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On an inverse problem for fractional evolution equation

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  • In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

    Mathematics Subject Classification: 35K05, 35K99, 47J06, 47H10x.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Reconstruction results at t = 0 from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

    Figure 2.  Reconstruction results at t = 0 from noisy measurement data: 2D drawing using QBV Method

    Figure 3.  Reconstruction results at $t = 0.05$ from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

    Figure 4.  Reconstruction results at $t = 0.25$ from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

    Figure 5.  Reconstruction results at t = 0 from noisy measurement data at $t=0$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using Quasi Reversibility method

    Figure 6.  Reconstruction results at t = 0 from noisy measurement data at $t=0$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using Quasi Reversibility method

    Figure 7.  The exact solution in Example 2 at t = 1.

    Figure 8.  Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $ \in =10^{-1}, \in =10^{-2}$ using Quasi Reversibility method

    Figure 9.  Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $ \in = 10^{-3}, \in = 10^{-4}$ using Quasi Reversibility method

    Table 1.   

    $ \in $t = 0t = 0.05t = 0.25
    err1 err2 err1err2 err1 err2
    1E-015.19E-025.11E-033.97E-011.48E-023.90E-011.26E-02
    1E-021.47E-031.45E-042.90E-023.68E-039.61E-023.12E-02
    1E-031.89E-051.86E-061.25E-024.63E-041.21E-023.92E-04
    1E-041.95E-071.92E-081.28E-034.77E-051.24E-034.03E-05
    1E-051.96E-061.93E-071.29E-044.78E-061.24E-044.04E-06
    1E-062.05E-072.12E-081.29E-054.78E-071.25E-054.04E-07
     | Show Table
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    Table 2.   

    $ \in $t = 0t = 1
    err1 err2 err1 err2
    1E-014.00E-013.04E-023.00E-011.68E-02
    1E-024.86E-023.33E-022.86E-031.84E-04
    1E-034.61E-033.25E-042.61E-041.80E-05
    1E-045.19E-042.43E-052.19E-051.34E-06
    1E-056.04E-056.88E-072.04E-043.81E-07
    1E-067.49E-068.43E-082.49E-054.66E-08
     | Show Table
    DownLoad: CSV
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