The two dimensional heat equation - Trinity University

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The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsAt the edges of the plate we impose some sort of boundaryconditions. The simplest are homogeneous Dirichletconditions:u(0,y,t) = u(a,y,t) = 0, 0 ≤ y ≤ b, t ≥ 0,u(x,0,t) = u(x,b,t) = 0, 0 ≤ x ≤ a, t ≥ 0. (2)Physically, these correspond to holding the temperature along theedges of the plate at 0.The way the plate is heated initially is given by the initialconditionu(x,y,0) = f(x,y), (x,y) ∈ R, (3)where R = [0,a]×[0,b].DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsSolving the 2D wave equation: homogeneous Dirichletboundary conditionsGoal: Write down a solution to the heat equation (1) subject tothe boundary conditions (2) and initial conditions (3).As usual, weseparate variables to produce simple solutions to (1) and(2), and thenuse the principle of superposition to build up a solution thatsatisfies (3) as well.DailedaThe 2D heat equation
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The two dimensional heat equation - Trinity University

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