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 $\displaystyle \mathbf{D}=\sum_{j=1}^n\mathbf{e}_j\partial_{x_j}$ 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 L\'evy-Feller type $$\partial_t\Phi_\alpha(\mathbf{x},t;\theta)=-(-\Delta)^{\frac{\alpha}{2}}\exp\left( \frac{i\pi\theta}{2} \mathcal{H}\right)\Phi_\alpha(\mathbb{x},t;\theta) $$ involving the fractional Laplacian $-(-\Delta)^{\frac{\alpha}{2}}$ of order $\alpha$, with $2m\leq \alpha <2m+2$ ($m\in \mathbb{N}$), 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 $|\theta|\leq \min\{\alpha-2m,2m+2-\alpha\}$. Such model problem permits us to obtain hypercomplex counterparts for the fundamental solutions of higher-order heat-type equations $\partial_t F_M(x,t)=\kappa_M(\partial_x)^M F_M(x,t)$ $(M=2,3,\ldots)$ 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.