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 Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

2016 ◽  
Vol 11 (10) ◽  
pp. 5705-5714
Author(s):  
Abeer Majed AL-Bugami
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Two Dimensional ◽  
Block Method ◽  
Numerical Examples ◽  
Block Methods ◽  
Runge Kutta ◽  
Method R

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

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 Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Ke ◽  
Guo Jiang ◽  
Mengting Deng
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Least Squares ◽  
Algebraic Equation ◽  
Volterra Integral Equation ◽  
Linear Algebraic Equation ◽  
Least Squares Method ◽  
Numerical Examples ◽  
Block Pulse Function

In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

scholarly journals Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 223
Author(s):  
Pedro González-Rodelas ◽  
Miguel Pasadas ◽  
Abdelouahed Kouibia ◽  
Basim Mustafa
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Volterra Integral Equation ◽  
Computational Cost ◽  
Basis Functions ◽  
Numerical Examples ◽  
Acceptable Accuracy ◽  
Convergence Results ◽  
Radial Basis ◽  
Low Computational Cost

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

scholarly journals Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

2015 ◽  
Vol 11 (5) ◽  
pp. 5220-5229
Author(s):  
Abeer Majed AL-Bugami
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Two Dimensional ◽  
Block Method ◽  
Numerical Examples ◽  
Block Methods ◽  
Runge Kutta ◽  
Method R

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

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 ON THE INVERSE PROBLEM OF FINDING THE RIGHT-HAND SIDE OF WAVE EQUATION WITH NONLOCAL CONDITION

2017 ◽  
pp. 16-25
Author(s):  
Hamlet Farman GULIYEV ◽  
◽  
Yusif Soltan GASIMOV ◽  
Hikmet Tahir TAGIYEV ◽  
Tunzale Maharram GUSEYNOVA ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Nonlocal Condition ◽  
Right Hand ◽  
The Right

An Inverse Problem in Elasticity With Partially Overprescribed Boundary Conditions, Part I: Theoretical Approach

1993 ◽  
Vol 60 (3) ◽  
pp. 595-600 ◽  
Author(s):  
Weichung Yeih ◽  
Tatsuhito Koya ◽  
Toshio Mura
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Linear Elasticity ◽  
Fredholm Integral Equation ◽  
Theoretical Approach ◽  
Regularization Method ◽  
Numerical Examples

A Cauchy problem in linear elasticity is considered. This problem is governed by a Fredholm integral equation of the first kind and cannot be solved directly. The regularization method, which has been originally employed by Gao and Mura (1989), is formulated from a different perspective in order to address some of the difficulties experienced in their formulation. The theoretical details are discussed in this paper. Numerical examples are treated to Part II.

Fast numerical method of solving 3D coefficient inverse problem for wave equation with integral data

2018 ◽  
Vol 26 (4) ◽  
pp. 477-492 ◽  
Author(s):  
Anatoly B. Bakushinsky ◽  
Alexander S. Leonov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Wave Equation ◽  
Wave Field ◽  
A Priori ◽  
Three Dimensional ◽  
Simulated Data ◽  
Three Dimensions ◽  
Unknown Coefficient ◽  
Computational Resources

Abstract An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

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