Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures

In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $L_{\mathrm{\infty }}$ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.