Use the Sarah-test for connection to the license server:

1+1

If you want to have specific assumptions used everytime you process an Integrate or Simplify command, you can change the value of the system variable $Assumptions.  The following is an example if we want to assume n is an integer throughout the session.

$Assumptions={Element[n,Integers]}

First example

Define a boundary condition function:

f[x_]=100

Compute expansion coefficients for this function:

a[n_]=(2/l/Sinh[n Pi/l w])Integrate[f[x]Sin[n Pi/l x],{x,0,l}]

Set values for l and w:

l=1

w=2

Plot FSS partial sum to check our coefficients:

Plot[Sum[a[n]Sinh[n Pi /l w]Sin[n Pi/l x],{n,1,100}],{x,0,l},PlotRange->All]

Clear[l,w]

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

l=1;
w=2;

u[x,1,4]

Plot[u[x,2,100],{x,0,1},PlotRange->All]

Plot[{u[x,1,100],u[x,1,1]},{x,0,1},PlotRange->All]

Plot3D[u[x,y,30],{x,0,1},{y,0,2},PlotRange->All]

ContourPlot[u[x,y,20],{x,0,1},{y,0,2},ColorFunction->"NeonColors",ContourLabels->Automatic]

Second example

Clear[l,w,a]

Define a boundary condition function:

f[x_]=200x(1-x)

Compute expansion coefficients for this function:

a[n_]=Simplify[(2/l/Sinh[n Pi/l w])Integrate[f[x]Sin[n Pi/l x],{x,0,l}]]

Set values for l and w:

l=1;
w=2;

Plot FSS partial sum to check our coefficients:

Plot[Sum[a[n]Sinh[n Pi /l w]Sin[n Pi/l x],{n,1,100}],{x,0,l},PlotRange->All]

Clear[l,w]

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

l=1;
w=2;

u[x,1,4]

Plot[u[x,2,100],{x,0,1},PlotRange->All]

Plot3D[u[x,y,30],{x,0,1},{y,0,2},PlotRange->All]

ContourPlot[u[x,y,20],{x,0,1},{y,0,2},ColorFunction->"NeonColors",ContourLabels->Automatic]