Spatiotemporal variable-order fractional modeling of long wave interactions: a spectral tau method with Mittag–Leffler-type memory

The long-wave interactions described by the generalized Hirota–Satsuma (GHS) coupled KdV system are studied here through a memory dependent model in which the time fractional order varies in both space and time. The fractional derivative is taken in the Atangana Baleanu Caputo (ABC) sense, whose Mittag Leffler kernel is non singular and captures the nonlocal nature of the dynamics. To solve this model, a fully spectral tau scheme is developed: Jacobi polynomials approximate the spatial behaviour, their shifted counterparts handle the temporal direction, and three purpose built operational matrices carry the computational load. These matrices act on the variable order ABC derivative, on the integer-order spatial derivatives, and on the space time Kronecker products generated by the nonlinear terms, so that the original fractional system collapses into a nonlinear algebraic one that requires no node-based evaluation. The scheme is supported by a complete theoretical treatment that covers approximation estimates in weighted Sobolev spaces, a stability result for the algebraic system, and a convergence theorem accompanied by an explicit error bound. Two test problems illustrate these findings, exhibiting exponential decay of the error and delivering better accuracy than a recent Chebyshev Gauss Lobatto collocation method even with fewer basis functions.