Question

The concentration C(,t) of a pollutant in a semi-infinite pipe diffuses according to the PDE D for 0<z<oo t>0, over where the diffusivity D is a given positive constant. An experiment is performed in which the initial concentration is zero everywhere, so that C(x,0) = 0 for all x 0. Once t > 0 a fixed concentration of C(0,t) = Co is imposed at x = 0, where Co is a constant, and it is also assumed that the concentration remains negligible far away, so C(z,t) → 0 as x → oo for all t >0. The objective here is to predict what will happen to the pollutant concentration C(x,t) at every point in the pipe, for all a20 when t > 0, assuming that the values of D and Co are known

here is the lecture notes that first question referring to. And I couldn't think of way to use the hint (phi = C - Co). can you please help me. you can only answer the first question.

Answer 1

MCL-Fo3 Similarity-Aolutions: Method to comert PDE tooDE:- Stretching tranyformabien the beatch parametors: Lot us assume -0TETi TX-Teas t*= teas Pusting in Tand 2-Tk For similar order xe casfficiests are eqpal ear-a ea3-a daz-a az-91 2ar az Lot as a a3=Ra : *= xea t te

Thus the paranseter Conwerts e ginen PDE is a variable hat DDE 6e A X Let thuo variakle where A is seme constondt Puctig in ¥e- = dT 3/2 Ax t myl A2 d2T = -dT d2T -2 RXA d

2oA (d/an) 2KA dz d(ayar) A ioa constant cmd hence LpaAa 1 we can assume A- Thuo Pareameter to conrert Vat PDE ODE From -d(nu Cap/uPP -Anc-22 From BC:- atxo, to Te Ti ce-22 and co

70 T-Ts enckion - e ) erf l Thus at = 0 a 2 TTR Ti-Ts =C 2 Ca (Ti-TA) -TA-T) ) TiIIG