The two dimensional heat equation - Trinity University

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