Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

Table 1

Comparison of the HAM solution with exact solution when ℏ=−1 and 𝑡=0,0.25,0.5,0.75,1, respectively.

𝑡	𝑥	Exact solution	HAM solution	Absolute error
0.05	−6; −3; 2; 5	−0.004693564364865844; −0.08634570715015544; −0.21807963830331248; −0.01396823161345761	−0.0046935643648653635; −0.08634570715015137; −0.21807963830330718; −0.013968231613456493	4.80518402845575×10^−16; 4.06619182768963 × 10^−15; 5.30131494258512 × 10^−15; 1.11716191852906 × 10^−15
0.25	−6; −3; 2; 5	−0.003846044713671389; −0.07186718166243183; −0.2522584503864486; −0.017007824315900595	−0.0038460447137292153; −0.07186718165065627; −0.25225845037331035; −0.017007824315379085	5.782613970994888 × 10^−14; 1.17755666328989×10^−11; 1.31382682511116×10^−11; 5.21509918582907×10^−13
0.5	−6; −3; 2; 5	−0.0029978573890808136; −0.056906053378639285; −0.29829291297695726; −0.02173245972250237	−0.002997857417138099; −0.05690604775947111; −0.2982929041406657; −0.02173245944445047	2.805728522778383×10^−11; 5.619168172432687×10^−9; 8.836291531810758×10^−9; 2.78051900948206×10^−10
0.75	−6; −3; 2; 5	−0.002336284005515009; −0.044899022133116494; −0.3462100376919465; −0.027731694033315973	−0.0023362850215908897; −0.04489882076408693; −0.3462095739849941; −0.02773168287526794	1.016075880551359×10^−9; 2.013690295621373×10^−7; 4.637069524471293×10^−7; 1.115804803414333×10^−8
1	−6; −3; 2; 5	−0.0018204295917628886; −0.03532791434845481; −0.3932319186044759; −0.03532556752567461	−0.001820442360243653; −0.03532541242658223; −0.3932238664829637; −0.03532541242658223	1.276848076445583×10^−8; 2.50192187×10^−7; 8.052121512225341×10^−7; 1.550990923818163×10^−7