Fourier Series Solution 13.5.8

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

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

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

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

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

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

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

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

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

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

uy200[x_,y_]=D[u[x,y,200],y];

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

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

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

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