Quasi‐Relativistic Heat Equation via Lévy Stable Distributions: Exact Solutions

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one‐sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one to obtain closed form solutions for a number of initial conditions. The resulting diffusion is slower than the non‐relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and kurtosis and is compared with the non‐relativistic case.