scholarly journals Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Boundary Data ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Parabolic Problems ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Lateral Boundary ◽  
Spatial Component

Abstract For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

scholarly journals An inverse source problem for a two dimensional time fractional diffusion equation with involution

2021 ◽  
Author(s):  
Batirkhan Turmetov ◽  
B. J. Kadirkulov
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Fourier Method ◽  
Dirichlet Condition ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Two Dimensional ◽  
Time Fractional Diffusion Equation

In this paper, we consider a two-dimensional generalization of the parabolic equation. Using the Fourier method, we study the solvability of the inverse problem with the Dirichlet condition and periodic conditions.

scholarly journals Determination of a nonlinear source term in a diffusion equation with final observations

2004 ◽  
Vol 2004 (14) ◽  
pp. 741-753 ◽  
Author(s):  
Gongsheng Li ◽  
Yichen Ma ◽  
Kaitai Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Integral Identity ◽  
Source Term ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Source Terms ◽  
Nonlinear Source ◽  
Quasilinear Diffusion

This paper deals with an inverse problem of determining a nonlinear source term in a quasilinear diffusion equation with overposed final observations. Applying integral identity methods, data compatibilities are deduced by which the inverse source problem here is proved to be reasonable and solvable. Furthermore, with the aid of an integral identity that connects the unknown source terms with the known data, a conditional stability is established.

On the stability of recovering two sources and initial status in a stochastic hyperbolic-parabolic system

2021 ◽  
Author(s):  
Bin Wu ◽  
Jijun Liu
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Hyperbolic Equation ◽  
Wave Field ◽  
Time Derivative ◽  
Stability Index ◽  
Carleman Estimates ◽  
Conditional Stability ◽  
Coupling System ◽  
The Stability

Abstract Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in a prior bound set. Due to the lack of information about the time derivative of wave field at final moment, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system.

Inverse source problem for parabolic equation with the condition of integral observation in time

2018 ◽  
Vol 26 (4) ◽  
pp. 523-539 ◽  
Author(s):  
Aleksey I. Prilepko ◽  
Vitaly L. Kamynin ◽  
Andrew B. Kostin
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Inverse Problems ◽  
Unique Solvability ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Sufficient Conditions ◽  
Inverse Source Problem ◽  
Source Determination ◽  
Integral Observation

Abstract We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufficient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulfilled.

A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

2020 ◽  
Vol 28 (4) ◽  
pp. 499-516
Author(s):  
Zewen Wang ◽  
Shuli Chen ◽  
Shufang Qiu ◽  
Bin Wu
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Source Term ◽  
Dimensional Space ◽  
Conditional Stability ◽  
Finite Dimensional Space ◽  
Initial Value ◽  
Equation Problem ◽  
Inverse Solutions ◽  
Direct Problems

AbstractThis paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

scholarly journals An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition

2016 ◽  
Vol 8 (1) ◽  
pp. 107-117
Author(s):  
N.Ye. Kinash
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Existence And Uniqueness ◽  
Boundary Data ◽  
Function Method ◽  
Neumann Boundary ◽  
Fixed Number ◽  
Initial Condition ◽  
Green Function Method ◽  
The Green Function

We consider an inverse problem of identifying the time-dependent coefficient $a(t)$ in a two-dimensional parabolic equation: $$u_t=a(t)\Delta u+b_1(x,y,t)u_x+b_2(x,y,t)u_y+c(x,y,t)u+f(x,y,t),$$ $(x,y,t)\in Q_T,$ with the initial condition, Neumann boundary data and the nonlocal overdetermination condition $$\nu_1(t)u(0,y_0,t)+\nu_2(t)u(h,y_0,t)=\mu_3(t),\quad t\in[0,T],$$ where $y_0$ is a fixed number from $[0,l].$ The conditions of existence and uniqueness of the classical solution to this problem are established. For this purpose the Green function method, Schauder fixed point theorem and the theory of Volterra intergral equations are utilized.

Numerical Study of Inverse Source Problem for Internal Degenerate Parabolic Equation

2020 ◽  
Vol 18 (01) ◽  
pp. 2050032
Author(s):  
M. Alahyane ◽  
I. Boutaayamou ◽  
A. Chrifi ◽  
Y. Echarroudi ◽  
Y. Ouakrim
Keyword(s):  
Optimal Control ◽  
Inverse Problem ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Augmented Lagrangian ◽  
Numerical Study ◽  
Augmented Lagrangian Method ◽  
Inverse Source Problem ◽  
Discrete Solution ◽  
Degenerate Parabolic

This paper is devoted to numerical analysis of an inverse source problem in a degenerate parabolic equation. The aims of this work are to show the well-posedness of the discrete inverse problem and its convergence to the continuous one. For this, we reformulate first the encountered inverse problem to a regularized optimal control one. Then, we approximate our optimal control problem by finite element method and we show the existence of the discrete solution and its convergence to the continuous one. Finally, in order to confirm the efficiency of the proposed scheme, some numerical results are obtained using the augmented Lagrangian method.

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