LAUSR

In this paper, we investigate the fractional backward differential formulas for the distributed-order differential equation with time delay. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional form. Next the transformed multi-term fractional equation is solved by discretizing in space by the fractional backward differential formulas method and in time by using the Crank–Nicolson scheme. We prove that proposed scheme is stable and convergent for $$\beta < \frac{5}{8}$$β<58 with the accuracy $$\mathrm{O}({h^2} + {\kappa ^2} + {\sigma ^2})$$O(h2+κ2+σ2). Finally, we give some examples to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis.