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Table 2 The RMS values (32) for \(p\in \{0,0.01\%,0.1\%\}\) and \(\beta \in \{0,10^{-12},10^{-11},10^{-10}, 10^{-9},10^{-8},10^{-7}\}\) for Tests 1 and 2

From: An inverse problem of reconstructing the time-dependent coefficient in a one-dimensional hyperbolic equation

Test 1 Test 2
p β RMS(α) Minimum value of \(\mathbb{F}\) or \(\mathbb{F}_{\beta }\) p β RMS(α) Minimum value of \(\mathbb{F}\) or \(\mathbb{F}_{\beta }\)
0 0 0.0868 \(\mathbb{F}=2.4\mbox{E}{-}30\) 0 0 0.0872 \(\mathbb{F}=8.1\mbox{E}{-}30\)
0.01% 0 3.4204 \(\mathbb{F}=1.3\mbox{E}{-}29\) 0.01% 0 3.3676 \(\mathbb{F}=8.5\mbox{E}{-}30\)
10−10 0.0790 \(\mathbb{F}_{\beta }=1.1\mbox{E}{-}7\) 10−12 0.2368 \(\mathbb{F}_{\beta }=7.1\mbox{E}{-}8\)
10−9 0.0776 \(\mathbb{F}_{\beta }=4.2\mbox{E}{-}7\) 10−11 0.1245 \(\mathbb{F}_{\beta }=2.3\mbox{E}{-}7\)
10−8 0.0982 \(\mathbb{F}_{\beta }=3.2\mbox{E}{-}6\) 10−10 0.2728 \(\mathbb{F}_{\beta }=1.5\mbox{E}{-}6\)
0.1% 0 34.2297 \(\mathbb{F}=1.3\mbox{E}{-}29\) 0.1% 0 33.9465 \(\mathbb{F}=2.1\mbox{E}{-}29\)
10−9 0.1573 \(\mathbb{F}_{\beta }=8.4\mbox{E}{-}6\) 10−11 0.8454 \(\mathbb{F}_{\beta }=6.7\mbox{E}{-}6\)
10−8 0.1100 \(\mathbb{F}_{\beta }=1.1\mbox{E}{-}5\) 10−10 0.5282 \(\mathbb{F}_{\beta }=8.6\mbox{E}{-}6\)
10−7 0.1701 \(\mathbb{F}_{\beta }=3.6\mbox{E}{-}5\) 10−9 0.6936 \(\mathbb{F}_{\beta }=1.6\mbox{E}{-}5\)