In the paper, a linearized compact finite difference scheme is presented for the semilinear fractional delay convection-reaction–diffusion equation. Firstly, the equation is transformed into an equivalent semilinear fractional delay reaction–diffusion equation by using a special transformation. Then, the temporal Caputo derivative is discreted by using approximation and the second-order spatial derivative is approximated by the compact finite difference scheme. The solvability, unconditional stability, and convergence in the sense of - and - norms are proved rigorously. Finally, numerical examples are carried out extensively to support our theoretical analysis.

The present work is mainly devoted to studying the fractional nonlinear Schrödinger equation with wave operator. We first derive two conserved quantities of the equation, and then develop a three-level linearly implicit difference scheme. This scheme is shown to be conserves the discrete version of conserved quantities. Using energy method, we prove that the difference scheme is unconditionally stable, and the difference solution converges to the exact one with second order accuracy in both the space and time dimensions. Numerical experiments are performed to support our theoretical analysis and demonstrate the accuracy, discrete conservation laws and effectiveness for long-time simulation.