scholarly journals On a nonlinear inverse problem in viscoelasticity

2009 ◽  
Vol 31 (3-4) ◽  
Author(s):  
H. D. Bui ◽  
S. Chaillat
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Boundary Data ◽  
Viscoelastic Body ◽  
Low Frequency ◽  
Cyclic Loads ◽  
Nonlinear Inverse Problem ◽  
Method Of Solution ◽  
The One

We consider an inverse problem for determining an inhomogeneity in a viscoelastic body of the Zener type, using Cauchy boundary data, under cyclic loads at low frequency. We show that the inverse problem reduces to the one for the Helmholtz equation and to the same nonlinear Calderon equation given for the harmonic case. A method of solution is proposed which consists in two steps: solution of a source inverse problem, then solution of a linear Volterra integral equation.

scholarly journals Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem

2021 ◽  
Vol 2092 (1) ◽  
pp. 012022
Author(s):  
Sergey I. Kabanikhin ◽  
Nikita S. Novikov ◽  
Maxim A. Shishlenin
Keyword(s):  
Inverse Problem ◽  
Integral Equations ◽  
Nonlinear Inverse Problem ◽  
Inverse Coefficient Problem ◽  
One Dimensional ◽  
The One ◽  
Linear Integral ◽  
Fast Inversion ◽  
Linear Integral Equations ◽  
Inverse Coefficient

Abstract In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

scholarly journals Inverse problem on the semi-axis: local approach

2011 ◽  
Vol 42 (3) ◽  
pp. 275-293
Author(s):  
S. A. Avdonin ◽  
B. P. Belinskiy ◽  
John V. Matthews
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Wave Equation ◽  
Volterra Integral Equation ◽  
Accurate Data ◽  
Data Preparation ◽  
Local Approach ◽  
Numerical Examples ◽  
Linear Version ◽  
The Right

We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

scholarly journals On the Analysis of Numerical Methods for Nonstandard Volterra Integral Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
H. S. Mamba ◽  
M. Khumalo
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Trapezoidal Rule ◽  
Collocation Methods ◽  
Point Collocation ◽  
The One ◽  
Theoretical Results

We consider the numerical solutions of a class of nonlinear (nonstandard) Volterra integral equation. We prove the existence and uniqueness of the one point collocation solutions and the solution by the repeated trapezoidal rule for the nonlinear Volterra integral equation. We analyze the convergence of the collocation methods and the repeated trapezoidal rule. Numerical experiments are used to illustrate theoretical results.

Remarks on approximate methods in geophysical inverse problems

1974 ◽  
Vol 337 (1608) ◽  
pp. 49-71 ◽  
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Inverse Problems ◽  
Linear Method ◽  
Dirichlet Kernel ◽  
Nonlinear Inverse Problem ◽  
Approximate Methods ◽  
Physical Constraints ◽  
Exactly Solvable ◽  
Approximate Expressions

Several remarks are made on well-known methods used in geophysics to analyse inverse problems. Approximate values are given for Backus and Gilbert kernels, together with an integral equation which enables one to derive then from a Dirichlet kernel. Ways to obtain approximate expressions of the Dirichlet kernels are then studied. Consequences of linearizing a nonlinear inverse problem are analysed both in general and with an exactly solvable example. This example shows situations in which the information derived by way of the linear method is irrelevant or misleading. Some simple remarks on the inference of physical constraints conclude the paper.

scholarly journals Problem of describing the function of a GPR source

2020 ◽  
Vol 100 (4) ◽  
pp. 71-80
Author(s):  
S.I. Kabanikhin ◽  
◽  
K.T. Iskakov ◽  
D.K. Tokseit ◽  
M.A. Shishlenin ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Electromagnetic Waves ◽  
Source Function ◽  
Homogeneous Medium ◽  
Delta Function ◽  
Pulse Characteristic ◽  
Pulse Source ◽  
Dirac Delta

In this paper, we consider the problem of determining the source h(t)δ(x) of electromagnetic waves from GPR data. The task of electromagnetic sensing is to find the pulse characteristic of the medium r(t) and consists in calculating the response of the medium to the pulse source of excitation δ(t) (Dirac Delta function). To determine the analytical expression of the impulse response of a homogeneous medium r(t), we use the method proposed in [1-2]. To determine h(t), the inverse problem is reduced to a system of Volterra integral equations. The source function h(τ), is defined as the solution of the Volterra integral equation of the first kind, f(t) = \int^t_0 r(t−τ)h(τ)dτ in which f(t) is the data obtained by the GPR at the observation points. The problem of calculating the function of the GPR source h(τ ) consists in numerically solving the inverse problem, in which the function of the source h(τ ) is unknown, and the electromagnetic parameters of the medium are known: the permittivity ε; the conductivity σ; the magnetic permeability µ and the response of the medium to a given excitation h(τ).

scholarly journals Synthesis of the parameters of the elastic-damping device in the excavator’s digging mechanism based on the use of the integral equation

2021 ◽  
Vol 326 ◽  
pp. 00026
Author(s):  
Nikolay Kuznetsov ◽  
Ivan Iov ◽  
Evgeniy Dolgih
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Inverse Problem ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Transition Process ◽  
Dynamic Loads ◽  
Natural Oscillations ◽  
Device Efficiency

The article is devoted to the issues of the synthesis of the parameters of an elastic-damping device built into the excavator’s digging mechanism to reduce dynamic loads. The determination of the parameters of this device was carried out by solving the inverse problem of dynamics using the Volterra integral equation of the second kind, which allowed linking these parameters to the nature of the transition process without finding the frequencies of natural oscillations. The transition from the differential equation of the oscillatory motion of the digging mechanism to the corresponding integral equation has been carried out, and the parameters of its resolvent and the elastic-damping device that ensure a decrease in dynamic loads are determined. The results of numerical simulation of the elastic-damping device efficiency based on the use of the characteristics of the real excavator’s digging mechanism are presented.

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