Solution of a partial differential equation related to queuing theory, heat conduction and quantum mechanics (Q1268532)

The object of this communication is to establish a relationship between some problems of queuing theory, heat conduction, quantum mechanics and the partial differential equation \[ \frac{\partial u}{\partial t}=k\left[\frac{\partial u^2}{\partial x^2}-x\frac{\partial u}{\partial x}+\frac{\lambda}{k}u\right],\qquad (-\infty<x<\infty), \] the solutions of which lead to the Chebyshev Hermite polynomials.