On a Priori Bounds in the Cauchy Problem for Elliptic Equations

1970 ◽  
Vol 1 (1) ◽  
pp. 82-89 ◽  
Author(s):  
L. E. Payne
Keyword(s):  
Cauchy Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
A Priori Bounds ◽  
The Cauchy Problem

scholarly journals An iterative method for the Cauchy problem for second-order elliptic equations

2018 ◽  
Vol 142-143 ◽  
pp. 216-223 ◽  
Author(s):  
George Baravdish ◽  
Ihor Borachok ◽  
Roman Chapko ◽  
B. Tomas Johansson ◽  
Marián Slodička
Keyword(s):  
Cauchy Problem ◽  
Iterative Method ◽  
Elliptic Equations ◽  
Second Order ◽  
The Cauchy Problem ◽  
Second Order Elliptic Equations

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 An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hao Cheng ◽  
Ping Zhu ◽  
Jie Gao
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Regularization Method ◽  
A Priori ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Regularization Parameters ◽  
The Cauchy Problem

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. Thea priorianda posteriorirules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.

The Cauchy problem for the system of Zakharov equations arising from ion-acoustic modes

1996 ◽  
Vol 126 (4) ◽  
pp. 811-820 ◽  
Author(s):  
Guo Boling ◽  
Yuan Guangwei
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Cauchy Data ◽  
Continuous Method ◽  
Initial Value ◽  
Acoustic Modes ◽  
Ion Acoustic ◽  
The Cauchy Problem

In this paper the initial value problem for a class of Zakharov equations arising from ion-acoustic modes is discussed. Without assuming the Cauchy data are small, we prove the existence and uniqueness of the global smooth solution for the problem via the so-called continuous method and delicate a priori estimates.

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