GATE 2024 MA – Question 65
In the upper half-disk 0<r<2,0<θ<π, u_rr+r⁻¹u_r+r⁻²u_θθ=0, with u(r,0)=u(r,π)=0 for 0≤r≤2 and u(2,θ)=k sin2θ. Given u(1,π/4)=2, find k to two decimals.
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Correct answer: 7.99 to 8.01
Explanation
For the regular Dirichlet solution at the origin, separation gives u=(k/4)r²sin2θ, matching the radius-two boundary. At r=1,θ=π/4, this is k/4. Set k/4=2, giving **8.00**. Regularity/continuity at the origin excludes the singular r⁻² mode.