Project ForgeEngineering Mathematics
Calculus
GATE 2022
Hard
Solution of ∇²T = 0 in a square domain (0<x<1 and 0<y<1) with boundary conditions: T(x,0)=x; T(0,y)=y; T(x,1)=1+x; T(1,y)=1+y is
- AT(x,y) = x − xy + y
- T(x,y) = x + y
- CT(x,y) = −x + y
- DT(x,y) = x + xy + y
Explanation
T(x,y)=x+y is linear, hence harmonic (∇²T=0 trivially), and satisfies all four boundary conditions exactly: T(x,0)=x, T(0,y)=y, T(x,1)=x+1=1+x, T(1,y)=1+y. By uniqueness of the Dirichlet problem for Laplace's equation, this is THE solution — option (B).
Source: GATE ME 2022 Session 1 (ME-1) (2022)
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