The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsSeparation of variablesAssuming thatu(x,y,t) = X(x)Y(y)T(t),plugging into the heat equation (1) and using the boundaryconditions (2) yields the separated equationsX ′′ −BX = 0, X(0) = 0, X(a) = 0, (4)Y ′′ −CY = 0, Y(0) = 0, Y(b) = 0, (5)T ′ −c 2 (B +C)T = 0. (6)DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsWe have already seen that the solutions to (4) and (5) areX m (x) = sinµ m x,µ m = mπa ,B = −µ2 mY n (y) = sinν n y, ν n = nπ b , C = −ν2 n,for m,n ∈ N. Using these values in (6) givesT mn (t) = e −λ2 mnt ,where√mλ mn = c√µ 2 m +ν2 n = cπ 2a 2 + n2b 2.DailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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