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Scientific reports of Bukhara State University

Abstract

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.

Results. 1) The direct problem is the Cauchy problem for heat equation. The integral term has the time convolution form of kernel and an elliptic operator of direct problem solution. 2) As additional infor-mation, the solution of the direct problem on the hyperplane y=0 is given. The problem reduces to an auxiliary problem which is more convenient for further consideration. Then the auxiliary problem is replaced by an equivalent system of Volterra-type integral equations with respect to unknown functions.

Conclusion. Applying the method of contraction mappings to this system in the Hölder class of functions, it is proved the main result of the paper representing a local existence and uniqueness theorem. The article is organized as follows. In Section 2, we reduce the problem (1)-(3) into an auxiliary problem where the additional condition contains the unknown k outside integral. In Section 3, we replace auxiliary problem by an equivalent system of integral equations with respect to unknown functions. In Section 4, we prove the main result which states the existence and uniqueness of solution of problem by a fixed point argument.

First Page

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Last Page

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