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In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L2 and L∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.

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In this study, the Rosenau-Korteweg-de Vries-Regular Longwave (Rosenau-KdV-RLW) equation has been converted into a partial differential equation system consisting of two equations using a splitting technique. Then, numerical solutions for the Rosenau-KdV-RLW equation system have been obtained using separately both cubic and quintic Bspline finite element collocation method. For the unknowns in those equations, B-spline functions at x-position and Crank-Nicolson type finite difference approaches at time positions are used. A test problem has been chosen to check the accuracy of the proposed discretized scheme. The basic conservation properties of the Rosenau-KdV-RLW equation have been shown to be protected by the proposed numerical scheme. The results are compared with the analytical solution of the problem and the results given in the literature. For the reliability of the method the error norms ‫ܮ‬ ଶ and ‫ܮ‬ ஶ are calculated. It is seen that the proposed method gives harmonious results with exact solutions.

In this study, a numerical scheme based on a collocation finite element method using quintic B‐spline functions for getting approximate solutions of nonhomogeneous Rosenau type equations prescribed by initial and boundary conditions is proposed. The numerical scheme is tested on four model problems with known exact solutions. To show how accurate results the proposed scheme produces, the error norms defined by L2 and L∞ are calculated. Additionally, the stability analysis of the scheme is done by means of the von Neuman method.