List of top Partial Differential Equations Questions on Wave and heat equations

The steady state solution for the heat equation \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad 0<x<2, \, t>0, \] with the initial condition \( u(x, 0) = 0, \, 0<x<2 \) and the boundary conditions \( u(0, t) = 1 \) and \( u(2, t) = 3, \, t>0 \) at \( x = 1 \) is
Let \( u(x, t) \) be the solution of the wave equation \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \, t>0, \] with the initial conditions \[ u(x, 0) = \sin x + \sin 2x + \sin 3x, \quad \frac{\partial u}{\partial t}(x, 0) = 0, \quad 0<x<\pi, \] and the boundary conditions \[ u(0, t) = u(\pi, t) = 0, \quad t \geq 0. \] Then, the value of \( u \left( \frac{\pi}{2}, \pi \right) \) is

If \( u(x,t) = A e^{-t} \sin x \) solves the following initial boundary value problem:
\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, 0 < x < \pi, t > 0, \] \[ u(0,t) = u(\pi,t) = 0, t > 0, \] \[ u(x,0) = \begin{cases} 60, & 0 < x \leq \frac{\pi}{2}, \\ 40, & \frac{\pi}{2} < x < \pi, \end{cases} \] then \( \pi A = \underline{\hspace{1cm}} \).

The function \( u(x, t) \) satisfies the initial value problem \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \, x \in \mathbb{R}, \, t > 0, \] \[ u(x, 0) = 0, \, \frac{\partial u}{\partial t} (x, 0) = 4xe^{-x^2}. \] Then \( u(5, 5) \) is: