A Method of Conversion of some Coefficient Inverse Parabolic Problems to a Unified Type of Integral-Differential Equation

2011 ◽  
Vol 222 ◽  
pp. 353-356
Author(s):  
Sharif E. Guseynov ◽  
Janis S. Rimshans ◽  
Jevgenijs Kaupuzs ◽  
Artur Medvid' ◽  
Daiga Zaime
Keyword(s):  
Differential Equation ◽  
Inverse Problems ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Linear Parabolic Equations ◽  
Different Types ◽  
Non Linear ◽  
Coefficient Inverse Problems ◽  
Integral Differential Equation

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

scholarly journals Some control and inverse problems for linear parabolic equations

2016 ◽  
pp. 280-290
Author(s):  
Н.Л. Гольдман
Keyword(s):  
Inverse Problems ◽  
Parabolic Equations ◽  
Initial Conditions ◽  
Control Function ◽  
Parabolic Problems ◽  
Duality Principle ◽  
Control Problems ◽  
One Dimensional ◽  
Linear Parabolic Equations

На основе принципа двойственности исследованы свойства решений задач управления и обратных задач для одномерных параболических уравнений. Такой подход позволяет обобщить для линейных параболических операторов с коэффициентами, зависящими от $(x,t)$, известный результат Лионса о плотности усредненных наблюдений в задачах управления с управляющим воздействием в начальном условии. Показано, что значение этих свойств плотности не ограничивается задачами управления. Рассмотрено использование таких свойств при изучении обратных параболических задач, в том числе при исследовании условий единственности их решения. Properties of solutions of control and inverse problems for one-dimensional parabolic equations with coefficients dependent on $(x,t)$ are studied. The proposed approach based on the duality principle allows one to generalize the known Lions' result on the density properties of averaged observations in control problems with a control function given in the initial conditions. It is shown that the significance of these density properties is not restricted to the control problems. Such properties are used to study inverse parabolic problems, in particular, to study the uniqueness conditions of their solutions.

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