GATE 2020 MA – Question 35

Partial Differential Equations · Heat and Wave Equations · 1 mark · Numerical answer

Let u solve $u_{tt}-u_{xx}=0$ on R for t > 0, with u(x,0) = f(x), uₜ(x,0) = 0 and twice continuously differentiable f. If f(−2) = 4, f(0) = 0 and u(2,2) = 8, find u(1,3).

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Correct answer: 10 to 10

Explanation

D’Alembert’s formula with zero initial velocity is $u(x,t)=[f(x+t)+f(x-t)]/2$. From u(2,2) = 8 obtain $[f(4)+f(0)]/2=8$, hence f(4) = 16. Then $u(1,3)=[f(4)+f(-2)]/2=(16+4)/2=10$.

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