scholarly journals Identical Approximation Operator and Regularization Method for the Cauchy problem of 2D Heat Conduction Equation

2020 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
Numerical Results ◽  
Conduction Equation ◽  
The Cauchy Problem ◽  
Ill Posed

In this paper, an identical approximate regularization method is extended to the Cauchy problem of two-dimensional heat conduction equation, this kind of problem is severely ill-posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries, and compared the identical approximate regularization solutions which are displayed in paper. The numerical results show that our method is effective, accurate and stable to solve the ill-posed Cauchy problems.

scholarly journals A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Convergence Estimate ◽  
Ill Posed

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

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.

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

2015 ◽  
Vol 7 (1) ◽  
pp. 31-42 ◽  
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.

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

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 705 ◽  
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.

scholarly journals METHOD OF LINES AND FINITE DIFFERENCE SCHEMES WITH THE EXACT SPECTRUM FOR SOLUTION THE HYPERBOLIC HEAT CONDUCTION EQUATION

2011 ◽  
Vol 16 (1) ◽  
pp. 220-232 ◽  
Author(s):  
Harijs Kalis ◽  
Andris Buikis
Keyword(s):  
Heat Conduction ◽  
Finite Difference ◽  
Difference Scheme ◽  
Heat Conduction Equation ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Numerical Results ◽  
Conduction Equation ◽  
Finite Difference Schemes ◽  
Hyperbolic Heat Conduction

This paper is concerning with the 1-D initial–boundary value problem for the hyperbolic heat conduction equation. Numerical solutions are obtained using two discretizations methods – the finite difference scheme (FDS) and the difference scheme with the exact spectrum (FDSES). Hyperbolic heat conduction problem with boundary conditions of the third kind is solved by the spectral method. Method of lines and the Fourier method are considered for the time discretization. Finite difference schemes with central difference and exact spectrum are analyzed. A novel method for solving the discrete spectral problem is used. Special matrix with orthonormal eigenvectors is formed. Numerical results are obtained for steel quenching problem in the plate and in the sphere with holes. The hyperbolic heat conduction problem in the sphere with holes is reduced to the problem in the plate. Some examples and numerical results for two typical problems related to hyperbolic heat conduction equation are presented.

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