List of top Engineering Mathematics Questions on Fourier series asked in GATE XE

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

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