The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsConclusionTheoremSuppose that f(x,y) is a C 2 function on the rectangle[0,a]×[0,b]. The solution to the heat equation (1) withhomogeneous Dirichlet boundary conditions (2) and initialconditions (3) is given byu(x,y,t) =∞∑m=1 n=1∞∑A mn sinµ m x sinν n y e −λ2mnt ,where µ m = mπa , ν n = nπ √b , λ mn = c µ 2 m +ν2 n , andA mn = 4ab∫ a∫ b00f(x,y)sin mπa x sin nπ by dy dx.DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsRemarks:ExampleWe have not actually verified that this solution is unique, i.e.that this is the only solution to problem.We will prove uniqueness later using the maximum principle.A 2×2 square plate with c = 1/3 is heated in such a way that thetemperature in the lower half is 50, while the temperature in theupper half is 0. After that, it is insulated laterally, and thetemperature at its edges is held at 0. Find an expression that givesthe temperature in the plate for t > 0.DailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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