In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the scalar part coincides with the original data. This is analogous to the question of the boundary correspondence of harmonic conjugates. Generalizing a representation of the Hilbert transform $\mathcal{H}$ in ${\mathbb{R}}^3$ given by T. Qian and Y. Yang (valid in ${\mathbb{R}}^n$), we define the Hilbert transform ${\mathcal{H}}_f$ associated to the main Vekua equation $DW=\left(Df/f\right)\overline{W}$ in bounded Lipschitz domains in ${\mathbb{R}}^3$. This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.