The Boundary Forced MKdV Equation

doi:10.1006/jcph.1994.1113

Journal of Computational Physics

Volume 113, Issue 1, July 1994, Pages 5-12

https://doi.org/10.1006/jcph.1994.1113 Get rights and content

Abstract

An unconditionally stable numerical algorithm for the modified Korteweg-de Vries equation based on the B-spline finite element method is described. The algorithm is validated through a single soliton simulation. In further numerical experiments forced boundary conditions u = U₀ are applied at the end x = 0 and the generated states of solitary waves are studied. By long impulse experiments these are shown to be generated periodically with period ΔT_B proportional to U^-3₀ and to have a limiting amplitude proportional to U₀. This limit is achieved by all waves, after the first, provided the experiment proceeds long enough. The temporal development of the derivatives U'(0, t), U"(0, t) and U'"(0, t ) is also periodic, with period ΔT_B. The effect of negative forcing is to generate a train of negative waves. The solitary wave states generated by applying a positive impulse followed immediately by an negative impulse, of equal amplitude and duration, is dependent on the period of forcing. The solitary waves generated by these various forcing functions possess many of the attributes of free solitons.

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Researchers used different analytical and numerical techniques for the solution of KdV equation (1) like Wang H, and Xiang C. [48] used Jacobi elliptic periodic function transform technique, Bakodah et al. [5] employed Adomian decomposition method based on convergence parameter, Syam [46] used Adomian decomposition method, Lakestani [20] has used B-Spline Functions, Korkmaz [18] has utilized cosine expansion based differential quadrature methods, El-Wakil et al. [7] have used computerized technique using symbolic computation, Abbassbanday and Zakaria [1] used homotopy analysis method, Chebyshev spectral method has been used by Helal [13], Seadawy [30] employed variational approximation method for explicit solutions as well as the direct algebraic method [31] of different types of KdV equations, while variational iteration method was employed by Wazwaz [49]. Many authors have solved equation (2) by different techniques like Geyikli and Kaya [11] employed both the finite element and the decomposition methods, Gardner et al. [10] utilized the B-spline finite element technique, El-Wakil et al. [7] has used computerized technique using symbolic computation, Shoucri et al. [44] used shape-preserving splines, Fontenelle et al. [9] presented recurrence relations for generating the terms for densities, Iqbal et al. [15] used direct algebraic method with new modification, while Mokhtari and Mohseni [24] have used meshless method based on global collocation using RBF, direct algebraic method, extended mapping method and Seadawy techniques [26,34–36,40–42]. To solve the model (3) numerically several methods can be found in the literature like Bakodah et al. [5] employed Adomian decomposition method based on convergence parameter, the direct method and leading order analysis technique were used by Zhang [51], Lu and Shi [21] have used Jacobi elliptic functions expansion method, modified functional variable method has been utilized by Djoudi and Zerarka [6], Kaya and Inan [16] employed decomposition method, an effective homogeneous balance method is used by Yu [50], unified algebraic method used by Fan [8], Seadawy and Rashidy [37] used a transformation method, while extended mapping method has been used by Krishnan and Peng [19].
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Regular ArticleThe Boundary Forced MKdV Equation

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Regular Article
The Boundary Forced MKdV Equation