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Theory and Modern Applications

Table 2 Absolute errors for \(N=4\) at equally spaced points \((x_{j}, t_{j})\) for Example 6.1

From: Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

\((x_{j}, t_{j})\)

\(\sigma _{1}(x, t)=1\)

\(\sigma _{2}(x, t)=0.875\)

\(\sigma _{3}(x, t)\)

\(\mathfrak{h}(t)=t \)

\(\mathfrak{h}(t)=e^{t^{2}}\)

\(\mathfrak{h}(t)=t \)

\(\mathfrak{h}(t)=e^{t^{2}}\)

\(\mathfrak{h}(t)=t\)

\(\mathfrak{h}(t)=e^{t^{2}}\)

(0.0, 0.0)

0.00

0.00

0.00

0.00

0.00

 

(0.1, 0.1)

2.90 × 10−22

1.70 × 10−22

1.90 × 10−21

3.00 × 10−22

1.90 × 10−21

4.24 × 10−21

(0.2, 0.2)

9.00 × 10−22

3.30 × 10−21

9.10 × 10−21

3.50 × 10−21

7.80 × 10−21

1.16 × 10−20

(0.3, 0.3)

5.00 × 10−21

1.80 × 10−20

1.90 × 10−20

1.30 × 10−20

1.30 × 10−20

1.00 × 10−21

(0.4, 0.4)

9.00 × 10−21

3.80 × 10−20

8.20 × 10−20

2.10 × 10−20

1.70 × 10−20

2.00 × 10−20

(0.5, 0.5)

2.00 × 10−20

5.00 × 10−20

1.20 × 10−19

3.00 × 10−20

1.00 × 10−20

3.00 × 10−20

(0.6, 0.6)

3.00 × 10−20

4.00 × 10−20

4.00 × 10−20

1.00 × 10−20

2.00 × 10−20

1.00 × 10−20

(0.7, 0.7)

3.00 × 10−20

1.00 × 10−20

2.90 × 10−19

0.00

8.00 × 10−20

1.00 × 10−19

(0.8, 0.8)

3.00 × 10−20

0.00

1.51 × 10−18

1.00 × 10−20

2.60 × 10−19

2.00 × 10−19

(0.9, 0.9)

2.00 × 10−20

5.00 × 10−20

5.68 × 10−18

1.00 × 10−20

9.70 × 10−19

1.40 × 10−19

(1.0, 1.0)

0.00

2.00 × 10−19

1.71 × 10−17

2.00 × 10−20

3.90 × 10−18

4.00 × 10−19