Fractional Differential Shift Integrators for Caputo-Type Initial Value Problems: Accuracy, Stability, and Complexity

Our focus in this document is on a newtypeof numerical technique called fractional differential shift (FDS) methods that solve a type of initial value problem known as Caputo-type fractional initial value problem (IVP). D _t ^α x(t)=F(t, x(t)) when t ∈ (0,T] and for 0 < α < 1.We propose an approach to solving this problem by representing the solution via the Volterra integral representation and approximating it using discrete convolution with a set of properly shifted weights based on the weakly singular kernel (t − τ) ^α−1 . We will show that this approach can be used to recover classical predictor-corrector methods, Grünwald-Letnikov meth ods, as specific examples, and we also demonstrate how the analysis of consistency, stability and computational cost can be unified under this framework. Local truncation error results and a global rate of convergence order, p, can be established under mild reg ularity assumptions on F. The selection of a shift parameter and treatment of the initial layer near time zero will determine how to assign a value for p explicitly. Further, through a stability analysis of the linear fractional test equation D _t ^α x = λx, we detail the stability regions in both the complex h ^α λ−; plane and for step sizes for all of the various finite difference schemes utilized. We provide an exhaustive account of how we implemented these schemes, contrasting the computational efficiency between traditional O(N ² ) convolution updates and the use of FFT and Short-Term Memory Approximations for O(NlogN) and O(N) implementations with similar accuracy levels. We provide numerical results from extensive benchmark problems that validate our findings through exam ples of oscillatory and stiff fractional differential equations.