Unconditional optimal error bounds of the fast nonuniform Alikhanov scheme for a nonlinear time-fractional biharmonic equation

In this work, we will consider the two-dimensional nonlinear time-fractional biharmonic equation with weakly singular solutions. By introducing an intermediate variable, the original problem is transformed into an equivalent system with two second-order differential equations. In order to reduce computational costs and storage requirements for this equivalent system, the numerical method of this paper applies the fast nonuniform Alikhanov scheme to discrete the Caputo derivative, the compact difference scheme in space, and the Newton linearization for the nonlinear term. By using an \(\alpha \) $\alpha$ -robust discrete fractional Gronwall inequality, the unconditional optimal convergence analysis of the proposed scheme in \(H^2\) $H^2$ -norm is given, and the obtained convergent result is \(\alpha \) $\alpha$ -robust. Finally, numerical experiments are given to further verify our theoretical analysis.