Fundamental Solutions For Discrete Dynamical Systems Involving The Fractional Laplacian

We prove representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem D𝛽t u(n, t) = −(−Δd)𝛼u(n, t) + g(n, t), where 0<β ≤ 2, 0<α ≤ 1, (Formula presented.), (−Δ_d)^α is the discrete fractional Laplacian, and (Formula presented.) is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.