The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type

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.

show abstract

“…Taking into account (20), (21), the coordinate representation of the kinetic energy (19) has the form…”

Section: Resultsmentioning

confidence: 99%

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

et al. 2021

View full text Add to dashboard Cite

show abstract

“…For the last 30 years, there has been much work related to problems of identification of memory kernel in these equations. Here, we mention some of them that are close to this work and the references therein for more details. In Durdiev and Durdiev and Totieva, the local in time existence and the uniqueness results for of some multidimensional inverse problems for the second‐order hyperbolic integro‐differential equations in the class of functions having certain smoothness in the time variable and analyticity with respect to the spatial variables were obtained.…”

Section: Setting Up the Problemmentioning

confidence: 86%

“…Here, we mention some of them that are close to this work and the references therein for more details. In Durdiev and Durdiev and Totieva, the local in time existence and the uniqueness results for of some multidimensional inverse problems for the second‐order hyperbolic integro‐differential equations in the class of functions having certain smoothness in the time variable and analyticity with respect to the spatial variables were obtained. Problems of determining the spatial part of the multidimensional kernel were investigated in the works of Durdiev and Safarov, Lorenzi and Romanov, and Romanov …”

Section: Setting Up the Problemmentioning

confidence: 99%

A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

Durdiev

Totieva

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

The inverse problem of determining 2D spatial part of integral member kernel in integro-differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial-boundary value problem for the half-space x > 0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain ( , t) ∈ R 2 , x > 0. Local existence and uniqueness theorem for the solution to the inverse problem is obtained.The theory and application of hyperbolic integro-differential equations play an important role in the mathematical modeling of many fields: physical, biological phenomens, and engineering sciences, in which it is necessary to take into consideration the effect of real problems. In many cases, PDEs of the electrodynamics and elasticity with integral convolution terms are reduced to one integro-differential equation; the main part of which is second-order hyperbolic operator. For the last 30 years, there has been much work related to problems of identification of memory kernel in these equations. Here, we mention some of them 1-8 that are close to this work and the references therein for more details. In Durdiev and Durdiev and Totieva, 3-5 the local in time existence and the uniqueness results for of some multidimensional inverse problems for the second-order hyperbolic integro-differential equations in the class of functions having certain smoothness in the time variable and analyticity with respect to the spatial variables were obtained. Problems of determining the spatial part of the multidimensional kernel were investigated in the works of Durdiev and Safarov, Lorenzi and Romanov, In this paper, using the method of the work of Romanov, 10 we focus on the inverse problem of recovering a spatial part of a kernel in the integral term of hyperbolic integro-differential equation. The presented results give a convenient approach for numerical solving of the inverse problem.Math Meth Appl Sci. 2019; :7440-7451.

show abstract

“…Unique solvability theorems were proved, and estimates for the conditional stability of the inverse problems under consideration were obtained. Here, also, we refer the reader to Durdiev and Totieva and the monograph, pp81‐130 (see also the references there), in which, for the special class of multidimensional kernels (smooth in the temporal variable and analytic in the part of the space variables), unique local solvability theorems were established using the method of scales of Banach spaces. Note that the idea to apply the method of scales of Banach spaces of analytic functions to the solution of multidimensional inverse problems is due to Romanov ,.…”

Section: Introductionmentioning

confidence: 99%

The problem of determining the one‐dimensional matrix kernel of the system of viscoelasticity equations

Durdiev

Totieva

2018

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

The integro‐differential system of viscoelasticity equations with a source of explosive type is considered. It is assumed that the coefficients of the equations depend only on one spatial variable. The problem of determining the matrix kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to a series of inverse problems for scalar hyperbolic equations. For each case, the inverse problem is replaced by an equivalent system of integral equations for unknown functions. The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied. The theorems of global unique solvability are proved, and the stability estimates of solution to the inverse problems are obtained.

show abstract

The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type

Cited by 30 publications

References 10 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 problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The problem of determining the one‐dimensional matrix kernel of the system of viscoelasticity equations

Contact Info

Product

Resources

About