Two-dimensional boundary value problem of the diffusion
The paper deals with a parabolic boundary value problem for the diffusion equation describing the concentration C(x,y;t) of suspended sediments in the domain (0,\(\infty)\times (0,H):\) \[ C_ t+U(x,y)C_ x=K_ 1(y)C_{xx}+(K_ 2(y)C_ y)_ y+VC_ y \] with an initial concentration at \(t=0\), prescribed time-dependent boundary values at \(x=0\), time-independent values at \(y=0\) and natural boundary conditions at \(y=H\). Under the assumption that \(K_ 2\) is constant asymptotic solutions are found; for a strictly increasing coefficient \(K_ 2\) estimates for the asymptotic solutions are given. In a simpler case of constant coefficients of the diffusion equation analytic and asymptotic solutions are found. Influences of the coefficients of the diffusion equation, the initial concentration and the boundary conditions on the asymptotic solutions are analysed.
scientific article; zbMATH DE number 3975448