Least squares residual power series solutions for Kawahara and Rosenau-Hyman nonlinear wave interactions with applications in fluid dynamics

The present study uses the least squares residual power series (LSRPS) method to obtain approximate solutions to the nonlinear fractional-order Kawahara and Rosenau- Hyman equations. This method combines the residual power series (RPS) technique and the least squares approach. The calculations are obtained using Caputo's sense as a basis. To obtain approximations of solutions, the well-known RPS method is first used. The functions are then proven to be linearly independent by checking the Wronskian determinant at fractional order. Next, a system of linear equations is generated and processed using the least squares approach. Using the least squares method, which uses fewer expansion terms than the classical RPS method, approximate solutions are determined. The problems presented below demonstrate how much faster the proposed method converges compared to the RPS method. Numerical results are presented to demonstrate the efficiency, accuracy, and rapid convergence of the method.