One-dimensional diffusion problem with not strengthened regular boundary conditions

Math Methods in App Sciences

Sarsenbi

2019

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

Section: Reduction To a Mathematical Problemmentioning

confidence: 85%

“…In this case, problems (5) to (8) and (1) to (4) coincide. Indeed, from (24) and Equation 1, we have…”

Section: Construction Of a Formal Solution Of The Problemmentioning

confidence: 99%

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Kirane

Math Methods in App Sciences

Sarsenbi

2019

“…is a partial solution of the inhomogeneous equation (21). Therefore, we obtain a unique solution of the Cauchy problem (21)- (22) in the following form:…”

Section: Uniqueness Of the Solution Of The Problemmentioning

confidence: 99%

On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

Advances in Mathematical Physics

Dildabek

Ivanova

2018

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

“…In [16] the initial-boundary value problem for the heat equation with a dynamictype boundary condition is considered. In [23] one family of problems simulating the determination of target components and density of sources from given values of the initial and final states is considered. In [1] an overdetermined initial-boundary value problem for a parabolic equation is considered.…”

Section: Introductionmentioning

confidence: 99%

Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions

Oralsyn

Ismailov

2018

Filomat

We investigate an inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. The boundary conditions of this problem are regular but not strengthened regular. The principal difference of this problem is: the system of eigenfunctions is not complete. But the system of eigen-and associated functions forming a basis. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method.