A Fast High-Order Nonpolynomial Spline Method for Nonlinear Time-Fractional Telegraph Model for Neutron Transport in a Nuclear Reactor

The aim of the present study is to develop a fast high-order method for a nonlinear time-fractional telegraph model for neutron transport in a nuclear reactor. The discretization of the model involves the utilization of a fast Alikhanov formula for the time-fractional derivative and a high-order nonpolynomial spline scheme for the spatial variable. The proposed algorithm is computationally efficient with computational cost of $𝒪\left(MN{\log }^{2}N\right)$ and a storage requirement of $𝒪\left(M{\log }^{2}N\right)$, where $M$ and $N$ represent the total number of grids in space and time, respectively. The stability and convergence of the developed method are established using the discrete energy approach in the $L_2$ norm. It is proved that the developed method converges with an order of $𝒪({\tau }^2,h^{4.5},\widetilde{ϵ}),$ where $\tau$ and $h$ represent the step sizes in time and space, respectively. The term $\widetilde{ϵ}$ represents the approximation error introduced by the fast sum-of-exponentials approximation. The numerical simulations validate the theoretical convergence outcome and showcase the effectiveness of the proposed approach.