scholarly journals An inverse problem of reconstructing the time-dependent coefficient in a one-dimensional hyperbolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. J. Huntul ◽  
Muhammad Abbas ◽  
Dumitru Baleanu
Keyword(s):  
Inverse Problem ◽  
Input Data ◽  
Time Dependent ◽  
One Dimensional ◽  
Noisy Input ◽  
Nonlinear Minimization ◽  
Ill Posed ◽  
Dependent Potential ◽  
The Stability ◽  
First Time

AbstractIn this paper, for the first time the inverse problem of reconstructing the time-dependent potential (TDP) and displacement distribution in the hyperbolic problem with periodic boundary conditions (BCs) and nonlocal initial supplemented by over-determination measurement is numerically investigated. Though the inverse problem under consideration is ill-posed by being unstable to noise in the input data, it has a unique solution. The Crank–Nicolson-finite difference method (CN-FDM) along with the Tikhonov regularization (TR) is applied for calculating an accurate and stable numerical solution. The programming language MATLAB built-in lsqnonlin is used to solve the obtained nonlinear minimization problem. The simulated noisy input data can be inverted by both analytical and numerically simulated. The obtained results show that they are accurate and stable. The stability analysis is performed by using Fourier series.

scholarly journals INVERSE PROBLEM OF A ONE-DIMENSIONAL MODEL IN MULTILAYER HEAT CONDUCTION

2020 ◽  
Vol 19 (1) ◽  
pp. 42
Author(s):  
G. C. Oliveira ◽  
S. S. Ribeiroa ◽  
G. Guimarães
Keyword(s):  
Inverse Problem ◽  
Measurement Errors ◽  
Input Data ◽  
Temperature Measurements ◽  
Dimensional Model ◽  
One Dimensional ◽  
Ill Posed ◽  
Measurements Errors ◽  
Chip Chip

The inverse problem in conducting heat is related to the determination of the boundary condition, rate of heat generation, or thermophysical properties, using temperature measurements at one or more positions of the solid. The inverse problem in conducting heat is mathematically one of the ill-posed problems, because its solution extremely sensitive to measurement errors. For a well-placed problem the following conditions must be satisfied: the solution must exist, it must be unique and must be stable on small changes of the input data. The objective of the work is to estimate the heat flux generated at the tool-chip-chip interface in a manufacturing process. The term "estimation" is used because in the temperature measurements, errors are always present and these affect the accuracy of the calculation of the heat flow.

scholarly journals Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mousa Huntul ◽  
Daniel Lesnic
Keyword(s):  
Inverse Problem ◽  
Free Boundary ◽  
Numerical Solution ◽  
Input Data ◽  
Free Boundary Problems ◽  
Time Dependent ◽  
Two Dimensional ◽  
Convection Coefficient ◽  
Content Type ◽  
Noisy Input

Purpose The purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. Design/methodology For the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Findings The numerical results demonstrate that accurate and stable solutions are obtained. Originality/value The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

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

2020 ◽  
Vol 28 (1) ◽  
pp. 43-52
Author(s):  
Durdimurod Kalandarovich Durdiev ◽  
Zhanna Dmitrievna Totieva
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equations ◽  
Stability Estimate ◽  
Continuous Functions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Weighted Norms ◽  
The Stability ◽  
The One ◽  
Global Unique Solvability

AbstractThe 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 kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to one inverse problem for scalar hyperbolic equations. This 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 theorem of global unique solvability is proved and the stability estimate of solution to the inverse problem is obtained.

scholarly journals Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 809-814 ◽  
Author(s):  
Makhmud Sadybekov ◽  
Gulaiym Oralsyn ◽  
Mansur Ismailov
Keyword(s):  
Inverse Problem ◽  
Heat Source ◽  
Input Data ◽  
Nonlocal Boundary ◽  
Fourier Method ◽  
Time Dependent ◽  
Regular Boundary ◽  
Associated Functions ◽  
Integral Overdetermination

We investigate an inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is not complete. But the system of eigen- and associated functions forming a basis. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method.

scholarly journals A note on inverse problem for strongly damped wave equation with Gaussian white noise

2018 ◽  
Vol 20 ◽  
pp. 02003
Author(s):  
Chu Duc Khanh ◽  
Nguyen Hoang Luc ◽  
Van Phan ◽  
Nguyen Huy Tuan
Keyword(s):  
White Noise ◽  
A Priori ◽  
Gaussian White Noise ◽  
True Solution ◽  
One Dimensional ◽  
Noise Data ◽  
Ill Posed ◽  
The One ◽  
First Time ◽  
Strongly Damped

In this paper, we study for the first time the inverse initial problem for the one-dimensional strongly damped wave with Gaussian white noise data. Under some a priori assumptions on the true solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Error estimates are given in both the L2– and Hp–norms.

scholarly journals Sparse Bayesian mass mapping with uncertainties: peak statistics and feature locations

2019 ◽  
Vol 489 (3) ◽  
pp. 3236-3250 ◽  
Author(s):  
M A Price ◽  
J D McEwen ◽  
X Cai ◽  
T D Kitching (for the LSST Dark Energy Science Collaboration)
Keyword(s):  
Inverse Problem ◽  
Large Scale ◽  
Signal To Noise Ratio ◽  
Statistical Interpretation ◽  
Confidence Levels ◽  
Ill Posed ◽  
Mass Mapping ◽  
Approximate Bayesian ◽  
Lower Limits ◽  
First Time

ABSTRACT Weak lensing convergence maps – upon which higher order statistics can be calculated – can be recovered from observations of the shear field by solving the lensing inverse problem. For typical surveys this inverse problem is ill-posed (often seriously) leading to substantial uncertainty on the recovered convergence maps. In this paper we propose novel methods for quantifying the Bayesian uncertainty in the location of recovered features and the uncertainty in the cumulative peak statistic – the peak count as a function of signal-to-noise ratio (SNR). We adopt the sparse hierarchical Bayesian mass-mapping framework developed in previous work, which provides robust reconstructions and principled statistical interpretation of reconstructed convergence maps without the need to assume or impose Gaussianity. We demonstrate our uncertainty quantification techniques on both Bolshoi N-body (cluster scale) and Buzzard V-1.6 (large-scale structure) N-body simulations. For the first time, this methodology allows one to recover approximate Bayesian upper and lower limits on the cumulative peak statistic at well-defined confidence levels.

scholarly journals Moving Least Squares Method for a One-Dimensional Parabolic Inverse Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Baiyu Wang
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
High Efficiency ◽  
Least Squares Method ◽  
Parabolic Problems ◽  
Moving Least Squares ◽  
One Dimensional ◽  
Moving Least Squares Method ◽  
The Stability

This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.

Sign in / Sign up

Export Citation Format

Share Document