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Theory and Modern Applications

Table 13 Problem 5 : MAEs and RMSEs using ( 2.21a )-( 2.21b ) with different values of C

From: A class of quasi-variable mesh methods based on off-step discretization for the solution of non-linear fourth order ordinary differential equations with Dirichlet and Neumann boundary conditions

λ

N

 

Uniform mesh ( C  = 1)

Quasi-variable mesh ( C  = 0.7)

MAE

RMSE

MAE

RMSE

1

16

u

1.59e−07

1.20e−08

1.01e−06

4.22e−08

\(u' \)

5.68e−07

3.72e−07

3.29e−06

1.59e−06

32

u

1.00e−08

2.13e−10

1.11e−07

1.20e−09

\(u' \)

3.55e−08

1.34e−08

3.52e−07

9.15e−08

64

u

6.29e−10

3.54e−12

1.29e−08

3.57e−11

order

4.00

5.91

3.10

5.07

\(u' \)

2.23e−09

4.49e−10

4.10e−08

5.47e−09

order

4.00

4.90

3.10

4.06

10

16

u

9.90e−07

1.24e−07

2.08e−06

1.58e−07

\(u' \)

4.63e−06

3.69e−06

8.43e−06

5.61e−06

32

u

6.20e−08

2.25e−09

1.82e−07

3.64e−09

\(u' \)

2.85e−07

1.40e−07

7.32e−07

2.70e−07

64

u

3.88e−09

3.79e−11

1.82e−08

9.09e−11

order

4.00

5.89

3.33

5.32

\(u' \)

1.77e−08

4.78e−09

7.18e−08

1.38e−08

order

4.01

4.87

3.35

4.30

100

16

u

5.57e−06

1.55e−06

7.17e−06

1.33e−06

\(u' \)

3.70e−05

3.70e−05

3.99e−05

3.99e−05

32

u

3.30e−07

3.14e−08

4.53e−07

2.65e−08

\(u' \)

2.18e−06

1.77e−06

2.55e−06

1.82e−06

64

u

2.03e−08

5.59e−10

3.36e−08

5.33e−10

order

4.02

5.81

3.75

5.64

\(u' \)

1.34e−07

6.76e−08

1.89e−07

7.79e−08

order

4.02

4.71

3.75

4.55