scholarly journals Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation

2007 ◽  
Vol 46 (5) ◽  
pp. 1849-1881 ◽  
Author(s):  
Assia Benabdallah ◽  
Patricia Gaitan ◽  
Jérôme Le Rousseau
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Diffusion Coefficients ◽  
Initial Conditions ◽  
Discontinuous Diffusion

scholarly journals Inverse problems for the heat equation

2017 ◽  
Vol 21 (6) ◽  
pp. 62-75
Author(s):  
A.R. Zaynullov
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Initial Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Of Solution ◽  
Spatial Coordinates ◽  
Criterion Of Uniqueness ◽  
Initial Boundary Value ◽  
The Right

The inverse problem of finding initial conditions and the right-hand side had been studied for the inhomogeneous heat equation on the basis of formulas for the solution of the first initial-boundary value problem. A criterion of uniqueness of solution of the inverse problem for finding the initial condition was found with Spectral analysis. The right side of the heat equation is represented as a product of two functions, one of which depends on the spatial coordinates and the other from time. In one task, along with an unknown solution is sought factor on the right side, depending on the time, and in another - a factor that depends on the spatial coordinates. For these tasks, we prove uniqueness theorems, the existence and stability of solution.

scholarly journals Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

scholarly journals Internal volumetric heat generation and heat capacity prediction during a material electromagnetic treatment process using hybrid algorithms

2018 ◽  
Vol 38 (1) ◽  
pp. 74-82
Author(s):  
Edgar García-Morantes ◽  
Iván Amaya-Contreras ◽  
Rodrigo Correa-Cely
Keyword(s):  
Inverse Problem ◽  
Heat Capacity ◽  
Electromagnetic Field ◽  
Heat Generation ◽  
Initial Conditions ◽  
Limited Range ◽  
First Case ◽  
Realistic Situation ◽  
Wide Range ◽  
Volumetric Heat Generation

This work considers the estimation of internal volumetric heat generation, as well as the heat capacity of a solid spherical sample, heated by a homogeneous, time-varying electromagnetic field. To that end, the numerical strategy solves the corresponding inverse problem. Three functional forms (linear, sinusoidal, and exponential) for the electromagnetic field were considered. White Gaussian noise was incorporated into the theoretical temperature profile (i.e. the solution of the direct problem) to simulate a more realistic situation. Temperature was pretended to be read through four sensors. The inverse problem was solved through three different kinds of approach: using a traditional optimizer, using modern techniques, and using a mixture of both. In the first case, we used a traditional, deterministic Levenberg-Marquardt (LM) algorithm. In the second one, we considered three stochastic algorithms: Spiral Optimization Algorithm (SOA), Vortex Search (VS), and Weighted Attraction Method (WAM). In the final case, we proposed a hybrid between LM and the metaheuristics algorithms. Results show that LM converges to the expected solutions only if the initial conditions (IC) are within a limited range. Oppositely, metaheuristics converge in a wide range of IC but exhibit low accuracy. The hybrid approaches converge and improve the accuracy obtained with the metaheuristics. The difference between expected and obtained values, as well as the RMS errors, are reported and compared for all three methods.

scholarly journals Identifying the heat sink

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
J. D. Audu ◽  
A. Boumenir ◽  
K. M. Furati ◽  
I. O. Sarumi
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Temperature Profile ◽  
Spectral Methods ◽  
Heat Sink ◽  
Evolution Equations ◽  
Identification Problem ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Pseudo Spectral

<p style='text-indent:20px;'>In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.</p>

Learning solutions to the source inverse problem of wave equations using LS-SVM

2019 ◽  
Vol 27 (5) ◽  
pp. 657-669 ◽  
Author(s):  
Ziku Wu ◽  
Chang Ding ◽  
Guofeng Li ◽  
Xiaoming Han ◽  
Juan Li
Keyword(s):  
Inverse Problem ◽  
Approximate Solution ◽  
Support Vector Machines ◽  
Wave Equations ◽  
Source Term ◽  
Initial Conditions ◽  
Support Vector ◽  
Vector Machines ◽  
Robustness To Noise ◽  
The Given

Abstract A method based on least squares support vector machines (LS-SVM) is proposed to solve the source inverse problem of wave equations. Contrary to the most existing methods, the proposed method provides a closed form approximate solution which satisfies the boundary conditions and the initial conditions. The proposed method can recover the unknown source term with the given additional conditions. Furthermore, it has reasonable robustness to noise. Numerical results show the proposed method can be used to solve the source inverse problem of wave equations.

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