Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method

Zhousheng Ruan; Sen Zhang; Sican Xiong

doi:10.3934/eect.2018032

Article Contents

2018, Volume 7, Issue 4: 669-682. Doi: 10.3934/eect.2018032

This issue Previous Article Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results Next Article

Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method

1.
Fundamental Science on Radioactive Geology and Exploration Technology Laboratory, East China University of Technology, Nanchang, Jiangxi 330013, China
2.
School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China
3.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Zhousheng Ruan
* Corresponding author: Zhousheng Ruan

Received: March 2017

Revised: May 2018

Published: September 2018

The first author is supported by National Natural Science Foundation of China (11661004, 11561003, 11761007), Natural Science Foundation of Jiangxi Province of China (20161BAB201034), Science and Technology Research Project of Education Department of Jiangxi Province (GJJ150568), Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province (20172BCB22019).

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Abstract

In this paper, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. Under some boundedness assumption, the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules, respectively. Based on the superposition principle, we propose a direct inversion algorithm in a parallel manner.

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Mathematics Subject Classification: Primary: 65M32, 35R30; Secondary: 65M30.

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Figure 1. Error curves for example 1

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Figure 2. Error surfaces for example 2

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Table 1. Inversional results for example 1 with different relative errors

$\delta$		0.05%	0.1%	0.2%	0.4%	0.8%	1.6%
$T_1$=0.1	$e(f, \delta)$	2.1%	2.9%	4.0%	5.5%	7.9%	10.1%
	$C_r$		0.34	0.33	0.31	0.38	0.24
$T_1$=0.2	$e(f, \delta)$	2.1%	3.1%	4.1%	5.9%	8.3%	11.0%
	$C_r$		0.38	0.28	0.35	0.33	0.28
$T_1$=0.4	$e(f, \delta)$	2.2%	3.2%	4.2%	6.2%	8.5 %	11.5%
	$C_r$		0.39	0.26	0.37	0.31	0.30
$T_1$=0.8	$e(f, \delta)$	2.2%	3.3 %	4.3%	6.3%	8.6 %	11.8%
	$C_r$		0.40	0.25	0.38	0.30	0.31
$T_1$=1	$e(f, \delta)$	2.2%	3.3 %	4.3%	6.4%	8.6 %	11.8%
	$C_r$		0.40	0.25	0.38	0.29	0.31

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Table 2. Inversional results for example 2 with different relative errors

$\delta$		0.05%	0.1%	0.2%	0.4%	0.8%	1.6%
$T_1$=0.1	$e(f, \delta)$	1.9%	2.8%	4.7%	6.9%	10.2%	14.7%
	$C_r$		0.41	0.50	0.37	0.39	0.36
$T_1$=0.2	$e(f, \delta)$	1.9%	2.9%	4.7%	6.9%	10.3%	14.8%
	$C_r$		0.41	0.49	0.37	0.39	0.35
$T_1$=0.4	$e(f, \delta)$	1.9%	2.9%	4.8%	7.0%	10.4 %	14.9%
	$C_r$		0.40	0.50	0.37	0.39	0.35
$T_1$=0.8	$e(f, \delta)$	1.9%	2.9 %	4.8%	7.0%	10.5 %	14.9%
	$C_r$		0.40	0.50	0.38	0.40	0.34
$T_1$=1	$e(f, \delta)$	1.9%	2.9 %	4.8%	7.0%	10.5 %	14.9%
	$C_r$		0.40	0.50	0.38	0.40	0.34

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