Coefficient Inverse Problems for Parabolic Type Equations and Their Application

✍ Scribed by P. G. Danilaev

• Publisher: De Gruyter
• Year: 2001
• Tongue: English
• Leaves: 128
• Series: Inverse and Ill-Posed Problems Series; 25
• Edition: Reprint 2014
• Category: Library

✦ Synopsis

As a rule, many practical problems are studied in a situation when the input data are incomplete. For example, this is the case for a parabolic partial differential equation describing the non-stationary physical process of heat and mass transfer if it contains the unknown thermal conductivity coefficient. Such situations arising in physical problems motivated the appearance of the present work.

In this monograph the author considers numerical solutions of the quasi-inversion problems, to which the solution of the original coefficient inverse problems are reduced.

Underground fluid dynamics is taken as a field of practical use of coefficient inverse problems. The significance of these problems for this application domain consists in the possibility to determine the physical fields of parameters that characterize the filtration properties of porous media (oil strata). This provides the possibility of predicting the conditions of oil-field development and the effects of the exploitation.

The research carried out by the author showed that the quasi-inversion method can be applied also for solution of "interior coefficient inverse problems" by reducing them to the problem of continuation of a solution to a parabolic equation. This reduction is based on the results of the proofs of the uniqueness theorems for solutions of the corresponding coefficient inverse problems.

✦ Table of Contents

Chapter 1.On the ill-posedness of coefficient inverse problems and the general approach to the study of them
Chapter 2. Determining the coefficient for the lowest term of equation
2.1.Setting of the problem. Determination of the coefficient
2.2.The difference quasi-inversion problem
2.3.A test example
Chapter 3. Determining of the coefficient for the leading terms of equation
3.1.Statement of the problem
3.2.The quasi-inversion problem and an estimate of stability of its solution
3.3.Simplification of equation of the quasi-inversion method
3.4.Simplification of the quasi-inversion problem
3.5.Finding the coefficient
3.6.Difference quasi-inversion problem
3.7.Numerical solution of the quasi-inversion problem
3.8.Results of solution of a test example problem
Chapter 4. Modification of the method of determining the coefficient of the leading terms in an equation
4.1.Modification method
4.2.Defining a coefficient
4.3.On deriving the main integro-differential equation
Chapter 5. Generalization of the developed algorithm for solving coefficient inversion problem
Chapter 6. On applications of coefficient inverse problems in underground fluid dynamics
6.1.Determining of filtration parameters of exploited non-homogeneous oil strata
6.2.Determining the filtration parameters in the case of non-linear filtration
6.3.The quasi-inversion problem for the considered cases
Summary
Bibliography