Fourier Series Solution 13.5.8

a=1
b=1
f[x_,a_]=x

Plot[f[x,1],{x,0,1},AspectRatio->Automatic]

A[j_,x_,a_]=Sin[j Pi x/a]

c[j_,a_,b_]=(2/a) Integrate[ f[x,a] Sin[j Pi x/a],{x,0,a}]/((j Pi/a) Cosh[j Pi b/a]+  Sinh[j Pi b/a])

B[j_,y_,a_,b_]=c[j,a,b] ((j Pi/a) Cosh[j Pi y/a]+  Sinh[j Pi y/a])

u[x_,y_,k_,a_,b_]:=Sum[A[j,x,a] B[j,y,a,b],{j,1,k}]

Plot[{f[x,1],u[x,1,2,1,1],u[x,1,4,1,1],u[x,1,10,1,1]},{x,0,1}]

Plot[{f[x,1],u[x,1,200,1,1]},{x,0,1},PlotRange->All,AxesOrigin->{0,0}]

Plot[{f[x,1],u[x,1,200,1,1]},{x,-2,2},PlotRange->All,AxesOrigin->{0,0}]

Plot3D[u[x,y,200,1,1],{x,0,1},{y,0,1},PlotRange->All]

uy200[x_,y_,a_,b_]=D[u[x,y,200,a,b],y];

Plot[{uy200[x,0,1,1]/2,u[x,0,200,1,1]},{x,0,1},PlotRange->All,PlotStyle->{Black,Red}]

Plot[{u[x,0,200,1,1], x(1-x)/6},{x,0,1},PlotRange->All,PlotStyle->{Black,Red}]