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The generalized equal width (GEW) wave equation is solved numerically by using lumped Galerkin approach with cubic B-spline functions. The proposed numerical scheme is tested by applying two test problems including single solitary wave and interaction of two solitary waves. In order to determine the performance of the algorithm, the error norms L2 and L∞ and the invariants I1, I2 and I3 are calculated. For the linear stability analysis of the numerical algorithm, von Neumann approach is used. As a result, the obtained findings show that the presented numerical scheme is preferable to some recent numerical methods.

In this article, a collocation algorithm based on quintic B-splines is proposed to find a numerical solution to the nonlinear Generalized Regularized Long Wave (GRLW) equation. Moreover, to analyze the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach to three test examples consisting of a single solitary wave, the collision of two solitary waves, and the growth of an undular bore is discussed. The accuracy of the method is demonstrated by calculating the error in L-2 and L-infinity norms and the conservative quantities l(1) , l(2) and l(3). The findings are compared with those previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to different parameters. (C) 2019 Sharif University of Technology. All rights reserved.

Collaborative filtering is a commonly used method to reduce information overload. It is widely used in recommendation systems due to its simplicity. In traditional collaborative filtering, recommendations are produced based on similarities among users/items. In this approach, the most correlated k neighbors are determined, and a prediction is computed for each user/item by utilizing this neighborhood. During recommendation process, a predefined k value as a number of neighbors is used for prediction processes. In this paper, we analyze the effect of selecting different k values for each user or item. For this purpose, we generate a model that determines k values for each user or item at the off-line time. Empirical outcomes show that using the dynamic k values during the k -nn algorithm leads to more favorable recommendations compared to a constant k value.

In this article, a numerical solution of the Kawahara equation is presented by septic B-spline collocation method. Applying the Von-Neumann stability analysis, the present method is shown to be unconditionally stable. The accuracy of the proposed method is checked by two test problems. L2 and L1 error norms and conserved quantities are given at selected times. The obtained results are found in good agreement with the some recent results.

In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}L2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_{\infty}}$$\end{document}L∞ and the conservative quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{1}}$$\end{document}I1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{2}}$$\end{document}I2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{3}}$$\end{document}I3 are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.

A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error normsL2andL∞and the invariantsI1,I2, andI3. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.