Multipole Expansion of the Fundamental Solution of a Fractional Degree of the Laplace Operator

A multipole expansion of the fundamental solution of the fractional power of the Laplace operator is constructed in terms of the Gegenbauer polynomials. Based on the decomposition constructed and the idea of the fast multipole method, we propose a numerical algorithm for solving the fractional differential generalization of the Poisson equation in the two-dimensional and three-dimensional spaces. © 2023, Springer Nature Switzerland AG.