On Fundamental Solutions of Higher-Order Space-Fractional Dirac equations

Abstract

Starting from the pseudo-differential decomposition $\mathbf{D}=(-\Delta {)}^{\frac{1}{2}}\mathcal{H}$ of the Dirac operator $\mathbf{D}=\sum _{j=1}^n{\mathbf{e}}_j{\partial }_{x_j}$ in terms of the fractional operator $(-\Delta {)}^{\frac{1}{2}}$ of order $1$ and of the Riesz-Hilbert type operator $\mathcal{H}$ we will investigate the fundamental solutions of the space-fractional Dirac equation of Lévy-Feller type $${\partial }_t{\Phi }_{\alpha }(\mathbf{x},t;\theta )=-(-\Delta {)}^{\frac{\alpha }{2}}\exp \left(\frac{i\pi \theta }{2}\mathcal{H}\right){\Phi }_{\alpha }(x,t;\theta )$$ involving the fractional Laplacian $-(-\Delta {)}^{\frac{\alpha }{2}}$ of order $\alpha$, with $2m\le \alpha <2m+2$ ($m\in \mathbb{N}$), and the exponentiation operator $\exp \left(\frac{i\pi \theta }{2}\mathcal{H}\right)$ as the hypercomplex counterpart of the fractional Riesz-Hilbert transform carrying the \textit{skewness parameter} $\theta$, with values in the range $|\theta |\le \min \{\alpha -2m,2m+2-\alpha \}$. Such model problem permits us to obtain hypercomplex counterparts for the fundamental solutions of higher-order heat-type equations ${\partial }_t{F}_M(x,t)={\kappa }_M({\partial }_x{)}^M{F}_M(x,t)$ $(M=2,3,\dots )$ in case where the even powers resp. odd powers ${\mathbf{D}}^{2m}=(-\Delta {)}^m$ ($M=\alpha =2m$) resp. ${\mathbf{D}}^{2m+1}=(-\Delta {)}^{m+\frac{1}{2}}\mathcal{H}$ ($M=\alpha =2m+1$) of $\mathbf{D}$ are being considered.

13 pages. Refs. correctly included (v2)