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Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 The absolute errors in the displacement u and the bending moment \(\pmb{u_{xx}}\) for the Euler-Bernoulli beam equation ( 68 ) for Example 1

From: A class of quasi-variable mesh methods based on off-step discretization for the numerical solution of fourth-order quasi-linear parabolic partial differential equations

Methods

Time

N  + 1

k

 

x  = 0.1

x  = 0.2

x  = 0.3

x  = 0.4

x  = 0.5

Proposed method (28a)-(28b)

0.02

20

0.00125

u

1.53 (−08)

2.91 (−08)

4.00 (−08)

4.70 (−08)

4.95 (−08)

\(u_{xx}\)

1.52 (−07)

2.90 (−07)

3.99 (−07)

4.69 (−07)

4.93 (−07)

0.02

40

0.00125

u

5.07 (−09)

9.64 (−09)

1.33 (−08)

1.56 (−08)

1.64 (−08)

\(u_{xx}\)

5.01 (−08)

9.53 (−08)

1.31 (−07)

1.54 (−07)

1.62 (−07)

0.05

20

0.005

u

4.79 (−07)

9.11 (−07)

1.25 (−06)

1.47 (−06)

1.55 (−06)

\(u_{xx}\)

1.08 (−05)

2.05 (−05)

2.82 (−05)

3.31 (−05)

3.48 (−05)

0.05

40

0.005

u

4.19 (−07)

7.96 (−07)

1.10 (−06)

1.29 (−06)

1.35 (−06)

\(u_{xx}\)

3.16 (−06)

6.02 (−06)

8.28 (−06)

9.74 (−06)

1.02 (−05)

Mohammadi [35]

0.02

20

0.00125

u

4.29 (−07)

2.51 (−07)

1.24 (−07)

1.38 (−07)

1.40 (−07)

0.02

40

0.00125

u

8.54 (−08)

6.23 (−08)

4.91 (−08)

5.07 (−08)

5.12 (−08)

0.05

20

0.005

u

2.96 (−06)

1.77 (−06)

1.64 (−06)

2.28 (−06)

2.65 (−07)

0.05

40

0.005

u

9.07 (−07)

7.84 (−07)

7.69 (−07)

8.27 (−07)

8.61 (−08)

Mittal and Jain [34]

0.02

181

0.005

u

1.50 (−07)

2.90 (−07)

3.90 (−07)

4.60 (−07)

4.90 (−07)

0.05

181

0.005

u

1.10 (−06)

2.09 (−06)

2.88 (−06)

3.38 (−06)

3.56 (−06)

Rashidinia and Mohammadi [33]

0.02

20

0.00125

u

4.47 (−07)

2.66 (−07)

1.39 (−07)

1.55 (−07)

1.57 (−07)

0.05

20

0.005

u

2.91 (−06)

1.73 (−06)

1.60 (−06)

2.23 (−06)

2.60 (−07)

Caglar and Caglar [32]

0.02

121

0.005

u

4.80 (−06)

9.70 (−06)

1.40 (−05)

1.90 (−05)

2.40 (−05)

0.02

191

0.005

u

5.20 (−06)

2.10 (−06)

3.10 (−06)

4.20 (−06)

5.20 (−06)

0.02

521

0.005

u

4.90 (−07)

9.90 (−07)

1.40 (−06)

1.90 (−06)

2.40 (−06)

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