GATE 2023 MA – Question 64

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

For 0<x<π,t>0 solve u_xx−u_tt=sin x sin(2t), with u(x,0)=u_t(x,0)=0 and u(0,t)=u(π,t)=0. Find u(π/2,3π/2) to two decimals.

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Correct answer: 0.64 to 0.70

Explanation

The forcing and boundary conditions suggest u=sin x·v(t). Substitution gives v″+v=−sin 2t, with v(0)=v′(0)=0. A particular solution is sin(2t)/3; the initial derivative requires −2 sin t/3. Thus v=(sin2t−2sint)/3. At the requested point u=(0−2(−1))/3=2/3=**0.67**.

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