Probabilistic representation of solution of the hyperbolic mass transfer equation with convection term (Q1603072)

The authors investigate the mathematical model of intensive processes of nonstationary mass-energy transfer with the presence of relaxation effects, \(\sigma (\partial^2c/\partial t^2)+\partial c/\partial t=D(\partial^2c/\partial x^2)+S(\partial c/\partial x)\). This equation is known as the hyperbolic mass transfer equation with convection term. Here \(c(x,t)\) is the physical field to be transferred, \(\sigma \) is the relaxation time of the medium, \(D\) is the diffusion coefficient, \(S\) is the drift velocity. This equation is considered under the initial condition, \(c(x,0)=\varphi (x)\), \((\partial c/\partial t)|_{t=0}=\psi (x)\), where \(\varphi (x)\) and \(\psi (x)\) are sufficiently regular functions. The authors establish a probabilistic model and through the theory of Markov processes and inverse Kolmogorov equation, it follows the existence of a solution for the considered problem.