(* First set up the linear system of equations for m and the coefficients c[n] *)

tab[la_,L_,e_,Pout_,max_]:=Join[{c[0]/2-la(1+m L/2+c[0]L^2/12+L^2Sum[c[n]/n^2/Pi^2,{n,1,max}]-c[0]/2 Log[1/e]),1+m L+c[0] L^2/4+L^2 Sum[c[n](1-(-1)^n)/n^2/Pi^2,{n,1,max}]-Pout},Table[c[n]-la(2m L(-1+(-1)^n)/n^2/Pi^2+c[0]L^2(-1)^n/n^2/Pi^2-L^2c[n]/n^2/Pi^2-BesselK[0,n Pi e/L]c[n]/(n Pi e/L)/BesselK[1,n Pi e/L]),{n,1,max}]]

(* Solve the system and evaluate the tube pressure P(z) and flux Q(z) *)

sol[la_,L_,e_,Pout_,max_]:=(ss=Solve[Evaluate[tab[la,L,e,Pout,max]==0],Join[{m},Table[c[n],{n,0,max}]]];Q=c[0]/2+Sum[c[n]Cos[n Pi z/L],{n,1,max}]/.ss[[1]];P=1+m z+c[0]z^2/4+L^2 Sum[c[n]/n^2/Pi^2(1-Cos[n Pi z/L]),{n,1,max}]/.ss[[1]];)

sol[1,1,0.1,0.1,50];P50=P;Q50=Q;

sol[1,1,0.1,0.1,100];P100=P;Q100=Q;

(* Now plot P(z) and Q(z) for some particular paremeter values.
Here we test the effect truncating the system at n=50 and n=100 *)

Plot[{P50,P100},{z,0,1},AxesLabel->{z,"P"}]

Plot[{Q50,Q100},{z,0,1},AxesLabel->{z,"Q"}]

(* The differences are negligible so the scheme has converged well *)

(* Next compute the net flux q *)

qsol[la_,L_,e_,Pout_,max_]:=c[0]/2/.Solve[Evaluate[tab[la,L,e,Pout,max]==0],Join[{m},Table[c[n],{n,0,max}]]][[1]]

(* Test the effect on q of varying ϵ *)

Table[{1/(Log[2]-k Log[10]),qsol[1,1,10^k,0.1,100]},{k,-20,-1,0.1}];

qp1=ListPlot[%,Joined->True,AxesLabel->{1/Log[2/ϵ],q}];

qp2=Plot[0.55 z,{z,0,0.1},PlotStyle->Dashed];

Show[qp1,qp2]

(* This shows that the solutions agrees with the asymptotic behaviour
q ∼ (((1+(Subscript[P, out])))/(2 log(2/ϵ))) as ϵ -> 0.
*)