Identification of a kernel in an evolutionary integral equation occurring in subdiffusion

2017 ◽  
Vol 25 (6) ◽  
Author(s):  
Jaan Janno ◽  
Kairi Kasemets
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Abstract Setting ◽  
Stability Of A Solution

AbstractAn inverse problem to determine a kernel in an evolutionary integral equation occurring in modeling of subdiffusion is considered. The existence, uniqueness and stability of a solution of the inverse problem are proved in an abstract setting. The results are global in time.

scholarly journals Inverse problem: acoustic potential vs acoustic length

1988 ◽  
Vol 123 ◽  
pp. 133-136
Author(s):  
Hiromoto Shibahashi
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Sound Velocity ◽  
Internal Structure ◽  
Asymptotic Method ◽  
The Sun ◽  
Quantization Rule ◽  
Rule Based ◽  
Deep Interior

By using the quantization rule based on the WKB asymptotic method, we present an integral equation to infer the form of the acoustic potential of a fixed ℓ as a function of the acoustic length. Since we analyze the acoustic potential itself by taking account of some factors other than the sound velocity and we can analyze the radial modes by this scheme as well as nonradial modes, this method improves the accuracy and effectiveness of the inverse problem to infer the internal structure of the Sun, in particular, the deep interior of the Sun.

scholarly journals Integral equation methods for the inverse problem with discontinuous wave speed

1996 ◽  
Vol 37 (7) ◽  
pp. 3218-3245 ◽  
Author(s):  
Tuncay Aktosun ◽  
Martin Klaus ◽  
Cornelis van der Mee
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Wave Speed ◽  
Discontinuous Wave ◽  
Integral Equation Methods

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 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.

A series expansion approach to the inverse problem

1998 ◽  
Vol 35 (2) ◽  
pp. 371-382 ◽  
Author(s):  
J. M. Angulo ◽  
M. D. Ruiz-Medina
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Random Field ◽  
Series Expansion ◽  
Random Fields ◽  
Stochastic Integral ◽  
Random Coefficients ◽  
Stochastic Integral Equation ◽  
Piezometric Head ◽  
Original Motivation

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

A series expansion approach to the inverse problem

1998 ◽  
Vol 35 (02) ◽  
pp. 371-382 ◽  
Author(s):  
J. M. Angulo ◽  
M. D. Ruiz-Medina
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Random Field ◽  
Series Expansion ◽  
Random Fields ◽  
Stochastic Integral ◽  
Random Coefficients ◽  
Stochastic Integral Equation ◽  
Piezometric Head ◽  
Original Motivation

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

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