Boundary Conditions

PDEs · Axiom Academy

How we specify the behavior at the edges of a domain 1 Dirichlet Boundary Conditions The Dirichlet boundary condition specifies the value of the solution at the boundary. This is like fixing the temperature at the ends of a rod. On the boundary of a domain, we specify: where g is a given function describing the boundary values. For the heat equation, Dirichlet conditions mean we're controlling the temperature at the boundary (e.g., holding the ends of a rod at fixed temperatures). The Neumann boundary condition specifies the derivative (or flux) at the boundary. This controls how fast heat flows in or out. On the boundary, we specify the normal derivative: where n is the outward normal vector and h is a given function. For the heat equation, Neumann conditions specify the heat flux at the boundary. When ∂u/∂n = 0, we have insulated (no-flux) boundaries. 3 Robin (Mixed) Boundary Conditions The Robin boundary condition combines both value and derivative in a linear relationship. This models convective heat transfer. On the boundary, we have a linear combination: where a , b , and c are constants or functions. Robin conditions model Newton's law of cooling : heat flow at the boundary is proportional to the temperature difference with the environment. 4 Comparing Boundary Conditions Let's see how the same heat rod behaves under different boundary conditions. Watch the three cases evolve simultaneously!

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