Abstract
In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including; single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, L2 and L∞ and invariants I1, I2 and I3 have been calculated. Our numerical results are compared with some of those available in the literature.
Highlights
Several physical processes for example dispersion of long waves in shallow water waves under gravity, bubble-liquid mixtures, ion acoustic plasma waves, fluid mechanics, nonlinear optics and wave phenomena in enharmonic crystals can be expressed by the Korteweg de-Vries (KdV) equation which was first introduced by Korteweg and de Vries [1]
Numerical solutions of the KdV equation were obtained using differential quadrature method based on cosine ex
Collocation finite element method based on quintic B-spline functions is applied to the generalized KdV equation by Ak et al [30]
Summary
Several physical processes for example dispersion of long waves in shallow water waves under gravity, bubble-liquid mixtures, ion acoustic plasma waves, fluid mechanics, nonlinear optics and wave phenomena in enharmonic crystals can be expressed by the KdV equation which was first introduced by Korteweg and de Vries [1]. Zabusky and Kruskal [4] were first obtained numerical solutions of the equation with finite difference method. Numerical solutions of the KdV equation were obtained using differential quadrature method based on cosine ex-. MKdV equation has been solved by using Galerkins’ method with quadratic B-spline finite elements by Biswas et al [18]. Collocation finite element method based on quintic B-spline functions is applied to the generalized KdV equation by Ak et al [30]. One of the primary mathematical models for describing the theory of water waves in shallow channels is the following Korteweg de-Vries (KdV) equation: Ut + εUU x + μU xxx = 0. We have numerically solved the GKdV equation using collocation method with septic B-spline finite elements. We have showed the suggested method is unconditionally stable applying the von-Neumann stability analysis
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