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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 L 2 and L∞ and the invariants I 1 , I 2 and I 3 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 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 and and the conservative quantities , and 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.

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 usedfor 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 on movie based dataset show that using the dynamic k values during the k-nn algorithm leads to more favorable recommendations compared to a constant k value.

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 norms 2 and ∞ and the invariants 1 , 2 , and 3 . 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.