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Problem of determining a multidimensional thermal memory in a heat conductivity equation


Abstract

We consider a multidimensional integro-differential equation of heat conductivity with time-convolution integral in the right hand-side. The direct problem is represented by the Cauchy problem of determining the temperature of the medium for a known initial distribution of heat. We study the inverse problem of determining the kernel, in the integral part, that depends on time and spatial variables, if a solution of the direct problem is known on the hyperplane $x_n=0$ for $t>0.$ With a use of the resolvent of the kernel, this problem is reduced to a study of a more convenient inverse problem. The later problem is replaced with an equivalent system of integral equations with respect to the unknown functions and, using a contractive mapping, we prove that the direct and the inverse problems have unique solutions.

Key words: Integro-differential equation, inverse problem, kernel, resolvent, Banach’s principle.


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Article Information

TitleProblem of determining a multidimensional thermal memory in a heat conductivity equation
SourceMethods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 219-226
MilestonesReceived 08/05/2019; Revised 03/07/2019
CopyrightThe Author(s) 2019 (CC BY-SA)

Authors Information

D. K. Durdiev
Bukhara State University, Uzbekistan, 200100, Bukhara, M. Iqbal, 11

Zh. Zh. Zhumayev
Bukhara State University, Uzbekistan, 200100, Bukhara, M. Iqbal, 11


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Citation Example

D. K. Durdiev and Zh. Zh. Zhumayev, Problem of determining a multidimensional thermal memory in a heat conductivity equation, Methods Funct. Anal. Topology 25 (2019), no. 3, 219-226.


BibTex

@article {MFAT1207,
    AUTHOR = {D. K. Durdiev and Zh. Zh. Zhumayev},
     TITLE = {Problem of determining a multidimensional thermal memory in a heat conductivity equation},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {25},
      YEAR = {2019},
    NUMBER = {3},
     PAGES = {219-226},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1207},
}


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