| Citation: |
Fan Yang, Lu-Lu Yan, Hao Liu, Xiao-Xiao Li. TWO REGULARIZATION METHODS FOR IDENTIFYING THE UNKNOWN SOURCE OF SOBOLEV EQUATION WITH FRACTIONAL LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 198-225. doi: 10.11948/20240065
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TWO REGULARIZATION METHODS FOR IDENTIFYING THE UNKNOWN SOURCE OF SOBOLEV EQUATION WITH FRACTIONAL LAPLACIAN
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Abstract
In this paper, an inverse source problem for the Sobolev equation with fractional Laplacian is investigated. We prove that this kind of problem is ill-posed and apply the Quasi-boundary regularization method and fractional Landweber iterative regularization method to solve this inverse problem. Based on the result of conditional stability, the error estimates between the exact solution and the regularization solution are given under the priori and posteriori regularization parameter selection rules. Finally, three examples are given to illustrate the effectiveness and feasibility of these methods.
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Keywords:
- Sobolev equation /
- inverse problem /
- identifying source term /
- regularization method
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References
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Fan Yang, Lu-Lu Yan, Hao Liu, Xiao-Xiao Li. TWO REGULARIZATION METHODS FOR IDENTIFYING THE UNKNOWN SOURCE OF SOBOLEV EQUATION WITH FRACTIONAL LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 198-225. doi: 10.11948/20240065
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Figure 1.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.1 $ $ \beta=1.1(a), 1.2(b), 1.3(c) $ $ \varepsilon=0.01, 0.005, 0.001. $ -
Figure 2.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.2 $ $ \beta=1.1(a), 1.2(b), 1.3(c) $ $ \varepsilon=0.01, 0.005, 0.001. $ -
Figure 3.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.3 $ $ \beta=1.1, 1.2, 1.3 $ $ \varepsilon=0.01, 0.005, 0.001. $ -
Figure 4.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.1 $ $ \beta=1.1(a), 1.2(b), 1.3(c) $ $ \varepsilon=0.01, 0.005, 0.001. $ -
Figure 5.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.2 $ $ \beta=1.1(a), 1.2(b), 1.3(c) $ $ \varepsilon=0.01, 0.005, 0.001. $ -
Figure 6.
The comparison of the exact solution
$ F(x) $ $ F^{m, \delta}(x) $ $ 5.3 $ $ \beta=1.1(a), 1.2(b), 1.3(c) $ $ \varepsilon=0.01, 0.005, 0.001. $