(* Let's let H=2,  and L=1  *)
(* Let's let the RIGHT wall have T=100 *)

(* Usual formula, bn = (2/H) integral(0 to H) t(y) sin(n Pi y/H) dy *)

(2/2)*Integrate[100*Sin[n Pi y/2],{y,0,2}]

b[n_] :=-((200 (-1+Cos[n π]))/(n π))

a[n_] := b[n]/Sinh[n Pi (1/2)]

t[x_,y_,nmax_] := 
Sum[a[n]*  Sinh[n Pi x/2]Sin[n Pi y/2],{n,1,nmax}]

(*Let's first just check I did my fourier series right, by checking that Temp[1,y] is indeed the function we started with *)
Plot[t[1,y,100],{y,0,2}, PlotRange -> {0,120}, PlotStyle->{Thick}]

(* Nice! Just what we wanted. 100 degrees! *)

Plot3D[t[x,y,30],{x,0,1},{y,0,2}, PlotRange -> {0,100}]

(* Did you notice you can grab the plot with your mouse and rotate it, to get a good look? Cool! *)

ContourPlot[t[x,y,30],{x,0,1},{y,0,2}, ContourLabels -> True, Contours -> {20,40,60,80}, ContourShading -> None, ContourStyle -> Black, PlotRange -> {0,100}, BaseStyle -> {FontSize ->24}]

ContourPlot[t[x,y,100],{x,0,1},{y,0,2}, ContourLabels -> True, Contours -> {1,90}, ContourShading -> None, ContourStyle -> Black, PlotRange -> {0,100}, BaseStyle -> {FontSize -> 24}]