Inverse Problem for an Integro-Differential Equation of Acoustics

Safarov, Zh.Sh.; Durdiev, D. K.

doi:10.1134/s0012266118010111

Cited by 21 publications

(3 citation statements)

References 3 publications

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“…Пусть вектор-функции φ(y,x 2 ,x 3 ), g(x 2 ,x 3 ,t), h(x 2 ,x 3 ,t), входящие в данные ( 6), ( 7), (8), финитны по x 2 , x 3 при каждом фиксированном y, t. Тогда, в силу гиперболичности системы (5), таким свойством обладает и решение задачи ( 5)-(7) (см. например [26, с.…”

Section: постановка задачиunclassified

“…В настоящее время достаточно хорошо изучены обратные задачи определения ядра из одного гиперболического интегро-дифференциального уравнения второго порядка. В работах [4][5][6][7][8] исследовались одномерные задачи нахождения свёрточного ядра: из общего волнового уравнения [5,6] и из уравнений вязкоупругости [7][8][9][10][11][12][13]. Основными результатами этих работ являются теоремы однозначной разрешимости поставленных задач.…”

Section: Introductionunclassified

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The problem of finding the kernels in the system of integro-differential acoustics equations

Durdiev,

Turdiev

2023

FEMJ

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For reduced to the canonical system of integro-differential equations of acoustics, a direct problem is posed, which consists in determining the velocity of the perturbed medium and the pressure and the inverse problem of finding the diagonal memory matrix. The problems are reduced to a closed system of integral equations of the second kind of the Volterra type with respect to the solution of the direct problem and unknowns of the inverse problem. The method of contraction mappings in the space of continuous functions with an exponential weighted norm is applied to this system. Existence and uniqueness theorems for solutions to problems in the global sense are proved.

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Section: постановка задачиunclassified

Section: Introductionunclassified

The problem of finding the kernels in the system of integro-differential acoustics equations

Durdiev,

Turdiev

2023

FEMJ

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“…In many cases equations that describe propagation of electrodynamic and elastic waves with integral convolution are reduced to one second-order hyperbolic integro-differential equation. Various problems of recovering the kernel of convolution integral in these equations were investigated [1][2][3][4][5][6][7][8][9][10][11][12]. Determination of time-and space-dependent kernels in parabolic integro-differential equations with several additional conditions was considered by many authors (see e.g., [13][14][15][16][17][18][19][20][21][22][23]).…”

Section: Introduction Formulation Of Problemmentioning

confidence: 99%

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Durdiev

Nuriddinov

2021

Journal of Siberian Federal University. Mathematics &Amp; Physics

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The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

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