scholarly journals A Modified Quasi-Reversibility Method for a Class of Ill-Posed Cauchy Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 627-642 ◽  
Author(s):  
Nadjib Boussetila ◽  
Faouzia Rebbani
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Linear Operator ◽  
Convergence Rate ◽  
A Priori ◽  
Cauchy Problems ◽  
Unbounded Linear Operator ◽  
Ill Posed ◽  
New Perturbation ◽  
Problem Data

Abstract The goal of this paper is to present some extensions of the method of quasi-reversibility applied to an ill-posed Cauchy problem associated with an unbounded linear operator in a Hilbert space. The key point to our proof is the use of a new perturbation to construct a family of regularizing operators for the considered problem. We show the convergence of this method, and we estimate the convergence rate under a priori regularity assumptions on the problem data.

scholarly journals Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 48
Author(s):  
Hongwu Zhang ◽  
Xiaoju Zhang
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Type Equation ◽  
Conditional Stability ◽  
Elliptic Type ◽  
Type Method ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
The Stability ◽  
Regularized Solution

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

scholarly journals The behavior of solutions of non-linear differential equations in Hilbert space. I

1988 ◽  
Vol 11 (1) ◽  
pp. 143-165 ◽  
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Case ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Degenerate Linear

This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi-uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon-Nirenberg is used to study the uniqueness of the Cauchy problem for the non-degenerate linear case.

Regularization method for an ill-posed Cauchy problem for elliptic equations

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Abderafik Benrabah ◽  
Nadjib Boussetila ◽  
Faouzia Rebbani
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Elliptic Pdes ◽  
Ill Posed

AbstractThe paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by

scholarly journals Regularization of a Cauchy problem for the heat equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 184-192
Author(s):  
Au Van Vo ◽  
Tuan Hoang Nguyen
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Heat Equation ◽  
Error Estimates ◽  
A Priori ◽  
Cauchy Data ◽  
Linear Source ◽  
Truncation Method ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we study a Cauchy problem for the heat equation with linear source in the form ut(x,t)= uxx(x,t)+f(x,t), u(L,t)=  φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data φε and Ψε satisfying ‖ φε - φ ‖+‖ Ψε - Ψ ‖ ≤ ε and that fε satisfying ‖ fε(x,. ) - f(x,.) ‖ ≤ ε .  We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.

scholarly journals Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Michael V. Klibanov ◽  
Nikolaj A. Koshev ◽  
Jingzhi Li ◽  
Anatoly G. Yagola
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Weight Function ◽  
Quasilinear Parabolic Equation ◽  
Initial Condition ◽  
Bounded Set ◽  
Ill Posed ◽  
Linear Pdes ◽  
Quasilinear Parabolic

AbstractWe solve numerically the side Cauchy problem for a 1-D parabolic equation. The initial condition is unknown. This is an ill-posed problem. The main difference with previous results is that our equation is quasilinear, whereas known publications on this topic work only with linear PDEs. The key idea is to minimize a weighted Tikhonov functional with the Carleman Weight Function (CWF) in it. Roughly, given a reasonable bounded set of any size in a reasonable Hilbert space, one can choose the parameter of the CWF in such a way that this functional becomes strictly convex on that set.

scholarly journals Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

2001 ◽  
Vol 63 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Continuous Selection ◽  
Fréchet Space ◽  
Inhomogeneous Equation ◽  
Frechet Space ◽  
Closed Linear Operator ◽  
Cauchy Problems

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

scholarly journals The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

2017 ◽  
Vol 2 (2) ◽  
pp. 143
Author(s):  
Susilo Hariyanto
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Linear Operator ◽  
Infinitesimal Generator ◽  
Original Problem ◽  
Semigroup Theory ◽  
Abstract Cauchy Problem ◽  
Direct Sum ◽  
Special Operator ◽  
Degenerate Cauchy Problem

<div style="text-align: justify;">In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.</div>

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