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GATE 2024 MA – Question 62

Partial Differential Equations · Heat and Wave Equations · 2 marks · Numerical answer

For u_tt−u_xx=0 on ℝ,t>0, u(x,0)=0 and initial velocity v(x)=x⁴(1−x)⁴ on 0<x<1, zero otherwise. Find α=inf{t>0:u(2,t)>0} to two decimals.

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Correct answer: 0.99 to 1.01

Explanation

D’Alembert gives u(2,t)=(1/2)∫_(2−t)^(2+t)v(s)ds. The velocity is positive exactly on (0,1), so this integral first becomes positive when 2−t<1, namely t>1. At t=1 the overlap is only an endpoint and the integral is zero, but the infimum is **1.00**.

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