Construction of a fundamental solution of a parabolic partial differential equation with piecewise constant coefficients (Q2706036)

The author of this paper constructs a fundamental solution of a parabolic partial differential equation \((L-\partial /\partial t)u = 0\), where \(L\) is a second-order elliptic differential operator with piecewise constant coefficients satisfying a supplementary condition. Such equations appear in some models describing the particles motion in a heterogeneous domain. A numerical method allowing an explicit formulation of the solution is established. The method is based on using a parametric function and Volterra series in order an expansion in functions series of the solution to be obtained. Using the Laplace transformation and a calculus technique of certain multiple integrals, the author presents a precise expression of the series coefficients.