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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).


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