Hybridization of the Analytic Banach’s Contraction Method Using the Particle Swarm Optimization Algorithm for Fractional System Solving Drinfeld-Sokolov-Wilson

In this manuscript, the fractional equations, which are one of the complex problems facing most researchers in the Drinfeld-Sokolov-Wilson system, were addressed, and solved using the analytical method (Banach’s contraction) and then using the smart method (particle swarm optimization), by choosing the best values for the parameters in the system, using the smart algorithm and improving the results for the system. The results obtained from the hybrid method (BCM – PSO), proved the efficiency of this method by finding Maximum Absolute Error (MAE) and Mean Square Error (MSE) compared with the exact solution. The tables for several different values of (α) show that the hybrid method is better than the analytical method.