A numerical application of the decomposition method for the combined KdV-MKdV equation

We consider solitary-wave solutions of the combined Korteweg de Vries-Modified Korteweg de Vries (KdV-MKdV) equation. We prove the convergence of Adomian decomposition method (ADM) applied to the combined KdV-MKdV equation. Then we obtain the exact solitary-wave solutions and numerical solutions of the combined KdV-MKdV equation for the initial conditions. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy are finally demonstrated for the combined KdV-MKdV equation.