The general analytical and numerical solution for the modified KdV equation with convergence analysis. (English) Zbl 1269.65101
The authors apply the homotopy perturbation method (HPM) (cf. [J.-H. He, Topol. Methods Nonlinear Anal. 31, No. 2, 205–209 (2008; Zbl 1159.34333)]) for solving the modified Kortweg-de Vries (mKdV) equation, which plays a crucial role in applied Mathematics and Physics as a nonlinear convection and alike diffusion problem. The mKdV equation has been solved numerically by the HPM and Runge-Kutta discontinuous Galerkin method (see [M. Asadzadeh et al., J. Mathematical Sciences 175, 228–248 (2011)]. Numerical and exact solutions are compared. Finally, the authors arrive at the conclusion that the HPM for mKdV can be a simple method instead of some complicated methods.
Reviewer: H. P. Dikshit (Bhopal)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
Keywords:
convergence; modified Kortweg-de Vries equation; commutative hypercomplex mathematics; homotopy perturbation method; Runge-Kutta discontinuous Galerkin methods; nonlinear convection-diffusion equation; numerical examplesCitations:
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