scholarly journals CARLEMAN'S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN

2021 ◽  
Vol 7 (2) ◽  
pp. 110
Author(s):  
Ermamat N. Sattorov ◽  
Zuxro E. Ermamatova
Keyword(s):  
Cauchy Problem ◽  
Bounded Domain ◽  
Poisson Equation ◽  
Function Method ◽  
Normal Derivative ◽  
The Poisson Equation ◽  
The Cauchy Problem

We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman--Yarmuhamedov function method.

scholarly journals Regularization and Error Estimate for the Poisson Equation with Discrete Data

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 422
Author(s):  
Nguyen Anh Triet ◽  
Nguyen Duc Phuong ◽  
Van Thinh Nguyen ◽  
Can Nguyen-Huu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Poisson Equation ◽  
Regularization Method ◽  
Random Noise ◽  
Two Dimensional ◽  
The Poisson Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Dimensional Domain

In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased.

scholarly journals THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E^n

2014 ◽  
Vol 15 (1) ◽  
Author(s):  
Ikbol E. Niyozov ◽  
O. I. Makhmudov
Keyword(s):  
Cauchy Problem ◽  
Bounded Domain ◽  
Analytical Continuation ◽  
System Of Equations ◽  
The Cauchy Problem

ABSTRACT: In this paper we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem. ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy.

scholarly journals HP estimation for the Cauchy problem for nonlinear elliptic equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 193-202
Author(s):  
Thang Duc Le
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Bounded Domain ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in Hp space under some priori assumptions on the exact solution.

scholarly journals An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

2019 ◽  
Vol 14 (2) ◽  
pp. 204
Author(s):  
Eduardo Hernandez-Montero ◽  
Andres Fraguela-Collar ◽  
Jacques Henry
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Laplace Equation ◽  
Boundary Data ◽  
A Priori ◽  
Normal Derivative ◽  
Cauchy Data ◽  
Dirichlet Data ◽  
Data Completion ◽  
The Cauchy Problem

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

scholarly journals The ∂ Cauchy-Problem on Weakly q-Convex Domains in CPn

2020 ◽  
Vol 44 (4) ◽  
pp. 581-591
Author(s):  
SAYED SABER
Keyword(s):  
Cauchy Problem ◽  
Projective Space ◽  
Complex Projective Space ◽  
Function Method ◽  
Function Theory ◽  
Convex Domains ◽  
The Cauchy Problem ◽  
Support Conditions ◽  
The Neumann Problem ◽  
Complex Projective

Let D be a weakly q-convex domain in the complex projective space ℂPn. In this paper, the (weighted) ∂-Cauchy problem with support conditions in D is studied. Specifically, the modified weight function method is used to study the L2 existence theorem for the ∂-Neumann problem on D. The solutions are used to study function theory on weakly q-convex domains via the ∂-Cauchy problem.

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