Analytical solutions of fractional Poisson's Equation with Riesz Derivatives

This paper provides an analytical framework for solving the fractional Poisson’s equation involving Riesz fractional derivatives of order α in a fractional dimensional space of dimension D. By employing the Fourier transform technique, we derive explicit solutions and establish a fundamental link between the order of fractional differentiation and the dimension of the space. A generalized Gauss’s law is formulated for fractional spaces, and the total electric flux is expressed as a function of α and D. Furthermore, a fractional multipole expansion using Gegen Bauer polynomials is introduced, enabling a compact representation of higher-order terms in fractional space. These results offer a significant extension of classical electromagnetic theory to fractional dimensions, providing a basis for modeling complex systems with anisotropic or confined geometries. The developed approach also opens the possibility of extending the analysis to fractional Helmholtz equations in future research.