The direct and inverse problem for two-dimensional turbulent diffusion

This paper examines the problem of the diffusion of a pollutant in the atmosphere considering the effect of the wind and turbulence. The following equation is considered: u(z).∂ c(z,x)/∂ x = ∂/∂ z ( K(z). ∂ c(z,x)/∂ z) (z and x in Ω where Ω = (z,x; z > 0, x> 0), with the initial condition c(z,0) = δ(z - Hs) and the boundary condition ∂ c(0,x)/∂ x= 0, where δ(z - Hs) is Dirac's δ distribution concentrated at the point z = Hs > 0, and where u(z) and K(z) are real valued piecewise constant functions with a finite number of jumps. For this problem an integral representation of c(z,x) is obtained; using this integral representation it is considered an inverse problem. That is from the knowledge of c(0,x) for x ≥ 0 it will be recovered the product of u(z)K(z) for z > 0. Finally the behaviour of c(0,x) is studied analytically in some particular cases