Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions
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摘要:
对Robin边界条件时间分数阶扩散方程的源项辨识问题进行了研究。这类问题是不适定的，因此提出了一种迭代型正则化方法，得到了源项辨识问题的正则近似解。给出了先验和后验参数选取规则下正则近似解和精确解之间的误差估计，数值算例验证了该方法的有效性。
Abstract:The source term identification for the time-fractional diffusion equation with Robin boundary conditions was studied. Since the ill-posedness of this problem, an iterative regularization method was constructed to calculate the regularized approximate solution of the source term. The error estimates between the regularized approximate solution and the exact solution were given under the priori and the posteriori regularization parameter choice rules. Numerical examples verify the effectiveness of the proposed method.
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图 1 算例1的精确解及其正则近似解：(a)先验规则；(b)后验规则
Figure 1. The exact solution and its regularized approximate solution of example 1: (a) the priori rule; (b) the posteriori rule
图 2 算例2的精确解及其正则近似解：(a)先验规则；(b)后验规则
Figure 2. The exact solution and its regularized approximate solution of example 2: (a) the priori rule; (b) the posteriori rule
图 3 算例3的精确解及其正则近似解：(a)先验规则；(b)后验规则
Figure 3. The exact solution and its regularized approximate solution of example 3: (a) the priori rule; (b) the posteriori rule
表 1 算例1在不同误差水平下的绝对误差和相对误差
Table 1. Absolute errors and relative errors under different error levels of example 1
$ \varepsilon $ 0.001 0.005 0.010 $ e{(f,\varepsilon )_{{\text{priori}}}} $ 0.344 4 0.649 1 0.729 8 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{priori}}}} $ 0.026 9 0.050 7 0.057 0 $ e{(f,\varepsilon )_{{\text{posteriori}}}} $ 0.294 5 0.557 0 0.681 2 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{posteriori}}}} $ 0.023 0 0.043 5 0.053 2 
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表 2 算例2在不同误差水平下的绝对误差和相对误差
Table 2. Absolute errors and relative errors under different error levels of example 2
$ \varepsilon $ 0.001 0.005 0.010 $ e{(f,\varepsilon )_{{\text{priori}}}} $ 0.200 9 0.343 5 0.531 3 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{priori}}}} $ 0.049 2 0.084 1 0.130 1 $ e{(f,\varepsilon )_{{\text{posteriori}}}} $ 0.197 7 0.299 0 0.368 4 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{posteriori}}}} $ 0.048 4 0.073 2 0.090 2
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表 3 算例3在不同误差水平下的绝对误差和相对误差
Table 3. Absolute errors and relative errors under different error levels of example 3
$ \varepsilon $ 0.001 0.005 0.010 $ e{(f,\varepsilon )_{{\text{priori}}}} $ 1.245 0 1.400 0 1.536 5 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{priori}}}} $ 0.249 0 0.280 1 0.307 3 $ e{(f,\varepsilon )_{{\text{posteriori}}}} $ 1.191 0 1.369 5 1.467 0 $ {e_{\text{r}}}{(f,\varepsilon )_{{\text{posteriori}}}} $ 0.238 2 0.273 9 0.293 4
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