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Table 1 Errors and order convergence for \(u_{\mathrm{ex}}^{1,\beta }\) using (41) with \(r= \frac{2-\beta }{\beta }\)

From: Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

\(r = \frac{2-\beta }{\beta }\) n = 32 n = 64 n = 128 n = 256 n = 512 n = 1028
β = 0.2 \(E_{\mathrm{max}_{I}}^{n}\) 2.479E−2 7.233E−3 2.087E−3 6.000E−4 1.724E−4 4.951E−5
\(\text{rate}_{{E_{\mathrm{max}_{I}}^{n}}}\) 1.777 1.793 1.798 1.799 1.800  
\(E_{\mathrm{max}}^{n}\) 5.647E−1 3.356E−1 1.823E−1 9.430E−2 4.748E−2 2.354E−2
\(\text{rate}_{E_{\mathrm{max}}^{n}}\) 0.751 0.880 0.951 0.990 1.012  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.700 0.746 0.767 0.777 0.783  
β = 0.4 \(E_{\mathrm{max}_{I}}^{n}\) 6.318E−2 2.163E−2 7.225E−3 2.393E−3 7.906E−4 2.609E−4
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.547 1.582 1.594 1.598 1.599  
\(E_{\mathrm{max}}^{n}\) 2.863E−1 1.468E−1 7.060E−2 3.209E−2 1.358E−2 5.086E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.964 1.056 1.138 1.240 1.417  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.565 0.581 0.589 0.593 0.596  
β = 0.6 \(E_{\mathrm{max}_{I}}^{n}\) 1.758E−1 7.328E−2 2.886E−2 1.110E−2 4.231E−3 1.607E−3
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.263 1.344 1.378 1.392 1.397  
\(E_{\mathrm{max}}^{n}\) 1.758E−1 7.328E−2 2.886E−2 1.110E−2 1.225E−2 1.296E−2
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 1.263 1.344 1.378 -0.142 -0.082  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.398 0.405 0.407 0.407 0.406  
β = 0.8 \(E_{\mathrm{max}_{I}}^{n}\) 5.185E−1 2.947E−1 1.479E−1 6.900E−2 3.100E−2 1.369E−2
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 0.815 0.994 1.100 1.154 1.180  
\(E_{\mathrm{max}}^{n}\) 5.185E−1 2.947E−1 1.479E−1 7.440E−2 7.131E−2 6.516E−2
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.815 0.994 0.992 0.061 0.130  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.214 0.217 0.218 0.216 0.215