On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

Batirkhan Turmetov; Valery Karachik; Batirkhan Turmetov; Valery Karachik

doi:10.3934/math.2024333

AIMS Mathematics

2024, Volume 9, Issue 3: 6832-6849. doi: 10.3934/math.2024333

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On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

Batirkhan Turmetov ^{1
,
,},
Valery Karachik ²

1.
Department of Mathematics, Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkistan, Kazakhstan
2.
Department of Mathematical Analysis, South Ural State University, Chelyabinsk, Russia

Received: 25 September 2023 Revised: 17 January 2024 Accepted: 24 January 2024 Published: 19 February 2024
MSC : 34K06, 34K08, 35G16

In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.
- inverse problem,
- nonlocal biharmonic operator,
- parabolic equation,
- eigenfunction,
- eigenvalue,
- Fourier method,
- existence of solution,
- uniqueness of solution
Citation: Batirkhan Turmetov, Valery Karachik. On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution[J]. AIMS Mathematics, 2024, 9(3): 6832-6849. doi: 10.3934/math.2024333

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Abstract

In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.

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