scholarly journals On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

2020 ◽  
Vol 30 (4) ◽  
pp. 572-584
Author(s):  
D.K. Durdiev ◽  
J.Z. Nuriddinov
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Variable Coefficient ◽  
Direct Problem ◽  
Local Existence ◽  
Problem Solution ◽  
Volterra Type ◽  
Integral Term ◽  
Local Existence And Uniqueness ◽  
Coefficient Of Thermal Conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

Heat distribution in a plane with a crack with a variable coefficient of thermal conductivity

2016 ◽  
Vol 98 (4) ◽  
pp. 285-307 ◽  
Author(s):  
Andrey V. Glushko ◽  
Aleksandr S. Ryabenko ◽  
Vera E. Petrova ◽  
Ekaterina A. Loginova
Keyword(s):  
Thermal Conductivity ◽  
Variable Coefficient ◽  
Heat Distribution ◽  
Coefficient Of Thermal Conductivity

scholarly journals An abstract inverse problem

1991 ◽  
Vol 4 (2) ◽  
pp. 117-128 ◽  
Author(s):  
M. Choulli
Keyword(s):  
Inverse Problem ◽  
Uniqueness Theorem ◽  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Existence And Uniqueness Theorem ◽  
Local Existence And Uniqueness ◽  
Abstract Integrodifferential Equation

In this paper we consider an inverse problem that corresponds to an abstract integrodifferential equation. First, we prove a local existence and uniqueness theorem. We also show that every continuous solution can be locally extended in a unique way. Finally, we give sufficient conditions for the existence and a stability of the global solution.

Combining Integral Transforms and Bayesian Inference in the Simultaneous Identification of Variable Thermal Conductivity and Thermal Capacity in Heterogeneous Media

2011 ◽  
Vol 133 (11) ◽  
Author(s):  
Carolina P. Naveira-Cotta ◽  
Helcio R. B. Orlande ◽  
Renato M. Cotta
Keyword(s):  
Thermal Conductivity ◽  
Inverse Analysis ◽  
Direct Problem ◽  
Heterogeneous Media ◽  
Integral Transform ◽  
Integral Transforms ◽  
Variable Thermal Conductivity ◽  
Thermal Capacity ◽  
Problem Analysis ◽  
Problem Solution

This work presents the combined use of the integral transform method, for the direct problem solution, and of Bayesian inference, for the inverse problem analysis, in the simultaneous estimation of spatially variable thermal conductivity and thermal capacity for one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field. The inverse problem analysis employs a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings sampling algorithm. Instead of seeking the functions estimation in the form of local values for the thermal conductivity and capacity, an alternative approach is employed based on the eigenfunction expansion of the thermophysical properties themselves. Then, the unknown parameters become the corresponding series coefficients for the properties eigenfunction expansions. Simulated temperatures obtained via integral transforms are used in the inverse analysis, for a prescribed concentration distribution of the dispersed phase in a heterogeneous media such as particle filled composites. Available correlations for the thermal conductivity and theory of mixtures relations for the thermal capacity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the inverse crime. Gaussian distributions were used as priors for the parameter estimation procedure. In addition, simulated results with different randomly generated errors were employed in order to test the inverse analysis robustness.

scholarly journals An Inverse Problem Solution for Thermal Conductivity Reconstruction

2021 ◽  
Vol 20 ◽  
pp. 187-195
Author(s):  
Tchavdar T. Marinov ◽  
Rossitza S. Marinova
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Boundary Value Problem ◽  
Thermal Conductivity Coefficient ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Nonlinear Coefficient ◽  
Higher Order ◽  
Quadratic Functional ◽  
Problem Solution

This work deals with the inverse problem of reconstructing the thermal conductivity coefficient of the (2+1)D heat equation from over–posed data at the boundaries. The proposed solution uses a variational approach for identifying the coefficient. The inverse problem is reformulated as a higher–order elliptic boundary–value problem for minimization of a quadratic functional of the original equation. The resulting system consists of a well–posed fourth–order boundary–value problem for the temperature and an explicit equation for the unknown thermal conductivity coefficient. The existence and uniqueness of the resulting higher–order boundary–value problem are investigated. The unique solvability of the inverse coefficient problem is proven. The numerical algorithm is validated and applied to problems of reconstructing continuous nonlinear coefficient and discontinuous coefficients. Accurate and stable numerical solutions are obtained.

scholarly journals Reconstruction of the thermal conductivity coefficient in the space fractional heat conduction equation

2017 ◽  
Vol 21 (1 Part A) ◽  
pp. 81-88 ◽  
Author(s):  
Rafał Brociek ◽  
Damian Słota
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Heat Conduction ◽  
Direct Problem ◽  
Heat Conduction Equation ◽  
Thermal Conductivity Coefficient ◽  
Conduction Equation ◽  
Liouville Fractional Derivative ◽  
Fractional Heat Conduction

In this paper an inverse problem for the space fractional heat conduction equation is investigated. Firstly, we describe the approximate solution of the direct problem. Secondly, for the inverse problem part, we define the functional illustrating the error of approximate solution. To recover the thermal conductivity coefficient we need to minimize this functional. In order to minimize this functional the Real Ant Colony Optimization (RealACO) algorithm is used. In the model we apply the Riemann-Liouville fractional derivative. The paper presents also some examples to illustrate the accuracy and stability of the presented algorithm.

scholarly journals On an Inverse Problem for the Stationary Equation with a Boundary Condition of the Third Type

2021 ◽  
Vol 14 (5) ◽  
pp. 659-666
Author(s):  
Alexander V. Velisevich ◽  
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Order Differential Equation ◽  
Local Existence ◽  
Unknown Coefficient ◽  
Lower Term ◽  
Stationary Equation ◽  
Integral Boundary ◽  
The Third ◽  
Local Existence And Uniqueness

The identification of an unknown coefficient in the lower term of elliptic second-order differential equation Mu + ku = f with the boundary condition of the third type is considered. The identification of the coefficient is based on integral boundary data. The local existence and uniqueness of the strong solution for the inverse problem is proved

scholarly journals INVERSE PROBLEM FOR INTEGRO-DIFFERENTIAL HEAT EQUATION WI TION WITH A VARIABLE COEFFICIEN ABLE COEFFICIENT OF THERM T OF THERMAL CONDUCTIVITY

2020 ◽  
Vol 4 (5) ◽  
pp. 3-12
Author(s):  
Durdimurod Kalandarovich Durdiev ◽  
◽  
Zhavlon Zafarovich Nuriddinov
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Heat Equation ◽  
Contraction Mapping ◽  
Cauchy Problems ◽  
Convolution Integral ◽  
Differential Heat ◽  
Memory Kernel ◽  
Coefficient Of Thermal Conductivity ◽  
Type Integral

Background. The inverse problem of finding a multidimensional memory kernel of a time convolution integral depending on a time variable t and (n-1)-dimensional spatial variable. 2-dimensional heat equation with a time-dependent coefficient of thermal conductivity is studied. Methods. The article is used Cauchy problems for the heat equation, resolvent methods for Volterra type integral equation and contraction mapping prinsiple.

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