Identification Problem for Strongly Degenerate Evolution Equations with the Gerasimov–Caputo Derivative

2020 ◽  
Vol 56 (12) ◽  
pp. 1613-1627
Author(s):  
V. E. Fedorov ◽  
M. Kostić
Keyword(s):  
Caputo Derivative ◽  
Evolution Equations ◽  
Identification Problem ◽  
Strongly Degenerate ◽  
Degenerate Evolution Equations

A class of inverse problems for fractional order degenerate evolution equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Vladimir E. Fedorov ◽  
Anna V. Nagumanova ◽  
Marko Kostić
Keyword(s):  
Inverse Problem ◽  
Fractional Order ◽  
Unique Solvability ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Bounded Operator ◽  
Sufficient Conditions ◽  
Order Degenerate ◽  
Strongly Degenerate ◽  
Degenerate Evolution Equations

AbstractThe criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter–Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases.

Identification problem for degenerate evolution equations of fractional order

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Vladimir E. Fedorov ◽  
Natalia D. Ivanova
Keyword(s):  
Fractional Order ◽  
Evolution Equations ◽  
Fluid Velocity ◽  
Homogeneous Equation ◽  
Fluid Pressure ◽  
Identification Problem ◽  
Fractional Differential ◽  
Degenerate Operator ◽  
Degenerate Evolution Equations ◽  
Linear Fractional Differential Equations

AbstractThe existence of a unique solution for the identification problem to linear fractional differential equations in Banach spaces were proved in the cases of the nondegenerate equation and of the equation with degenerate operator at the Caputo derivative. The degenerate case was studied for the overdetermination on the phase space and on the degeneracy subspace of the corresponding homogeneous equation. Abstract results are used to the investigation of the identification problem for time-fractional order Sobolev’s system of equations in the cases of the overdetermination on the fluid velocity and on the fluid pressure gradient.

scholarly journals Identifying the heat sink

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
J. D. Audu ◽  
A. Boumenir ◽  
K. M. Furati ◽  
I. O. Sarumi
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Temperature Profile ◽  
Spectral Methods ◽  
Heat Sink ◽  
Evolution Equations ◽  
Identification Problem ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Pseudo Spectral

<p style='text-indent:20px;'>In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.</p>

scholarly journals A Class of Fractional Degenerate Evolution Equations with Delay

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1700
Author(s):  
Amar Debbouche ◽  
Vladimir E. Fedorov
Keyword(s):  
Unique Solvability ◽  
Evolution Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Systems Of Equations ◽  
Fractional Differential ◽  
Initial Boundary Value ◽  
Degenerate Type ◽  
Degenerate Evolution Equations ◽  
Equations With Delay

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays.

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