Exact radially symmetric solutions to n -dimensional space–time fractional diffusion equations: Resolution of singularities via generalized Hankel transforms

We present, for the first time, exact radially symmetric solutions to the n-dimensional space–time fractional diffusion equation with point source initial conditions, employing the most general fractional framework: the Riemann–Liouville temporal derivative of order μ > 0 and the Hilfer–Riesz spatial derivative of order 2β. Through systematic analysis of the literature, we establish that while various authors have studied fractional diffusion in multiple dimensions, the explicit analytical solution for arbitrary dimension n with delta function initial conditions using generalized Hankel transforms has not been previously derived. We provide multiple equivalent representations: as integrals involving Bessel functions, as Wright functions with dimension-dependent parameters, and as Fox H-functions. Our solution demonstrates that the transition from one-dimensional to higher-dimensional formulations fundamentally resolves the singularities and sign-indefiniteness that plague fractional diffusion models. The key insight lies in the transformation from Fourier to generalized Hankel transforms of order (n − 2)/2, where the associated Bessel functions provide natural regularization. We prove analytically that for all dimensions n ≥ 2, the hypersingular behavior ∼|x|−μ/β present in one-dimensional Wright function solutions is completely eliminated, with the solution remaining bounded at the origin. Furthermore, we establish through complete monotonicity arguments that the n-dimensional solution maintains strict positivity across all parameter regimes, including the previously problematic superdiffusive cases where β > 1 or μ > 1. The general framework encompasses all previously studied special cases while providing a unified approach to fractional diffusion across diverse applications, including anomalous transport in complex media, Lévy flights, turbulent dispersion, and non-Gaussian stochastic processes. This study establishes the mathematical foundation for physically realizable fractional models in arbitrary dimensions.