The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsWe must solve the heat equation problem (1) - (3) with{50 if y ≤ 1,f(x,y) =0 if y > 1.The coefficients in the solution areA mn =42·2∫ 2 ∫ 20∫ 20f(x,y)sin mπ2 x sin nπ 2∫ 12 x dx sin nπ 2 y dy= 50 sin mπ0 0( 2(1+(−1) m+1 )() 2(1−cosnπ= 50πm πn= 200 (1+(−1) m+1 )(1−cos nπ 2 )π 2 .mny dy dx2 ))DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsSincethe solution isλ mn = π 3√m 24 + n24 = π √m62 +n 2u(x,y,t) = 200π 2 ∞ ∑m=1 n=1∞∑( (1+(−1) m+1 )(1−cos nπ 2 )mn×sin nπ 2 y e−π2 (m 2 +n 2 )t/36 ) .sin mπ2 xDailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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