Special solutions to the space fractional diffusion problem

Abstract

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