scholarly journals On an Inverse Problem for the Stationary Equation with a Boundary Condition of the Third Type

2021 ◽  
Vol 14 (5) ◽  
pp. 659-666
Author(s):  
Alexander V. Velisevich ◽  
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Order Differential Equation ◽  
Local Existence ◽  
Unknown Coefficient ◽  
Lower Term ◽  
Stationary Equation ◽  
Integral Boundary ◽  
The Third ◽  
Local Existence And Uniqueness

The identification of an unknown coefficient in the lower term of elliptic second-order differential equation Mu + ku = f with the boundary condition of the third type is considered. The identification of the coefficient is based on integral boundary data. The local existence and uniqueness of the strong solution for the inverse problem is proved

scholarly journals On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

2020 ◽  
Vol 30 (4) ◽  
pp. 572-584
Author(s):  
D.K. Durdiev ◽  
J.Z. Nuriddinov
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Variable Coefficient ◽  
Direct Problem ◽  
Local Existence ◽  
Problem Solution ◽  
Volterra Type ◽  
Integral Term ◽  
Local Existence And Uniqueness ◽  
Coefficient Of Thermal Conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

scholarly journals Analysis of 2 + 1 diffusive–dispersive PDE arising in river braiding

2016 ◽  
Vol 27 (5) ◽  
pp. 756-780
Author(s):  
SALEH TANVEER ◽  
CHARIS TSIKKOU
Keyword(s):  
Boundary Condition ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Local Existence ◽  
Dispersive Equation ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Local Existence And Uniqueness ◽  
Formula Group ◽  
Image Position

We present local existence and uniqueness results for the following 2 + 1 diffusive–dispersive equation due to P. Hall arising in modelling of river braiding: $$\begin{equation*} u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left ( u^2 \right )_{xyy} = 0 \end{equation*}$$ for (x,y) ∈ [0, 2π] × [0, π], t > 0, with boundary condition uy=0=uyyy at y=0 and y=π and 2π periodicity in x, using a contraction mapping argument in a Bourgain-type space Ts,b. We also show that the energy ∥u(·, ·, t)∥2L2 and cumulative dissipation ∫0t∥uy (·, ·, s)∥L22dt are globally controlled in time t.

scholarly journals An abstract inverse problem

1991 ◽  
Vol 4 (2) ◽  
pp. 117-128 ◽  
Author(s):  
M. Choulli
Keyword(s):  
Inverse Problem ◽  
Uniqueness Theorem ◽  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Existence And Uniqueness Theorem ◽  
Local Existence And Uniqueness ◽  
Abstract Integrodifferential Equation

In this paper we consider an inverse problem that corresponds to an abstract integrodifferential equation. First, we prove a local existence and uniqueness theorem. We also show that every continuous solution can be locally extended in a unique way. Finally, we give sufficient conditions for the existence and a stability of the global solution.

scholarly journals The Regularity of the Solutions of Inverse Problems for the Pseudoparabolic Equation

2001 ◽  
Vol 14 (4) ◽  
pp. 414-424
Author(s):  
Anna Sh. Lyubanova ◽  
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Continuous Dependence ◽  
Input Data ◽  
Order Term ◽  
Unknown Coefficient ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
Additional Information ◽  
The Third

The paper discusses the regularity of the solutions to the inverse problems on finding unknown coefficients dependent on t in the pseudoparabolic equation of the third order with an additional information on the boundary. By the regularity is meant the continuous dependence of the solution on the input data of the inverse problem. The regularity of the solution is proved for two inverse problems of recovering the unknown coefficient in the second order term and the leader term of the linear pseudoparabolic equation

AN INVERSE PROBLEM FOR THE HYPERBOLIC HEAT EQUATION

1992 ◽  
Vol 02 (01) ◽  
pp. 113-120
Author(s):  
E.G. SAVATEEV ◽  
L.M. DE SOCIO
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
One Dimensional ◽  
Constructive Solution ◽  
Local Existence And Uniqueness ◽  
Diffusive Term ◽  
Hyperbolic Heat Equation

In this paper we prove a theorem of local existence and uniqueness for the solution of the hyperbolic heat equation in the case where the coefficient of the diffusive term is unknown. The problem is one-dimensional in space and the ratio of the two characteristics times, upon which the physics depends, is small. The demonstration relies on a constructive solution.

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