The Aboodh–Temimi–Ansari Technique for the Time-Fractional Drinfeld–Sokolov–Wilson System: Convergence Analysis and Numerical Validation

The Drinfeld–Sokolov–Wilson (DSW) system originates in the theory of affine Lie algebras and the Korteweg–de Vries hierarchy, so that its integrability reflects an underlying algebraic symmetry that persists in rescaled form under fractionalization, and it also models dispersive shallow-water waves. This work applies the hybrid Aboodh–Temimi–Ansari technique to the time-fractional DSW system under the Caputo derivative, a scheme that requires neither Adomian nor He polynomials. We show that its iterates are exactly the Picard iterates of an equivalent fractional Volterra integral equation. Under a Lipschitz condition on the nonlinear operator, this yields the existence and uniqueness of a local solution by a Banach fixed-point argument, together with explicit bounds on the truncation error. We also show that the dispersive term of the DSW system prevents this condition from holding in the classical spaces of bounded smooth functions, so that the well-posedness of the fractional system itself remains open. Two initial value problems built from known kink and soliton profiles are solved, recovering the exact classical solution when the fractional order equals one. After two corrections, the approximations agree with the exact profiles to several significant figures, with the residual error concentrated where the nonlinear coupling is largest.