AN ILL-POSED PROBLEM FOR THE HEAT EQUATION

2009 ◽  
Vol 19 (09) ◽  
pp. 1631-1641 ◽  
Author(s):  
L. E. PAYNE ◽  
G. A. PHILIPPIN
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Outer Boundary ◽  
Slight Modification ◽  
Cauchy Data ◽  
The Cauchy Problem ◽  
Ill Posed

The Cauchy problem for the heat equation in which Cauchy data are prescribed on the outer boundary of a domain with cavity and no data are given on the inner boundary is known to be ill-posed. By a slight modification of the boundary conditions a new problem is introduced whose solution depends continuously on the data in L2.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

scholarly journals The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions

1956 ◽  
Vol 8 ◽  
pp. 426-431 ◽  
Author(s):  
E. P. Miles ◽  
Ernest Williams
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Cauchy Data ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Image Position

We shall discuss solutions of linear partial differential equations of the form1where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;2j = 1,2, ߪ,s − 1,whenever the function P is annihilated by a finite iteration of the operator Φ.

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 An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

2011 ◽  
Vol 9 (4) ◽  
pp. 878-896 ◽  
Author(s):  
Houde Han ◽  
Leevan Ling ◽  
Tomoya Takeuchi
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Regularization Method ◽  
Boundary Data ◽  
Outer Boundary ◽  
Cauchy Problems ◽  
Electrical Field ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractDetecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

scholarly journals No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

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.

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