The problem of determining the piezoelectric module of electroviscoelasticity equation

Abstract

We consider the problem of finding the piezoelectric module e(x 3 ), x 3 > 0, occurring in the system of integro-differential electroviscoelasticity equations. The medium density and the Lamé parameters are assumed to be function of one variable. The integrand K(t), t ∈ [0, T ] is known. As additional information, the Fourier transform of the second component of the displacement vector function for x 3 = 0 is specified. The theorems on the existence of a unique solution of the inverse problem and the stability theorem are the research results.

Memory kernel reconstruction problems in the integro‐differential equation of rigid heat conductor

“…As result, we conclude that if 𝜎 and 𝜌 are taken from conditions 𝜎 > max(𝜎 1 , 𝜎 2 , 𝜎 3 , 𝜎 4 , 𝜎 5 , 𝜎 6 ) and 𝜌 ∈ (0, min(𝜌 1 , 𝜌 2 , 𝜌 3 , 𝜌 4 )), then the operator A carries out contracting mapping the ball S(g 0 , 𝜌) into itself, and according to Banach theorem in this ball, it has a unique fixed point; that is, there exists a unique solution of operator Equation (18). The proof of the theorem is complete.…”

One-dimensional inverse coefficient problems of anisotropic viscoelasticity

“…(2) 1 (ỹ)ỹ ′ (2) . Приводя дроби к общему знаменателю, переходя к разностям аналогично методике работ [1], [10] (в силу громоздкости процедуры выкладки опускаются), учитывая (5.33), (5.34), а также оценки ∥D ) , и применяя лемму Гронуолла к (5.32), получаем требуемую оценку (2.41). Теорема 6 доказана.…”

“…Коэффициентные обратные задачи для гиперболических интегро-дифференциальных уравнений вязкоупругости рассматривались, например, в работах [6][7][8][9][10][11]. В частности, отметим [6], где для системы уравнений вязкоупругости с граничным условием Неймана специального вида решается обратная задача определения четырех неизвестных: плотности ρ(x 3 ), коэффициентов Ламэ λ(x 3 ), µ(x 3 ), ядра K(t) при x 3 > 0, t ≥ 0.…”

A 2D kernel determination problem in a visco‐elastic porous medium with a weakly horizontally inhomogeneity

“…In many cases, the equations describing the propagation of electrodynamic and elastic waves with integral convolution terms are reduced to one second‐order hyperbolic integro‐differential equations. One‐dimensional and multidimensional problems of recovering the kernel of convolution integral in these equations were investigated in previous studies 1-15 . Besides, the works 16-19 that studied the 3D rotating flow of a nanoliquid were examined in the presence of Darcy–Forchheimer porous space and homogeneous–heterogeneous reactions.…”

“…The main feature, appropriate to other works 3-14 and to the present work, is the use of a source localized on the boundary of the considered space domain; this source initiates the physical process of wave transmission. This feature essentially increases the meaning of the investigation for applications.…”