A lumped Galerkin finite element method for the generalized Hirota-Satsuma coupled KdV and coupled MKdV equations.

*(English)*Zbl 1434.65284Summary: In the present study, a Lumped Galerkin finite element method using quadratic B-splines has been applied to the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) and coupled modified Korteweg-de Vries (mKdV) equations. The numerical solutions of discretized equations using Lumped Galerkin finite element method have been obtained using the fourth order Runge-Kutta method which is widely used for the solution of ordinary differential equation system. The numerical solutions obtained for various space and time values have been compared with exact ones using the error norms \(L_2\) and \(L_{\infty}\). Lumped Galerkin finite element method is an effective one which can be applied to a wide range of nonlinear evolution equations.

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65D07 | Numerical computation using splines |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

##### Keywords:

the generalized Hirota-Satsuma coupled KdV and coupled mKdV; finite element method; Galerkin method; B-spline; Runge-Kutta
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\textit{N. M. Yagmurlu} et al., Tbil. Math. J. 12, No. 3, 159--173 (2019; Zbl 1434.65284)

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