Discretized fractional substantial calculus

This paper discusses the properties and the numerical discretizations of the fractional substantial integral $$I_s^\nu f(x)=\frac{1}{\Gamma(\nu)} \int_{a}^x{\left(x-\tau\right)^{\nu-1}}e^{-\sigma(x-\tau)}{f(\tau)}d\tau,\nu>0, $$ and the fractional substantial derivative $$D_s^\mu f(x)=D_s^m[I_s^\nu f(x)], \nu=m-\mu,$$ where $D_s=\frac{\partial}{\partial x}+\sigma=D+\sigma$, $\sigma$ can be a constant or a function without related to $x$, say $\sigma(y)$; and $m$ is the smallest integer that exceeds $\mu$. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error $\mathcal{O}(h^p)$$ (p=1,2,3,4,5)$ are theoretically proved and numerically verified.