Dirichlet Boundary Conditions

PDEs · Axiom Academy

Solving the heat equation with fixed temperature endpoints 1 Dirichlet Boundary Conditions The Dirichlet boundary condition specifies the value of the solution at the boundary of the domain. For a heat equation on a rod of length L, this means fixing the temperatures at both ends. Definition: Dirichlet BC for Heat Rod For a rod of length L with endpoints at x = 0 and x = L: where T₁ and T₂ are constant temperatures maintained at the endpoints. Imagine holding the left end of a metal rod at temperature T₁ (perhaps in ice water at 0°C) and the right end at temperature T₂ (perhaps in boiling water at 100°C). The Dirichlet boundary conditions enforce these fixed temperatures at all times. 2 Heat Diffusion with Fixed Endpoints Let's visualize how heat diffuses in a rod when the endpoints are held at fixed temperatures. Watch as the temperature profile evolves from an initial distribution to a steady state. Notice how the endpoints remain fixed throughout the evolution, while the interior temperature adjusts to form a smooth profile. Eventually, the system reaches a steady state where the temperature distribution no longer changes with time. To solve the heat equation with Dirichlet boundary conditions, we use the powerful technique of separation of variables . We assume the solution can be written as a product of functions: Substituting this into the heat equation and dividing by u(x,t) gives:

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