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# Regularized solution for a biharmonic equation with discrete data

• In this work, we focus on the Cauchy problem for the biharmonic equation associated with random data. In general, the problem is severely ill-posed in the sense of Hadamard, i.e, the solution does not depend continuously on the data. To regularize the instable solution of the problem, we apply a nonparametric regression associated with the Fourier truncation method. Also we will present a convergence result.

Mathematics Subject Classification: 35K05, 35K99, 47J06, 47H10.

 Citation: • • Figure 1.  The exact and regularized solutions at $y = 0$ (a) and its errors (b)

Figure 2.  The exact and regularized solutions at $y = 0.1$ (a) and its errors (b)

Figure 3.  The exact and regularized solutions at $y = 0.2$ (a) and its errors (b)

Figure 4.  The exact and regularized solutions at $y = 0.3$ (a) and its errors (b)

Figure 5.  The exact solution $u$ (a) and the regularized solution $\widehat v$ (b)

Table 1.  The errors between the exact solution and the regularized solution at $y \in \{0.1, \, 0.2, \, 0.3\}$

 Errors $n = 20$ $n=50$ $n=100$ $\mathrm{Err}(0.1)$ 0.016631443540398 0.017785494891671 0.016885191385811 $\mathrm{Err}(0.2)$ 0.018572286169882 0.016934217726270 0.017559567667446 $\mathrm{Err}(0.3)$ 0.012505879576731 0.012581413283707 0.015940532272324
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Tables(1)

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