一类半线性椭圆方程柯西问题的正则化方法

Abstract

Abstract:

In this paper, we construct and use a generalized fractional Tikhonov regularization method to study a Cauchy problem for semi-linear elliptic equation. Based on one nonlinear integral equation that the constructed regularization solution satisfies, we firstly prove the existence, uniqueness and stability for it. And then we give and prove the convergence for regularized method under an a-priori assumption on the exact solution. Ultimately, the regularized solution is calculated by designing an iteration algorithm, and we verify the stability and feasibility for proposed method by the corresponding computational results.

Key words: Cauchy problem, Semilinear elliptic equation, Regularization method, Convergence estimate, Numerical simulation

CLC Number:

O175.25

Hongwu Zhang. One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation[J].Acta mathematica scientia,Series A, 2022, 42(1): 45-57.

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References 30

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$q$	1.1	1.4	1.7	2	2.5	3	3.5
$\epsilon_{y=0.6}(u)$	0.0043	0.0069	0.0093	0.0122	0.0205	0.0296	0.0349
$\epsilon_{y=1}(u)$	0.0027	0.0059	0.0090	0.0122	0.0212	0.0302	0.0352

$\gamma$	0.5	0.8	1.1	1.5	1.8	2.5	3
$\epsilon_{y=0.6}(u)$	0.0052	0.0072	0.0095	0.0141	0.0192	0.0310	0.0353
$\epsilon_{y=1}(u)$	0.0034	0.0062	0.0090	0.0144	0.0200	0.0318	0.0358

$m$	1	2	3	4	5	6	7	8	9	10
$\epsilon_{y=0.6}(u)$	0.0380	0.0380	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353
$\epsilon_{y=1}(u)$	0.0382	0.0382	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358

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One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation

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