Numerical treatment of non-linear fourth-order distributed fractional sub-diffusion equation with time-delay

In this paper, we proposed a linearized second-order numerical technique for non-linear fourth-order distributed fractional sub-diffusion equation with time delay. Time fractional derivative is represented using Caputo derivative and further approximated using L2−1σ formula which gives second-order temporal convergence and compact difference operator is employed for spatial dimensions. The proposed method is unconditionally stable and convergent to the analytical solution with the order of convergence O(τ2+h4+(Δα)4) improvement in earlier work. Non-linear terms are linearized with the help of Taylor’s series. At the last, we provided a few examples to show the efficiency of the compact difference scheme to support the theoretical results and also presented comparison with L1-approximation of Caputo fractional derivative.