Special solutions to the space fractional diffusion problem

Abstract We derive a fundamental solution $${{\mathscr {E}}}$$ E to space-fractional diffusion problem on the half-line. The equation involves Caputo derivative. establish properties of as well formulas for solutions Dirichlet and fixed slope problems in terms convolution with data. also study integrability derivatives given this way. present conditions, which are sufficient uniqueness solutions. Finally, we show infinite speed signal propagation.