal problem for Boussinesq type integro-differential equation with degenerate kernel.**
*(Russian.
English summary)*
Zbl 1463.35011

Summary: In the three-dimensional domain a Boussinesq type linear integro-differential equation of the fourth order with a restore coefficient and a degenerate kernel is considered. The solution of this integro-differential equation is considered in the class of continuously differentiable functions. First, we study the classical solvability of a nonlocal direct boundary value problem for the considered Boussinesq integro-differential equation with a parameter in the integral term. The method of separation of variables and the method of a degenerate kernels are used. A countable system of algebraic equations is obtained. The solution of this algebraic system of equations for regular values of the spectral parameter in the integral term of a given equation allows us to construct a solution of a non-local direct boundary value problem for an integro-differential equation in the form of a Fourier series. A criterion for the unique solvability of a direct boundary value problem is established for fixed values of the restore function. Using the Cauchy-Bunyakovsky inequality and the Bessel inequality, we prove the absolute and uniform convergence of the obtained Fourier series. The continuity of all the derivatives of the solution of the direct boundary value problem for a given equation is also proved. Further, with the help of an additional integral condition, the restore function is uniquely determined in the form of a Fourier series. The criterion of continuity of second order derivatives of the restore function with respect to space variables is established. Based on the found values of the restore function, the main unknown function is uniquely determined as a solution to the inverse problem for the considering integro-differential equation. In addition, the stability with respect to restore function of the solution of an integro-differential equation is studied.

### MSC:

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

35M10 | PDEs of mixed type |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

### Keywords:

Boussinesq type integro-differential equation; fourth-order equation; degenerate kernel; integral conditions; one valued solvability
PDFBibTeX
XMLCite

\textit{T. K. Yuldashev}, Vladikavkaz. Mat. Zh. 21, No. 2, 67--84 (2019; Zbl 1463.35011)

### References:

[1] | Akhtyamov A. M., Ayupova A. R., “On Solving the Problem of Diagnosing Defects in a Small Cavity in the Rod”, #Middle Volga Mathematical Society Journal, #12:3 (2010), 37-42 (in Russian) · Zbl 1299.74099 |

[2] | Turbin M. V., “Investigation of Initial-Boundary Valye Problem for the Herschel-Bulkley Mathematical Fluid Model”, #Proceedings of Voronezh State University. Series: Physics. Mathematics, 2013, no. 2, 246-257 (in Russian) · Zbl 1475.35292 |

[3] | Whitham G. B., #Linear and Nonlinear Waves, John Wiley & Sons, 1974 · Zbl 0373.76001 |

[4] | Benney D. J., Luke J. C., “Interactions of permanent waves of finite amplitude”, #J. Math. Phys., #43 (1964), 309-313 · Zbl 0128.44601 · doi:10.1002/sapm1964431309 |

[5] | Gordeziani D. G., Avalishvili G. A., “On the Constructing of Solutions of the Nonlocal Initial Boundary Value Problems for One-Dimensional Medium Oscillation Equations”, #Matematicheskoe Modelirovanie, #12:1 (2000), 94-103 (in Russian) · Zbl 1027.74505 |

[6] | Pul’kina L. S., “A Nonlocal Problem for a Hyperbolic Equation with Integral Conditions of the 1st Kind with Time-Dependent Kernels”, #Russian Mathematics, #56:10 (2012), 26-37 · Zbl 1263.35006 · doi:10.3103/s1066369x12100039 |

[7] | Il’in V. A., “The Solvability of Mixed Problems for Hyperbolic and Parabolic Equations”, #Russian Mathematical Surveys, #15:2 (1960), 85-142 · Zbl 0116.29802 · doi:10.1070/rm1960v015n02abeh004217 |

[8] | Lazhetich N. L., “On the Existence of a Classical Solution of a Mixed Problem for a Second-Order One-Dimensional Hyperbolic Equation”, #Differential Equations, #34:5 (1998), 683-695 · Zbl 0999.35054 |

[9] | Chernyatin V. A., #Obosnovanie Metoda Fur’e v Smeshannom Zadache dlya Uravneniy v Chastnykh Proizvodnykh, Moscow State University, M., 1991, 112 pp. (in Russian) |

[10] | Kononenko L. I., “Direct and Inverse Problems for a Singular System with Slow and Fast Variables in Chemical Kinetics”, #Vladikavkaz Math. J., #17:1 (2015), 39-46 (in Russian) · Zbl 1474.34314 · doi:10.23671/VNC.2015.1.7291 |

[11] | Kostin A. B., “The Inverse Problem of Recovering the Source in a Parabolic Equation Under a Condition of Nonlocal Observation”, #Sbornik: Mathematics, #204:10 (2013), 1391-1434 · Zbl 1292.35329 · doi:10.1070/SM2013v204n10ABEH004344 |

[12] | Prilepko A. I., Tkachenko D. S., “Properties of Solutions of a Parabolic Equation and the Uniqueness of the Solution of the Inverse Source Problem with Integral Overdetermination”, #Computational Mathematics and Mathematical Physics, #43:4 (2003), 537-546 · Zbl 1078.35138 |

[13] | Yuldashev T. K., “On the Inverse Problem for the quasilinear Partial Differential Equation of the First Order”, #Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2012, no. 2(18), 56-62 (in Russian) · Zbl 07592904 |

[14] | Yuldashev T. K., “Inverse Problem for Nonlinear Partial Differential Equation with High Order Pseudoparabolic Operator”, #J. Samara State Tech. Univ. Ser. Phys. Math. Sci., 2012, no. 3(28), 17-29 (in Russian) · Zbl 1326.35420 · doi:10.14498/vsgtu1041 |

[15] | Yuldashev T. K., “Inverse Problem for a Nonlinear Integral and Differential Equation of the Third Order”, #Vestnik Samarskogo Universiteta. Estestvenno-Nauchnaya Seriya, 2013 №9/1(110), 58-66 (in Russian) · Zbl 1320.45010 |

[16] | Yuldashev T. K., “Inverse Problem for a Third Order Fredholm Integro-Differential Equation with Degenerate Kernel”, #Vladikavkaz Math. J., #18:2 (2016), 76-85 (in Russian) · Zbl 1445.45004 · doi:10.23671/VNC.2016.2.5921 |

[17] | Polozhiy G. N., #Uravneniya Matematicheskoy Fiziki, Vysshaya Shkola, M., 1964, 560 pp. (in Russian) |

[18] | Nikol’skiy S. M., #Kurs Matematicheskogo Analiza, v. 1, Nauka, M., 1990, 528 pp. (in Russian) |

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.