GATE 2024 MA – Question 23

Partial Differential Equations · Heat and Wave Equations · 1 mark · Multiple choice

u_t−4u_xx=0 on 0<x<1,t>0, zero endpoint values, u(x,0)=sinπx. Let g(t)=∫₀¹u(x,t)²dx. Choose its behavior.

Show answer and explanation

Correct answer: (A) Decreases to 0

Explanation

The initial data is one eigenmode: $u=e^{-4\pi^2t}\sin\pi x$. Therefore $g(t)=\frac12e^{-8\pi^2t}$, strictly decreasing to zero. **A**.

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