Open Access

# 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.

### Article Information

 Title Problem of determining a multidimensional thermal memory in a heat conductivity equation Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 219-226 MathSciNet MR4016209 Milestones Received 08/05/2019; Revised 03/07/2019 Copyright The 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

### 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},
MRNUMBER = {MR4016209},
URL = {http://mfat.imath.kiev.ua/article/?id=1207},
}

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