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Theory and Modern Applications

Table 1 The discrete \(L^{2}\) error and convergence rates in time for schemes (8) and (16) for Example 1 with \(h=\pi /900\) and \(\Delta \alpha =1/300\)

From: Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo–Fabrizio fractional derivative

\(v_{1}\), \(v_{2}\)

τ

\(\Vert e^{N} \Vert ^{\#}_{2}\)

Order

\(\Vert e^{N} \Vert ^{*}_{2}\)

Order

0.1, 0.4

1/22

3.1795e−04

 

3.1810e−04

 

1/23

7.9364e−05

2.0022

7.9543e−05

1.9997

1/24

1.9701e−05

2.0102

1.9887e−05

1.9999

1/25

4.7846e−06

2.0418

4.9718e−06

2.0000

0.5, 0.9

1/22

0.0288

 

0.0288

 

1/23

0.0074

1.9605

0.0074

1.9605

1/24

0.0019

1.9615

0.0019

1.9615

1/25

4.6265e−04

2.0380

4.6265e−04

2.0380

0.2, 0.8

1/22

0.0124

 

0.0124

 

1/23

0.0031

2.0000

0.0031

2.0000

1/24

7.7660e−04

1.9970

7.8234e−04

1.9864

1/25

1.8990e−04

2.0319

1.9569e−04

1.9992