The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsThe general solution satisfies the Laplace equation (7) inside therectangle, as well as the three homogeneous boundary conditionson three of its sides (left, right and bottom).We now determine the values of B n to get the boundary conditionon the top of the rectangle. This requiresf 2 (x) = u(x,b) =∞∑n=1B n sinh nπbasin nπ a x,which is the Fourier sine series for f 2 (x) on 0 < x < a.Appealing to previous results, we can now summarize our findings.DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsConclusionTheoremIf f 2 (x) is piecewise smooth, the solution to the Dirichlet problemis∇ 2 u = 0, 0 < x < a, 0 < y < b,u(x,0) = 0, u(x,b) = f 2 (x), 0 < x < a,u(0,y) = u(a,y) = 0, 0 < y < b.u(x,y) =where µ n = nπ a and B n =∞∑B n sinµ n x sinhµ n y,n=12asinh nπba∫ a0f 2 (x)sin nπ a x dx.DailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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