Theory and Modern Applications
Table 3 Errors and corresponding observation orders at \(t=1\), \(\alpha =1.8\)
From: Second-order numerical methods for the tempered fractional diffusion equations
λ | h = τ | \(\gamma _{3}=-0.009\) | \(\gamma _{3}=0\) | \(\gamma _{3}=0.019\) | |||
|---|---|---|---|---|---|---|---|
\(\|e\|_{L_{2}}\) | Order | \(\|e\|_{L_{2}}\) | Order | \(\|e\|_{L_{2}}\) | Order | ||
1/50 | 1/10 | 7.0968e−03 | 8.2095e−03 | 1.0559e−02 | |||
1/20 | 1.7653e−03 | 2.0073 | 2.0554e−03 | 1.9979 | 2.6681e−03 | 1.9846 | |
1/40 | 4.4073e−04 | 2.0019 | 5.1397e−04 | 1.9997 | 6.6861e−04 | 1.9966 | |
1/80 | 1.1015e−04 | 2.0004 | 1.2850e−04 | 1.9999 | 1.6724e−04 | 1.9992 | |
0 | 1/10 | 7.6081e−03 | 8.7279e−03 | 1.1093e−02 | |||
1/20 | 1.8930e−03 | 2.0069 | 2.1846e−03 | 1.9983 | 2.8006e−03 | 1.9858 | |
1/40 | 4.7265e−04 | 2.0018 | 5.4624e−04 | 1.9998 | 7.0166e−04 | 1.9969 | |
1/80 | 1.1812e−04 | 2.0005 | 1.3656e−04 | 2 | 1.7550e−04 | 1.9993 | |
2 | 1/10 | 3.2345e−03 | 3.6657e−03 | 4.5644e−03 | |||
1/20 | 8.2615e−04 | 1.9691 | 9.5022e−04 | 1.9478 | 1.2121e−03 | 1.9129 | |
1/40 | 2.0755e−04 | 1.9929 | 2.3977e−04 | 1.9866 | 3.0865e−04 | 1.9735 | |
1/80 | 5.1949e−05 | 1.9983 | 6.0075e−05 | 1.9968 | 7.7317e−05 | 1.9971 | |