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 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 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 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 Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 667 ◽  
Author(s):  
Hongwu Zhang ◽  
Xiaoju Zhang
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Regularization Parameter ◽  
A Priori ◽  
Conditional Stability ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Tikhonov Method ◽  
Convergence Results ◽  
The Cauchy Problem

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.

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.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

scholarly journals Find A General Solution Of An Equation Of The Hyperbolic Type With A Second-Order Singular Coefficient And Solve The Cauchy Problem Posed For This Equation.

2021 ◽  
Vol 25 (1) ◽  
pp. 80
Author(s):  
Karimova Shalola Musayevna ◽  
Melikuzieva Dilshoda Mukhtorjon qizi
Keyword(s):  
Cauchy Problem ◽  
General Solution ◽  
Type Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Singular Coefficient ◽  
The Cauchy Problem

This paper presents a general solution of a hyperbolic type equation with a second-order singular coefficient and a solution to the Cauchy problem posed for this equation.

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 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.

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