Some applications of the least squares-residual power series method for fractional generalized long wave equations

Abstract

This article examines a new effective method called the least squares-residual power series method (LS-RPSM) and compares this method with the RPSM. The LS-RPSM assembles the least-squares process with the residual power series method. These techniques are applied to investigate the linear and nonlinear time-fractional regularized long wave equations (TFRLWEs). The RLW models define the shallow water waves in oceans and the internal ion-acoustic waves in plasma. Firstly, we apply the well-known RPSM to acquire approximate solutions. In the next step, the Wronskian determinant is searched in fractional order to show that the functions are linearly independent. After these operations, a system of linear equations is obtained. In the last step, the least-squares algorithm is used to find the necessary coefficients. When this article is examined, it can be said that LS-RPSM is more useful because it requires using fewer terms than the required number of terms when applying the RPSM. Additionally, the experiments show that this method converges better than RPSM.