识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法

摘要/Abstract

摘要：

该文研究具有Riemann-Liouville时间分数阶导数的Rayleigh-Stokes方程未知源识别问题.首先证明这个问题是不适定的，并应用分数阶Landweber正则化方法求解此反问题.基于条件稳定性结果，在先验和后验正则化参数选取规则下，分别给出精确解与正则解之间的误差估计.最后通过数值例子说明此方法求解此类反问题的有效性和可行性.

关键词: Rayleigh-Stokes方程, 反问题, 识别源项, 分数阶Landweber迭代正则化方法

Abstract:

In this paper, the inverse problem of identifying the unknown sources for the Rayleigh-Stokes equation with a Riemann-Liouville fractional derivative in time is considered. We prove that such a problem is ill-posed and apply the fractional Landweber method to solve this inverse problem. Based on the results of conditional stability, under the priori and posteriori regularization parameters choice rules, the error estimates between the exact solution and the regularization solution are given respectively. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of these methods.

Key words: Rayleigh-Stokes equation, Inverse problem, Identifying unknown source problem, Fractional Landweber iterative regularization method

中图分类号:

O175

杨帆,王乾朝,李晓晓. 识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法[J]. 数学物理学报, 2021, 41(2): 427-450.

Fan Yang,Qianchao Wang,Xiaoxiao Li. Fractional Landweber Iterative Regularization Method to Identify Source Term for the Rayleigh-Stokes Equation[J]. Acta mathematica scientia,Series A, 2021, 41(2): 427-450.

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参考文献 46

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α				α = 0.2	α = 0.5	α = 0.9
η(f)	ε=0.01	Landweber	BD	0.0099	0.0083	0.0075
			C-N	0.0072	0.0052	0.0041
		分数阶Landweber	BD	0.0088	0.0076	0.0061
			C-N	0.0063	0.0048	0.0039

	ε=0.005	Landweber	BD	0.0042	0.0039	0.0034
			C-N	0.0041	0.0035	0.0025
		分数阶Landweber	BD	0.0038	0.0035	0.0021
			C-N	0.0031	0.0028	0.0011

	ε= 0.001	Landweber	BD	0.0025	0.0019	7.9678e-04
			C-N	0.0020	0.0011	5.5690e-04
		分数阶Landweber	BD	0.0018	0.0010	7.1943e-04
			C-N	0.0011	8.7631e-04	4.6982e-04

α				α = 0.2	α = 0.5	α = 0.9
迭代步数(m)	ε = 0.01	Landweber	BD	24027	9658	30
			C-N	25556	31634	37652
		分数阶Landweber	BD	19683	6514	21
			C-N	21423	23963	29685

	ε = 0.005	Landweber	BD	46820	21349	34
			C-N	27941	30652	39541
		分数阶Landweber	BD	39870	12981	29
			C-N	19685	23916	33921

	ε = 0.001	Landweber	BD	79686	38679	44
			C-N	35719	42387	52802
		分数阶Landweber	BD	68765	27695	38
			C-N	26985	34796	44348

α				α = 0.2	α = 0.5	α = 0.9
CPU时间(unit: s)	ε = 0.01	Landweber	BD	480.54	193.16	0.60
			C-N	511.12	632.68	753.04
		分数阶Landweber	BD	393.66	130.28	0.42
			C-N	428.46	479.26	593.70

	ε = 0.005	Landweber	BD	936.40	426.98	0.68
			C-N	558.82	613.04	790.82
		分数阶Landweber	BD	797.40	259.62	0.58
			C-N	393.70	478.32	678.42

	ε = 0.001	Landweber	BD	1593.72	773.58	0.88
			C-N	714.38	847.74	1056.04
		分数阶Landweber	BD	1375.30	553.90	0.76
			C-N	539.70	695.92	886.96

α				α = 0.2	α = 0.5	α = 0.9
η(f)	ε = 0.01	Landweber	BD	0.0243	0.0128	0.0088
			C-N	0.1039	0.0864	0.0747
		分数阶Landweber	BD	0.0208	0.0106	0.0075
			C-N	0.0931	0.0705	0.0628

	ε = 0.005	Landweber	BD	0.0155	0.0116	0.0076
			C-N	0.0838	0.0725	0.0634
		分数阶Landweber	BD	0.0191	0.0103	0.0063
			C-N	0.0786	0.0596	0.0413

	ε = 0.001	Landweber	BD	0.0033	0.0019	8.6476e-04
			C-N	0.0589	0.0362	0.0295
		分数阶Landweber	BD	0.0021	0.0011	6.5483e-04
			C-N	0.0412	0.0268	0.0105

α				α = 0.2	α = 0.5	α = 0.9
迭代步数(m)	ε = 0.01	Landweber	BD	22348	6987	29
			C-N	3717	5816	7425
		分数阶Landweber	BD	16534	3768	19
			C-N	2674	4168	6123

	ε = 0.005	Landweber	BD	34258	12368	33
			C-N	6670	8126	11198
		分数阶Landweber	BD	22369	6879	26
			C-N	4396	6021	8934

	ε = 0.001	Landweber	BD	79463	39654	41
			C-N	15285	86956	131965
		分数阶Landweber	BD	54986	21685	33
			C-N	13210	69663	113824