PurposeThe purpose of this paper is to investigate the numerical solutions of the Burgers' and modified Burgers' equation using sextic B‐spline collocation method.Design/methodology/approachCrank‐Nicolson central differencing scheme has been used for the time integration and sextic B‐spline functions have been used for the space integration to the modified and time splitted modified Burgers' equation.FindingsIt has been found that the proposed method is unconditionally stable and obtained results are consistent with some earlier published studies.Originality/valueSextic B‐spline collocation method for the Burgers' and modified Burgers' equation is given.

Quintic B-spline collocation schemes for numerical solution of the regularized long wave (RLW) equation have been proposed. The schemes are based on the CrankNicolson formulation for time integration and quintic B-spline functions for space integration. The quintic B-spline collocation method over finite intervals is also applied to the time-split RLW equation and space-split RLW equation. After stability analysis is applied to all the schemes, the results of the three algorithms are compared by studying the propagation of the solitary wave, interaction of two solitary waves and wave undulation.2000 Mathematics subject classification: 65D07, 65N30, 65N35, 76B25.

Both time- and space-splitted Burgers' equations are solved numerically. Cubic B-spline collocation method is applied to the time-splitted Burgers' equation. Quadratic B-spline collocation method is used to get numerical solution of the space-splitted Burgers' equation. The results of both schemes are compared for some test problems.

In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second-and third-order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L ∞ error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results.

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