Analytical solutions of fractional Poisson’s Equation with Riesz Derivatives

Somvanshi, Piyusha (2025) Analytical solutions of fractional Poisson’s Equation with Riesz Derivatives. International Journal of Science and Research Archive, 16 (1). pp. 1595-1600. ISSN 2582-8185

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.