The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 2 Errors and corresponding observation orders with \(\lambda=1\), and the parameters \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\) are selected in \(S_{1}^{\alpha}(\gamma_{1},\gamma_{2},\gamma_{3})\)

α	h	τ	T-ML₂		T-WSGL (\(\gamma_{1}=0.61\))
α	h	τ	∥ε(h,τ)∥	Rate	∥ε(h,τ)∥	Rate
1.2	\(\frac{1}{10}\)	\(\frac{1}{10}\)	1.9919e − 05	–	1.9478e − 05	–
	\(\frac{1}{20}\)	\(\frac{1}{20}\)	4.4251e − 06	2.1703	4.3499e − 06	2.1628
	\(\frac{1}{40}\)	\(\frac{1}{40}\)	1.0291e − 06	2.1044	1.0137e − 06	2.1014
	\(\frac{1}{80}\)	\(\frac{1}{80}\)	2.4771e − 07	2.0546	2.4421e − 07	2.0534
	\(\frac{1}{160}\)	\(\frac{1}{160}\)	6.0724e − 08	2.0283	5.9887e − 08	2.0278

α	h	τ	T-ML₂		T-WSGL (\(\gamma_{1}=0.76\))
α	h	τ	∥ε(h,τ)∥	Rate	∥ε(h,τ)∥	Rate
1.5	\(\frac{1}{10}\)	\(\frac{1}{10}\)	1.3017e − 05	–	1.3016e − 05	–
	\(\frac{1}{20}\)	\(\frac{1}{20}\)	3.3092e − 06	1.9759	3.3016e − 06	1.9791
	\(\frac{1}{40}\)	\(\frac{1}{40}\)	8.4075e − 07	1.9768	8.3865e − 07	1.9770
	\(\frac{1}{80}\)	\(\frac{1}{80}\)	2.1209e − 07	1.9870	2.1157e − 07	1.9869
	\(\frac{1}{160}\)	\(\frac{1}{160}\)	5.3289e − 08	1.9928	5.3160e − 08	1.9927

α	h	τ	T-ML₂		T-WSGL (\(\gamma_{1}=0.92\))
α	h	τ	∥ε(h,τ)∥	Rate	∥ε(h,τ)∥	Rate
1.8	\(\frac{1}{10}\)	\(\frac{1}{10}\)	1.4954e − 05	–	1.4668e − 05	–
	\(\frac{1}{20}\)	\(\frac{1}{20}\)	3.3806e − 06	2.1451	3.6011e − 06	2.0262
	\(\frac{1}{40}\)	\(\frac{1}{40}\)	8.1335e − 07	2.0553	9.0055e − 07	1.9996
	\(\frac{1}{80}\)	\(\frac{1}{80}\)	2.0006e − 07	2.0235	2.2513e − 07	2.0001
	\(\frac{1}{160}\)	\(\frac{1}{160}\)	4.9655e − 08	2.0104	5.6275e − 08	2.0002