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(Solved): Let u=u(x,t) denote the concentration of a chemical pollutant. We study its diffusion in the semi-in ...
Let u=u(x,t) denote the concentration of a chemical pollutant.
We study its diffusion in the semi-infinite region x>0.
Initial condition
At time t=0 the region contains no pollutant:
u(x,0)=0,x>0.
Boundary condition at the origin
For all t>0 a unit negative concentration gradient is imposed at x=0 :
u_(x)(0,t)=-1
Far-field condition
The concentration vanishes infinitely far from the boundary:
u(x,t) ->0, as x->+\infty
Assuming a constant diffusivity equal to 1 , the process is governed by the one-dimensional diffusion (heat) equation
u_(t)=u_(xx) ,x>0,t>0.
Use the Laplace transform in the time variable and the corresponding inversion formulae to solve this boundary-initial value problem and obtain an explicit expression for the concentration u(x,t).