Space-Time Fractional Bessel Diffusion Equation

This study introduces a novel space-time fractional diffusion equation with the capacity to model a diverse spectrum of diffusion processes. The equation incorporates Caputo-type time derivatives with an arbitrary order $$\beta$$ and introduces a spatial-fractional operator known as the fractional Bessel operator. The fundamental solution of this equation is obtained utilizing harmonic analysis on the Kingman Bessel hypergroup. We present both Mellin-Barne integral and series representations for the fundamental solution and establish its positivity through the application of the Bernstein characterization of completely monotone functions and the positive definiteness property with respect to the Bessel operator.