Sahil wants points

TheMeansOfProduction

Use Mathematica to find a numerical solution to the following system: $u_t=u_{xx}$, $u(x,0)=\sin(5x)$, $u(0,t)=0$, $u(1,t)=1$.

SighHillDoveN

Using the code:

heatSystem = {
   D[u[x, t], t] == D[u[x, t], x, x],
   u[x, 0] == Sin[5 x],
   u[0, t] == 0, u[1, t] == 1};
u[x_, t_] = NDSolveValue[heatSystem,
  u[x, t], {x, 0, 1}, {t, 0, 1}]
Manipulate[Plot[u[x, t], {x, 0, 1},
  ColorFunction -> "TemperatureMap", 
  ColorFunctionScaling -> False,
  PlotRange -> {-1, 1}],
 {t, 0, 0.15}]
pics = Table[Plot[u[x, t], {x, 0, 1},
    ColorFunction -> "TemperatureMap", 
    ColorFunctionScaling -> False,
    PlotRange -> {-1, 1}],
   {t, 0, 0.5, 0.01}];
Export["anim.gif", pics]
u[0.5, 1]

The graph of the solution is:

The solution to the system $u_{t}=u_{xx}, u(x,0)=sin(5x), u(0,t)=0, u(1,t)=1$ at the midpoint when t=1 is:

$u(0.5,1)=0.0894617$