Let $u(x, t)$ satisfy the IVP: $\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}, x \in \mathbb{R}, t > 0$ and $$u(x, 0) = \begin{cases} 1, \ \ 0\leq x \leq 1\\ 0, \ \ \text{elsewhere}\end{cases}$$ Then the value of $\lim_{t \to 0^+}u(1, t)$ is?
My attempt: Solving by method of separation of variables to get $u(x, t) = (a\cos x+b\sin x)e^{-t}$ and initial condition implies $a\cos x+b\sin x=1$ for $0\leq x \leq 1$. How to proceed further?