The inverse problem for determining the right part of the pseudo-parabolic equationIn this paper the inverse problem of determining a solution and an unknown right-hand side that depends only on spatial variable for the linear pseudo-parabolic equation of the third order is investigated. In inverse problems, together with the initial and boundary conditions also consider an additional information, the need for which is due to the presence of unknown coefficients or the right side of the equation. In this paper, as additional information the integral overdetermination condition is considered. Inverse problems of determining the right-hand side of a differential equation arise in the mathematical modeling of some physical processes in the case when, in addition to solving the equation, it is necessary to restore the action of external sources. Today, studies of direct and inverse problems for pseudo-parabolic equations are rapidly developing due to the needs of modeling and process control in thermophysics, hydrodynamics and continuum mechanics. Similar pseudo-parabolic equations to considered in this paper arise in the description of heat and mass transfer processes, processes of motion of non-Newtonian fluids, wave processes, and in many other areas. Using series expansion, the existence and uniqueness theorems of classical solutions to this problem are proved. The result of this work is a solution presented in the series form, which allows the necessary numerical calculations to be performed with a given accuracy.
This paper deals with the unique solvability of some inverse problems
for nonlinear Navier-Stokes-Voigt (Kelvin-Voigt) system with memory that
governs the flow of incompressible viscoelastic non-Newtonian fluids.
The inverse problems that study here, consist of determining a time
dependent intensity of the density of external forces, along with a
velocity and a pressure of fluids. As an additional information, two
types of integral overdetermination conditions over space domain are
considered. The system supplemented also with an initial and one of the
boundary conditions: stick and slip boundary conditions. For all inverse
problems, under suitable assumptions on the data, the global and local
in time existence and uniqueness of weak and strong solutions were
established.
In this paper, the inverse problem for a linear system of Kelvin-Voigt equations with memory describing the dynamics of a viscoelastic incompressible non-newtonian fluid is considered. In the inverse problem under consideration, along with the solution (velocity and fluid pressure) of the equation, it is also required to find the unknown (intensity of the external force) on the right side, which depends only on the time variable. Definitions of weak and strong solutions are given. Weak and strong solutions of the set inverse problems satisfy the boundary condition of sliding at the boundary. The sliding boundary condition gives a mathematical and physical character to the study of a linear system of Kelvin-Voigt equations with memory. The applicability of the Faedo-Galerkin method for this type of system of equations is analyzed. With the help of the Faedo-Galerkin method, the global theorem of the existence of solutions to the presented inverse problem is proved in a weak and strong generalized sense. To prove the theorem of the existence of a solution "as a whole"in time, it is associated with obtaining a priori estimates, the constants in which depend only on the data of the problem and the magnitude of the time interval. And also the uniqueness theorem of the solutions of the considered inverse problems for a linear system of Kelvin-Voigt equations with memory is obtained and proved.
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