Discretized fractional substantial calculus

Chen, Minghua¹ ; Deng, Weihua¹

¹ School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 373-394

Résumé

This paper discusses the properties and the numerical discretizations of the fractional substantial integral

$$I_s^{\nu }f\left(x\right)=\frac{1}{\Gamma \left(\nu \right)}{\int }_a^x{\left(x-\tau \right)}^{\nu -1}{\mathrm{e}}^{-\sigma (x-\tau )}f\left(\tau \right)\mathrm{d}\tau ,\nu >0,$$

and the fractional substantial derivative

$$D_s^{\mu }f\left(x\right)=D_s^m\left[I_s^{\nu }f\left(x\right)\right],\nu =m-\mu ,$$

where $D_s=\frac{\partial }{\partial x}+\sigma =D+\sigma$, $\sigma$ can be a constant or a function not related to $x$, say $\sigma \left(y\right)$; 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 $𝒪\left(h^p\right)$$(p=1,2,3,4,5)$ are theoretically proved and numerically verified.

Détail

Reçu le : 2014-01-05
DOI : 10.1051/m2an/2014037

Classification : 26A33, 65L06, 42A38, 65M12
Keywords: Fractional substantial calculus, fractional linear multistep methods, fourier transform, stability and convergence