
This article is cited in 1 scientific paper (total in 1 paper)
Mathematics
On a problem of determining the righthand side of the partial integrodifferential equation
A. O. Mamytov^{} ^{} Osh State University, Osh, Kyrgyz Republic
Abstract:
As it is known, in the inverse problem, apart from the soughtfor “basic” solution of the problem (i. e., the solution of the direct problem), the components of the direct problem are unknown. It is required to find these unknown components, so they will be also included in the solution of the inverse problem. To determine these components in the inverse problem, some additional information on the solution of the direct problem is added to the given equations. The additional information is called the inverse problem data. In the proposed article, the specific fourthorder partial integrodifferential equation with the known initial and boundary conditions is considered. For simplicity, the homogeneous boundary conditions have been examined, since with the help of a linear transformation, the always inhomogeneous boundary conditions can be reduced to the homogeneous ones. The righthand side of the equation contains $n$ unknown functions: $\varphi_{i}(t)$, $i = 1,2,\dots,n$. To determine these unknown functions: $\varphi_{i}(t)$, $i = 1,2,\dots,n$ in the inverse problem there is additional information on the solution of the direct problem, i.e., the values of the soughtfor “basic” solution to the problem in the inner segments of the investigated region are known, i. e., $u(t,x_{i}) = g_{i}(t)$, $t\in [0,T]$, $x_{i}\in(0,1)$, $i = 1, 2,\dots, n$. The problem is investigated in a rectangle located in the first quarter of the Cartesian coordinate system. To solve the inverse problem, an algorithm has been elaborated and sufficient conditions for the existence and the uniqueness of the solution of the inverse problem for the restoration of the righthand side in a fourthorder partial integrodifferential equation have been found. When solving the inverse problem, the methods of transformations, Green's function, solutions of systems of linear Volterra integral equations have been used. As a result, the inverse problem has been reduced to a system of $(n+1)$ linear Volterra integral equations of the second kind, the solution of which for small $0<T$ exists and is unique. The considered inverse problem can be called the inverse source problem.
Keywords:
inverse source problem, fourthorder partial integrodifferential equation, system of Volterra integral equations, Green's function, resolvent.
Received: 25.05.2021
Citation:
A. O. Mamytov, “On a problem of determining the righthand side of the partial integrodifferential equation”, Vestn. YuzhnoUral. Gos. Unta. Ser. Matem. Mekh. Fiz., 13:3 (2021), 31–38
Linking options:
https://www.mathnet.ru/eng/vyurm488 https://www.mathnet.ru/eng/vyurm/v13/i3/p31

