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Table 1 Comparison of various iterations for \(\lambda =1\) and \(\alpha _{k}=\beta _{k}=0.9\) for \(k=0\) to 10

From: Numerical solution of Bratu’s boundary value problem based on Green’s function and a novel iterative scheme

t

Exact-Solution

Mann–Green

Ishikawa–Green

New

0.1

0.049846700

3.92372 × 10−9

2.59312 × 10−11

5.48173 × 10−16

0.2

0.089189935

7.48453 × 10−9

4.88960 × 10−11

1.04083 × 10−15

0.3

0.117609095

1.03321 × 10−8

6.69472 × 10−11

1.41553 × 10−15

0.4

0.134790252

1.21720 × 10−8

7.84832 × 10−11

1.66533 × 10−15

0.5

0.140539214

1.28083 × 10−8

8.24510 × 10−11

1.74860 × 10−15

0.6

0.134790252

1.21720 × 10−8

7.84832 × 10−11

1.69309 × 10−15

0.7

0.117609095

1.03321 × 10−8

6.69472 × 10−11

1.33227 × 10−15

0.8

0.089189935

7.48453 × 10−9

4.88959 × 10−11

9.99201 × 10−16

0.9

0.089189935

3.92372 × 10−9

2.59312 × 10−11

4.64906 × 10−16