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Theory and Modern Applications

Table 16 Problem 7 : MAEs and RMSEs using ( 2.7a )-( 2.7b ) and ( 2.21a )-( 2.21b ) with \(\pmb{C=1}\)

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

N

 

Second order

Fourth order

 

MAE

RMSE

MAE

RMSE

10

u

7.1411e−06

4.8598e−06

1.5183e−08

1.0322e−08

\(u' \)

2.4886e−05

1.6832e−05

4.8261e−08

3.5393e−08

v

3.3399e−06

2.2408e−06

2.0788e−08

1.4229e−08

\(v' \)

1.1264e−05

8.1372e−06

7.3344e−08

4.6608e−08

20

u

1.8060e−06

1.1822e−06

9.4738e−10

6.2681e−10

\(u'\)

6.4868e−06

4.1284e−06

3.0174e−09

2.1522e−09

v

8.5494e−07

5.5949e−07

1.2768e−09

8.4967e−10

\(v' \)

2.6592e−06

1.9574e−06

4.7506e−09

2.9017e−09

40

u

4.5147e−07

2.9169e−07

5.9265e−11

3.8611e−11

\(u'\)

1.6208e−06

1.0204e−06

1.8898e−10

1.3259e−10

v

2.1498e−07

1.3896e−07

8.0073e−11

5.2627e−11

\(v' \)

6.7508e−07

4.8234e−07

3.0237e−10

1.8155e−10

Order

u

2.0046

2.0092

4.00

4.02

Order

\(u' \)

2.0020

2.0087

4.00

4.02

Order

v

2.0253

2.0288

4.00

4.01

Order

\(v' \)

2.0268

2.0305

3.97

4.00