How can I solve this impulsive heat equation please?

Questions
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1

I need to solve the following impulsive heat equation:

$$
\left\{\begin{array}{ll}
\partial_{t} \psi(x,t)-\partial_{xx} \psi(x,t)=0, & (x,t)\in (0,1) \times((0, 2) \backslash\{1\}) \\
\psi(0,t)= \psi(1,t)=0, & t \in  (0, 2) \\
\psi(x, 0)= x (1-x), & x \in (0,1) \\
\psi(x, 1)=\psi\left(x, 1^{-}\right)+4, & x \in (0,1)
\end{array}\right.
$$

1^{-} denotes the limit to the left!

This is the code I tried in Mathematica, but it’s not giving the results

                               (* problem *)
homogen = If[x = 1, {f[x, t] == Limit[f[x, t], t -> 1, Direction -> "FromBelow"] + 4}, {D[f[x, t], {t, 1}] - D[f[x, t], {x, 2}] == 0}];
                         (*Initial conditions *)
ic = {f[x, 0] == x*(1 - x)};
                   (* Dirichlet boundary conditions*)
bc = {f[0, t] == 0, f[1, t] == 0};
                          (*solution*)
sol = DSolve[{homogen, ic, bc}, f[x, t], {x, 0, 1}, {t, 0, 2}]
2

If[x = 1 ? Perhaps you meant If[x == 1? Try using Piecewise rather than If.

3

thank you for your reply, but it’s not working!!

4

You will have to be more specific than

but it’s not working!!

Post the code you tried.