Green's Function for the
1D Heat Equation

g[x_,xi_,t_,L_,k_]=Sum[(2/L) Sin[j Pi xi/L] Sin[j Pi x/L] Exp[-(j Pi/L)^2 t],{j,1,k}]

gten[x_,xi_,t_]=g[x,xi,t,1,10];

Plot[gten[x,1/4,1],{x,0,1}]

Plot[200 gten[x,1/4,1],{x,0,1},AspectRatio->Automatic]

Plot[gten[x,1/4,1/10],{x,0,1}]

Plot[gten[x,1/4,1/100],{x,0,1}]

Animate[Show[Plot[gten[x,1/4,t],{x,0,1},PlotRange->{0,10}],ParametricPlot[{1/4,t},{t,0,10}]],{t,1/100,1/10}]

Animate[Show[Plot[gten[x,1/4,t],{x,0,1},PlotRange->{0,10}],ParametricPlot[{1/4,t},{t,0,10}]],{t,1/200,1/10}]

Animate[Show[Plot[gten[x,1/4,t],{x,0,1},PlotRange->{0,10}],ParametricPlot[{1/4,t},{t,0,10}]],{t,1/500,1/10}]

gfifty[x_,xi_,t_]=g[x,xi,t,1,50];

Animate[Show[Plot[gfifty[x,1/4,t],{x,0,1},PlotRange->{0,10}],ParametricPlot[{1/4,t},{t,0,10}]],{t,1/500,1/10}]

Animate[Show[Plot[gfifty[x,1/4,t],{x,0,1},PlotRange->{0,10}],ParametricPlot[{1/4,t},{t,0,10}]],{t,1/1000,1/100}]