Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains

• Main Author: Arsen Pskhu
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Series: Mathematics
• Subjects: diffusion-wave equation; boundary-value problem; green function; double-layer potential; fractional derivative; dzhrbashyan–nersesyan operator
• Online Access: https://www.mdpi.com/2227-7390/8/4/464

We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan−Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann−Liouville and Caputo derivatives.