Numerical Solutions of Coupled Nonlinear Evolution Equations via El-gendi Nodal Galerkin Method
M. El-Kady *
Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt.
Salah M. El-Sayed
Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Egypt.
Heba. E. Salem
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Coupled mKdV equation, generalized Hirota-Satsuma Coupled KdV equation, El-gendi nodal Galerkin method, Legendre and Chebyshev cardinal functions