For u tt−u xx=0 on 0 0, u(x,0)=sin x+sin2x+sin3x,u t(x,0)=0 and u(0,t)=u(π,t)=0, find u(π/2,π).

GATE 2022 MA – Question 46

Partial Differential Equations · Heat and Wave Equations · 2 marks · Multiple choice

For u_tt−u_xx=0 on 0<x<π,t>0, u(x,0)=sin x+sin2x+sin3x,u_t(x,0)=0 and u(0,t)=u(π,t)=0, find u(π/2,π).

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Correct answer: (B) 0

Each eigenmode evolves as sin(nx)cos(nt), so u=Σₙ₌₁³sin(nx)cos(nt). At (π/2,π), the three values are −1,0,+1 respectively. Their sum is **0, B**.

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