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Table 1 The absolute errors of the approximate and exact solutions for Example 1

From: Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

x

\(\boldsymbol{\varphi_{m,n}(x)}\) : The present method

\(\boldsymbol{\varphi_{h}(x)}\) : The difference method

(2,2)

(2,4)

(4,2)

(4,4)

h  = 0.1

h  = 0.05

0.10

1.8921e − 3

1.3455e − 4

4.0629e − 4

8.1435e − 6

1.7128e − 4

4.2829e − 5

0.20

3.7502e − 3

2.6842e − 4

7.8165e − 4

1.5896e − 5

3.0443e − 4

7.6114e − 5

0.30

5.4142e − 3

3.9569e − 4

1.0209e − 3

2.0120e − 5

3.9356e − 4

9.8358e − 5

0.40

6.4733e − 3

4.8748e − 4

1.2069e − 3

2.3084e − 5

4.3201e − 4

1.0798e − 4

0.50

6.1339e − 3

4.5086e − 4

1.2227e − 3

2.3171e − 5

4.1224e − 4

1.0301e − 4

0.60

4.6732e − 3

2.8768e − 4

1.0183e − 3

1.7900e − 5

3.2672e − 4

8.1720e − 5

0.70

3.0518e − 3

1.2016e − 4

6.9688e − 4

1.1472e − 5

1.6693e − 4

4.1672e − 5

0.80

7.6299e − 4

6.2865e − 5

9.8850e − 4

3.8926e − 6

7.3872e − 5

1.8509e − 5

0.90

3.5063e − 3

3.2838e − 4

2.0438e − 3

2.2064e − 5

4.0280e − 4

1.0080e − 4

1.00

1.2268e − 2

9.0172e − 4

3.5652e − 3

4.6343e − 5

8.2447e − 4

2.0603e − 4