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Table 2 Numerical results of problem two at an exact value \(\alpha = 1\)

From: Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg–de Vries equation

\(t\)

Value of (x)

Numerical solution

Exact solution

Abs error

\(t = 0\)

0

1

1

0

2

 − 0.4161468365

 − 0.4161468365

0

4

 − 0.6536436209

 − 0.6536436209

0

6

0.9601702867

0.9601702867

0

8

 − 0.1455000338

 − 0.1455000338

0

10

 − 0.8390715291

 − 0.8390715291

0

\(t = 1\)

0

0.5403023059

0.5403023059

\({3}{\text{.186028260 }} \times 10^{ - 11}\)

2

0.5403023057

0.5403023059

\({2 } \times 10^{ - 10}\)

4

 − 0.9899924967

 − 0.9899924966

\({1} \times 10^{ - 10}\)

6

0.2836621855

0.2836621855

0

8

0.7539022544

0.7539022543

\({1} \times 10^{ - 10}\)

10

 − 0.9111302620

0.9111302619

\({1} \times 10^{ - 10}\)

\(t = 2\)

0

 − 0.4161468396

 − 0.4161468365

\({3}{\text{.05285761300}} \times 10^{ - 9}\)

2

1.000000023

1

\({2}{\text{.3 }} \times 10^{ - 8}\)

4

 − 0.4161468534

 − 0.4161468365

\({1}{\text{.69 }} \times 10^{ - 8}\)

6

 − 0.6536436313

 − 0.6536436209

\({1}{\text{.04 }} \times 10^{ - 8}\)

8

0.9601703110

0.9601702867

\({2}{\text{.43 }} \times 10^{ - 8}\)

10

 − 0.1455000450

 − 0.1455000338

\({1}{\text{.12 }} \times 10^{ - 8}\)

\(t = 3\)

0

 − 0.9899944948

 − 0.9899924966

\(1.9982 \times 10^{ - 6}\)

2

0.5403127921

0.5403023059

\({104862} \times 10^{ - 5}\)

4

0.5402955751

0.5403023059

\({6}{\text{.7308}} \times 10^{ - 6}\)

6

 − 0.9899973839

0.9899924966

\({4}{\text{.8873}} \times 10^{ - 6}\)

8

0.2836729814

0.2836621855

\({1}.07959 \times 10^{ - 5}\)

10

0.7538981557

0.7539022543

\(4.0986 \times 10^{ - 6}\)

  1. Table shows that the absolute error drastically increases as \(t\) increases. When \(t = 0\), the absolute error is zero; when \(t = 1\), the error ranges from \(\times 10^{ - 11}\) to \(\times 10^{ - 10}\); when \(t = 2\), the error increases to an order which ranges from \(\times 10^{ - 8}\) to \(\times 10^{ - 9}\); and at \(t = 3\), the absolute error alternate between \(\times 10^{ - 6}\) to \(\times 10^{ - 6}\).