f1[x_,n_]:=Sin[((n π x)/a)]
"define an coefficients ";
acoff[n_]=Simplify[((2 (∫_0^(a/2))x f1[x,n]\[DifferentialD]x)/a)+((2 (∫_(a/2)^a)(-x+a) f1[x,n]\[DifferentialD]x)/a)]

v[x_,nmax_]=Sum[acoff[n] f1[x,n],{n,1,nmax,2}];

Plot[Evaluate[v[x,100]/.a->1],{x,0,1}]

"now we define full 2-D series, according to the  text we have ";
phi[x_,y_,nmax_]=Sum[acoff[n] f1[x,n] (E^(-((n π y)/a))),{n,1,nmax,2}]

ContourPlot[Evaluate[phi[x,y,100]/.a->1],{x,0,1},{y,0,1},ContourShading->False,Contours->30,PlotPoints->50,ContourSmoothing->Automatic,PlotRange->{0,1}]

"looking at 3-D perspective ";
Plot3D[Evaluate[phi[x,y,100]/.a->1],{x,0,1},{y,0,1},PlotRange->{0,0.5},ViewPoint->{-1,2,1.0}]