GATE 2022 MA – Question 61

Partial Differential Equations · Second Order Equations and Laplace Equation · 2 marks · Numerical answer

A harmonic u on 1<x<2,1<y<2 has u_x(1,y)=y,u_x(2,y)=5,u_y(x,1)=αx²/7,u_y(x,2)=x. If a solution exists, find α.

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Correct answer: 15 to 15

Explanation

Integrate Δu=0 over the rectangle. The outward-normal flux is ∫₁²[5−y]dy+∫₁²[x−αx²/7]dx=7/2+3/2−α/3. Set this to zero: 5−α/3=0, so **α=15**. The signs at the left and bottom edges are negative because their outward normals point in negative coordinate directions.

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