Skip to main content

Table 3 Errors and order convergence for \(u_{\mathrm{ex}}^{1,\beta }\) using (42) 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 \(E_{\mathrm{max}_{I}}^{n}\) 1.992E−2 5.794E−3 1.670E−3 4.801E−4 1.379E−4 3.961E−5
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.782 1.795 1.798 1.800 1.800  
\(E_{\mathrm{max}}^{n}\) 5.719E−1 3.434E−1 1.894E−1 9.953E−2 5.104E−2 2.584E−2
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.736 0.859 0.928 0.964 0.982  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.921 0.981 0.994 1.004 1.005  
β = 0.4 \(E_{\mathrm{max}_{I}}^{n}\) 3.875E−2 1.307E−2 4.346E−3 1.437E−3 4.745E−4 1.566E−4
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.567 1.589 1.596 1.599 1.600  
\(E_{\mathrm{max}}^{n}\) 3.110E−1 1.691E−1 8.825E−2 4.511E−2 2.281E−2 1.147E−2
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.879 0.938 0.968 0.984 0.992  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 0.999 1.020 1.023 1.020 1.016  
β = 0.6 \(E_{\mathrm{max}_{I}}^{n}\) 7.709E−2 3.042E−2 1.171E−2 4.465E−3 1.696E−3 6.432E−4
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.341 1.377 1.391 1.397 1.399  
\(E_{\mathrm{max}}^{n}\) 1.927E−1 1.011E−1 5.184E−2 2.627E−2 1.323E−2 6.639E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 0.930 0.963 0.981 0.990 0.995  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 1.024 1.037 1.038 1.034 1.030  
β = 0.8 \(E_{\mathrm{max}_{I}}^{n}\) 1.551E−1 7.262E−2 3.268E−2 1.444E−2 6.326E−3 2.761E−3
\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\) 1.095 1.152 1.179 1.191 1.196  
\(E_{\mathrm{max}}^{n}\) 1.551E−1 7.262E−2 3.303E−2 1.668E−2 8.391E−3 4.210E−3
\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\) 1.095 1.137 0.985 0.991 0.995  
\(\text{rate}_{E_{\mathrm{conv}}^{n}}\) 1.016 1.031 1.036 1.037 1.037