Abstract
The least squares-residual power series method and the residual power series method are two effective methods used to investigate approximate solutions of fractional equations. The least squares-residual power series method combines the least-squares method with the residual power series method. These methods are used to solve the linear and nonlinear time-fractional regularized long wave equations. These equations describe the nature of ion-acoustic waves in plasma and shallow water waves in oceans. Initially, we used the classic residual power series method to find analytical solutions. Then, the expression of fractional Wronskian is examined to obtain the linear independence of the functions. Later, a linear equation system is written by using an approximate solution to obtain unknown coefficients. Eventually, the least-squares method is applied to obtain the unknown coefficients. The least squares-residual power series method is more advantageous for including fewer expansion terms than the residual power series method. Examining the physical interpretations, it can be said that the absolute error for the solution obtained by the least squares-residual power series method is closer to zero.