I am attempting to solve the heat equation $\frac{\partial T}{\partial t}=\nabla^2T$, where $T=T(x,y,z,t)$, subject to the following boundary conditions:

$\frac{\partial T}{\partial x}|_{x=10}=\frac{\partial T}{\partial x}|_{x=-10}=0$

$\frac{\partial T}{\partial y}|_{y=10}=0$

$T(0,0,z_{crit}(t),t)=f(t)$ where $z_{crit}(t)=t$ and $f(t)=e^{-t}$

$\frac{\partial T}{\partial y}|_{y=0}=\frac{\partial T}{\partial z}|_{z=0}=0$

But my attempted code has not worked properly and I encounter frequently many different errors.

Is there any way to fix this issue?

f[t_] := Exp[-t]
eqns = {
  D[c[x, y, z, t], t] == Laplacian[c[x, y, z, t], {x, y, z}], 
  Derivative[1, 0, 0, 0][c][10, y, z, t] == 0, 
  Derivative[1, 0, 0, 0][c][-10, y, z, t] == 0, 
  Derivative[0, 1, 0, 0][c][x, 10, z, t] == 0,
  c[0, 0, t, t] == f[t]

Thanks in advance.



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