| Citation: |
Lu-Lu Yan, Fan Yang, Xiao-Xiao Li. THE FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE IN A SPACE-FRACTIONAL ALLEN-CAHN EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2257-2282. doi: 10.11948/20230364
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THE FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE IN A SPACE-FRACTIONAL ALLEN-CAHN EQUATION
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Abstract
In this paper, we consider the inverse problem for identifying the source term and initial value simultaneously in a space-fractional Allen-Cahn equation. This problem is ill-posed, i.e., the solution of this problem does not depend continuously on the data. The fractional Tikhonov method is used to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solutions and the exact solutions are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our method.
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References
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Lu-Lu Yan, Fan Yang, Xiao-Xiao Li. THE FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE IN A SPACE-FRACTIONAL ALLEN-CAHN EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2257-2282. doi: 10.11948/20230364
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Figure 1.
The exact solution
$ f(x) $ $ f^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.7 $ -
Figure 2.
The exact solution
$ \varphi(x) $ $ \varphi^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.7 $ -
Figure 3.
The exact solution
$ f(x) $ $ f^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.55 $ -
Figure 4.
The exact solution
$ \varphi(x) $ $ \varphi^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.55 $ -
Figure 5.
The exact solution
$ f(x) $ $ f^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.45 $ -
Figure 6.
The exact solution
$ \varphi(x) $ $ \varphi^{\delta}_{\mu}(x) $ $ \alpha=0.3 $ $ \alpha=0.45 $