The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsSuperpositionAssembling these results, we find that for any pair m,n ≥ 1 wehave the normal modeu mn (x,y,t) = X m (x)Y n (y)T mn (t) = sinµ m x sinν n y e −λ2 mn t .For any choice of constants A mn , by the principle of superpositionwe then have the general solution to (1) and (2)u(x,y,t) =∞∑m=1 n=1∞∑A mn sinµ m x sinν n y e −λ2mnt .DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsInitial conditionsWe must determine the values of the coefficients A mn so that oursolution also satisfies the initial condition (3). We needf(x,y) = u(x,y,0) =∞∑∞∑n=1 m=1A mn sin mπa x sin nπ b ywhich is just the double Fourier series for f(x,y). We know thatif f is a C 2 function thenA mn = 4ab∫ a ∫ b00f(x,y)sin mπa x sin nπ bdy dx.DailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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