To study short-time fractional series solutions of nonlinear time–fractional KdV-type systems and to determine, through the equation residual, where a truncated series remains usable.
The modified Riemann–Liouville derivative is used. Applying FRDT gives recurrence relations for the coefficients of powers tkα. These recurrences are written out for the generalized fractional Ito's system, the generalized fractional Drinfeld–Sokolov system and the fractional Kaup–Kupershmidt equation. The truncated series are then substituted back into the original equations to compute residual norms.
Explicit coefficient recurrences are obtained for the three models. In the Banachspace setting used in the paper, with bounded spatial operators and a locally Lipschitz nonlinear part, the coefficients satisfy a majorant estimate and the series converges locally in time. For an N-term truncation, the residual has order O(t(N+1)α) 1 near t = 0. The computations show that the useful time interval depends on both N and α, and is shortest for the Kaup–Kupershmidt case.
The paper gives explicit FRDT coefficient recurrences for three fractional KdV-type models and uses the residual of the original equation to mark the interval on which a truncated fractional series should still be regarded as a valid approximation.
