Application of Bernstein collocation method for solving the generalized regularized long wave equations

Abstract

The regularized and the modified regularized long wave (RLW and MRLW) equations are solved numerically by the Bernstein polynomials in both the space and time directions based on Kronecker product. In this paper, we applied a fully different Bernstein collocation method than the other methods which used Bernstein polynomials to solve the problems. The approximate solution is defined by the Bernstein polynomials in all directions. A general form for any m derivative of any Bernstein polynomials is constructed. A general matrix form for the vector of any m derivative of any Bernstein polynomials is also constructed. Convergence study for the proposed numerical scheme is investigated. To determine the conservation properties of the RLW and MRLW equations, three invariants of motion (I1, I2 and I3) are computed. To test the accuracy, two error norms (‖E‖2 and ‖E‖∞) are evaluated. Numerical outcomes and comparisons with other techniques for the single and the interaction of two solitary waves for RLW and MRLW equations are presented.

Keywords

• Bernstein collocation method
• RLW equation
• MRLW equation
• Kronecker product
• Partial differential equations