A Cauchy problem for the asymmetric Parabolic equation in polar coordinates with the perturbed diffusivity

The inverse problem for the heat equation plays an important area of study and application. Up to now, the backward heat problem (BHP) in Cartesian coordinates has been arisen in many articles, but the BHP in different domains such as polar coordinates, cylindrical one or spherical one is rarely considered. This paper’s purpose is to investigate the BHP on a disk, especially, the problem is associated with the perturbed diffusivity and the space-dependent heat source. In order to solve the problem, the authors apply the separation of variables method, associated with the Bessel’s equation and Bessel’s expansion. Based on the exact solution, the regularized solution is constructed by using the modified quasi-boundary value method. As a result, a Holder type of convergence rate has been obtained. In addition, a numerical experiment is given to illustrate the flexibility and effectiveness of the used method.

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On an asymmetric backward heat problem with the space and time-dependent heat source on a disk

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2017-0013 ◽

2019 ◽

Vol 27 (1) ◽

pp. 103-115

Author(s):

Triet Minh Le ◽

Quan Hoang Pham ◽

Phong Hong Luu

Keyword(s):

Exact Solution ◽

Heat Source ◽

Stable Solution ◽

Time Dependent ◽

Polar Coordinates ◽

Heat Problem ◽

Space And Time ◽

Boundary Value Method ◽

Ill Posed ◽

Backward Heat Problem

Abstract In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method.

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A Regularization of the Backward Problem for Nonlinear Parabolic Equation with Time-Dependent Coefficient

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2012/943621 ◽

2012 ◽

Vol 2012 ◽

pp. 1-20

Author(s):

Pham Hoang Quan ◽

Le Minh Triet ◽

Dang Duc Trong

Keyword(s):

Exact Solution ◽

Parabolic Equation ◽

Numerical Experiment ◽

Time Dependent ◽

Backward Problem ◽

Nonlinear Parabolic ◽

Boundary Value Method ◽

Ill Posed ◽

Sharp Error ◽

Regularized Solution

We study the backward problem with time-dependent coefficient which is a severely ill-posed problem. We regularize this problem by combining quasi-boundary value method and quasi-reversibility method and then obtain sharp error estimate between the exact solution and the regularized solution. A numerical experiment is given in order to illustrate our results.

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HP estimation for the Cauchy problem for nonlinear elliptic equation

Science and Technology Development Journal - Natural Sciences ◽

10.32508/stdjns.v1it5.553 ◽

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.

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Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Mathematics ◽

10.3390/math7080705 ◽

2019 ◽

Vol 7 (8) ◽

pp. 705 ◽

Cited By ~ 15

Author(s):

Fan Yang ◽

Ping Fan ◽

Xiao-Xiao Li

Keyword(s):

Cauchy Problem ◽

Exact Solution ◽

Helmholtz Equation ◽

Regularization Method ◽

Three Dimensional ◽

Wave Number ◽

Modified Helmholtz Equation ◽

The Cauchy Problem ◽

Ill Posed ◽

Regularized Solution

In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

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A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.12-m12113 ◽

2015 ◽

Vol 7 (1) ◽

pp. 31-42 ◽

Cited By ~ 2

Author(s):

Jingjun Zhao ◽

Songshu Liu ◽

Tao Liu

Keyword(s):

Cauchy Problem ◽

Exact Solution ◽

Heat Conduction ◽

Heat Conduction Equation ◽

Kernel Method ◽

Conduction Equation ◽

Two Dimensional ◽

Ill Posed ◽

Well Posed ◽

Regularized Solution

AbstractIn this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.

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Regularization of a Cauchy problem for the heat equation

Science and Technology Development Journal - Natural Sciences ◽

10.32508/stdjns.v1it5.552 ◽

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.

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