Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation ∂tu(x,t)=D∂x2u(x,t)−∂x[F(x,t)u(x,t)]−κu(x,t)∂xu(x,t)+Φ(x,t), where F(x,t)F(x,t) is an external force and Φ(x,t)Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k1(t)−k2(t)xF(x,t)=k1(t)−k2(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein–Uhlenbeck process and depending on the choice of k1(t)k1(t) and k2(t)k2(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ1(t)+Φ2(t)xΦ(x,t)=Φ1(t)+Φ2(t)x and Φ(x,t)=Φ3x−Φ4/x3Φ(x,t)=Φ3x−Φ4/x3 are incorporated in the previous analysis and exact solutions are obtained for both cases.