The long time error estimates for the second order backward difference approximation to sub-diffusion equations with boundary time delay and feedback gain.

We consider time fractional diffusion initial boundary value problem with boundary time delay and feedback gain involving two Caputo fractional derivatives in time. When both the time delay and feedback gain are taken as appropriately small such that the characteristic equation has no solution in the location of the right half plane of the imaginary axis, we derive the solution’s existence, uniqueness and representation. The second-order backward difference time discrete schemes are investigated in time discretization. The lt∞(0,∞;L2) error analysis of the numerical schemes is derived when both the time delay and feedback gain are appropriately small such that the characteristic equation of the discrete problem has no solution in a strip of the right half plane of the imaginary axis, and the order of boundary Caputo fractional derivative is equal to the order or half order of interior domain Caputo fractional derivative. The numerical examples illustrate its accuracy, efficiency and robustness, and are consistent with the theoretical results.