Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

Durdiev, D. K.; Rahmonov, A. A.

doi:10.1134/s0040577918060090

Cited by 36 publications

(15 citation statements)

References 3 publications

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“…From the conditions (22), it follows that the center of mass of the load does not lie in the plane of the centerline or the centrifugal moments of inertia J zx . J zy are different from zero, then the flexural-torsional and flexural-longitudinal vibrations are related to each other even if the corresponding unknowns in equations ( 12), ( 13) are separated.…”

Section: Resultsmentioning

confidence: 99%

Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties

et al. 2021

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This article is dedicated to the operation and management of systems of machine-building and aviation enterprises, systems of production, transport, storage of oil and gas, issues of control of technological processes are of great importance. Control of technological processes is carried out by monitoring the pressure and other parameters. These measuring instruments must have high reliability and the necessary accuracy. In this connection, there is a sharp increase in interest in determining the dynamic parameters of the elements of measuring devices. The main elements of such devices are monomeric tubular springs (Bourdon tubes). The paper considers the natural and forced steady-state oscillations of a thin curved rod interacting with a liquid. Based on the principle of possible displacements, a resolving system of partial differential equations and the corresponding boundary conditions are obtained. The problem is solved numerically by the Godunov orthogonal run method, and the Muller method and the Eigen frequencies found are compared with the experimental results. As a result, for a given axial perturbation, it was possible to select such an effect, in the orthogonal direction, that the amplitude of the longitudinal vibrations of the rod at the first resonance decreased by 20 times. The described vibration damping effect is due to the interrelation of transverse and longitudinal vibrations and is fundamentally impossible in the case of a straight rod.

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Section: Resultsmentioning

confidence: 99%

Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties

et al. 2021

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“…Besides, we use formulas () and (), which follow from the definition of the functions q , g and relation (). Under the condition of the theorem 2.1, the equivalence of the system of integral equations () to () and the inverse problem in equations () to () can be proved in the standard way 13 . The lemma is proved.…”

Section: Reducing the Inverse Problem To An Equivalent System Of Intementioning

confidence: 94%

“…We note that the direct and inverse dynamical problems for these equations for k ( x , t )=0 were studied in previous works 22,23 . This formulation of the inverse problem without taking into account the dependence of the kernel K on x and the derivative with respect to x on Σ 1 is equal to zero was considered in Durdiev and Rahmonov 13 . When solving, the principle of contracting mapping in the space of continuous functions with weight norm was applied to the system of integral equations, which is equivalent to the inverse problem.…”

Section: Introduction and Setting Up The Problemmentioning

confidence: 98%

“…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.…”

Section: Introduction and Setting Up The Problemmentioning

confidence: 99%

“…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.…”

Section: Introduction and Setting Up The Problemmentioning

confidence: 99%

See 2 more Smart Citations

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

Durdiev

Rahmonov

2020

Math Methods in App Sciences

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We consider a system of hyperbolic integro‐differential equations of SH waves in a visco‐elastic porous medium. In this work, it is assumed that the visco‐elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial‐boundary problem: the initial data is equal to zero, and the Neumann‐type boundary condition is specified at the half‐plane boundary and is an impulse function. As additional information, the oscillation mode of the half‐plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K0(t)+ϵxK1(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K0,K1 up to a correction of the order of O(ϵ2).

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

Durdiev

Zhumaev

2020

Math Methods in App Sciences

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The inverse problems of determining the energy–temperature relation α(t) and the heat conduction relation k(t) functions in the one‐dimensional integro‐differential heat equation are investigated. The direct problem is the initial‐boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problems, the solution of the direct problem for x = x0 is given. At the beginning, an auxiliary problem, which is equivalent to the original problem, is introduced. Then the auxiliary problem is reduced to an equivalent closed system of Volterra‐type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions with weighted norms, we prove the main result of the article, which is a global existence and uniqueness theorem of inverse problem solutions.

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Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

Cited by 36 publications

References 3 publications

Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties

Manometric Tubular Springs Oscillatory Processes Modeling with Consideration of its Viscoelastic Properties

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

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

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