Inverse Boundary-Value Problem for an Integro-Differential Boussinesq-Type Equation with Degenerate Kernel

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INVERSE BOUNDARY-VALUE PROBLEM FOR AN INTEGRO-DIFFERENTIAL BOUSSINESQ-TYPE EQUATION WITH DEGENERATE KERNEL UDC 517.968

T. K. Yuldashev

Abstract. 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 unique solvability of the problem and establish a criterion of stability of solutions with respect to recovery functions. Keywords and phrases: inverse boundary-value problem, integro-differential Boussinesq-type equation, degenerate kernel, unique solvability. AMS Subject Classification: 35R30

1. Statement of the problem. Mathematical modeling of various processes occurring in the real world often leads to the study of direct and inverse problems for equations that have no analogs in classical mathematical physics. From the point of view of physical applications, higher-order partial differential equations are of great interest. Numerous problems of gas dynamics, elasticity theory, theory of plates and shells etc. can be reduced to higher-order partial differential equations (see [3, 4, 30]). Higher-order partial differential and integro-differential equations were studied by many authors (see, e.g., [1, 2, 5, 7–9, 11, 13, 15–17, 19, 21–24, 29]). Partial differential equations of the Boussinesq type have many applications in mathematical physics (see [20]). Theory of inverse problems is an actively developing direction of the modern theory of differential equations. Inverse problems were examined, for example, in [6, 10, 12, 14, 25, 26]). The method of degenerate kernel for partial integro-differential equations was developed in [27, 28]. In this paper, we propose a method of analysis of an inverse boundary-value problem for a fourthorder, nonlinear integro-differential equation of the Boussinesq type with degenerate kernel. In a domain Ω, consider the equation ∂ 2 U (t, x) ∂ 4 U (t, x) − −μ ∂t2 ∂t2 ∂x2

T 0

 ∂ 2 U (s, x) K(t, s) ds = η(t)β(x) + f x, β(x), ∂x2

T

 H(θ)U (θ, x)dθ

(1)

0

with the boundary conditions

  U (t, x)t=0 = ϕ1 (x), U (t, x)t=T = ϕ2 (x),   U (t, x)x=0 = U (t, x)x=l = 0,

x ∈ Ωl ,

(2)

t ∈ ΩT

(3)

and the additional conditions T Θ(s)U (s, x)ds = ψ(x), 0

 Ut (t, x)t=T = r(x),

x ∈ Ωl ,

x ∈ Ωl ,

(4) (5)

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2505–0847 

847

where η(t) ∈ C(ΩT ), f (x, β, γ) ∈ C(Ωl × R × R), β(x) ∈ C(Ωl ) is the first recovery function, H(t) ∈ C(ΩT ), ϕ2 (x) is the second recovery function, K(t, s) =

m 

0 < ai (t), bi (s) ∈ C(ΩT ),

ai (t)bi (s),

ψ(

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