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Table 2 Errors and order convergence for \(u_{\mathrm{ex}}^{2,\beta }\) using 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 \(E_{\mathrm{max}_{I}}^{n}\) 2.204E−2 6.439E−3 1.858E−3 5.344E−4 1.535E−4 4.410E−5
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.775 1.793 1.798 1.799 1.800  
\(E_{\mathrm{max}}^{n}\) 4.902E−2 2.435E−2 1.203E−2 5.917E−3 2.899E−3 1.413E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 1.009 1.017 1.024 1.030 1.036  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.638 0.659 0.681 0.700 0.716  
β = 0.4 \(E_{\mathrm{max}_{I}}^{n}\) 5.527E−2 1.897E−2 6.342E−3 2.101E−3 6.943E−4 2.291E−4
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.543 1.581 1.594 1.598 1.599  
\(E_{\mathrm{max}}^{n}\) 5.527E−2 1.897E−2 8.380E−3 3.611E−3 1.421E−3 1.058E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 1.543 1.179 1.214 1.346 0.426  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.523 0.539 0.553 0.564 0.573  
β = 0.6 \(E_{\mathrm{max}_{I}}^{n}\) 1.515E−1 6.342E−2 2.502E−2 9.630E−3 3.671E−3 1.394E−3
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.256 1.342 1.377 1.391 1.397  
\(E_{\mathrm{max}}^{n}\) 1.515E−1 6.342E−2 2.502E−2 9.630E−3 6.739E−3 5.411E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 1.256 1.342 1.377 0.515 0.317  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.375 0.383 0.389 0.393 0.395  
β = 0.8 \(E_{\mathrm{max}_{I}}^{n}\) 4.380E−1 2.510E−1 1.265E−1 5.908E−2 2.656E−2 1.173E−2
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 0.803 0.989 1.098 1.153 1.179  
\(E_{\mathrm{max}}^{n}\) 4.380E−1 2.510E−1 1.265E−1 5.908E−2 2.697E−2 2.394E−2
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.803 0.989 1.098 1.131 0.172  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.199 0.202 0.203 0.204 0.204