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

• * Corresponding author: Zhousheng Ruan
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).
• 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.

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

 Citation: • • Figure 1.  Error curves for example 1

Figure 2.  Error surfaces for example 2

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

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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Tables(2)

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