List of top Engineering Mathematics Questions on Method of separation of variables asked in GATE XE

Let $u(x,t)$ be the solution of the initial boundary value problem: \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, 0<x<\pi, \; t>0, \] \[ u(x,0) = 2 \sin\!\left(\tfrac{3x}{2}\right)\cos\!\left(\tfrac{x}{2}\right), 0<x<\pi, \] \[ u(0,t) = u(\pi,t) = 0, t>0. \] Then the value of $\lim_{t\to\infty u\!\left(\tfrac{3\pi}{4},t\right)$ is equal to (rounded off to two decimal places).}

Let $u(x,t)$ be the solution of the initial–boundary value problem \[ \frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}=0, x\in(0,2),\ t>0, \] \[ u(x,0)=\sin(\pi x), x\in(0,2),\qquad u(0,t)=u(2,t)=0. \] Then the value of $e^{\pi^2}\!\left(u\!\left(\tfrac{1}{2},1\right)-u\!\left(\tfrac{3}{2},1\right)\right)$ is ________________ (in integer).