The current work highlights the following issues: a brief survey of the development in the theory of fractional differential equations has been raised. A very recent technique based on the generalized Taylor series called—residual power series (RPS)—is introduced in detailed manner. The time-fractional foam drainage equation is considered as a target model to test the validity of the RPS method. Analysis of the obtained approximate solution of the fractional foam model reveals that RPS is an alternative method to be added for the fractional theory and computations and considered to be a significant method for exploring nonlinear fractional models.