In this research the solution of coupled modified Korteweg-de Vries equation (mKdV) and the generalized Hirota–Satsuma coupled KdV equation by using El-gendi nodal Galerkin (EGG)
approaches are presented. El-gendi nodal Galerkin (EGG) (EGG) approaches consist of two approaches, the first is El-gendi Chebyshev nodal Galerkin (ECG) and the second approach is
called El-gendi Legendre nodal Galerkin (ELG). In these new approaches spaces of the solution and the weak form to the system are presented. The resulted systems of ODES are solved by the fourth order Runge-Kutta solver. The convergence and the stability of these new methods are analyzed numerically. Numerical results are presented and compared with the results obtained
by pseudo-spectral method.