14. Boundary and Initial Conditions

We have set up a differential equation, with \(T\) as the dependent variable. The solution will give us \(T(x,y,z)\). Solution depends on boundary conditions (BC) and initial conditions (IC).

How many BC’s and IC’s? Heat equation is second order in spatial coordinate. Hence, 2 BC’s are needed for each coordinate.

  • 1D problem: 2 BC in \(x\)-direction.

  • 2D problem: 2 BC in \(x\)-direction, 2 in \(y\)-direction.

  • 3D problem: 2 in \(x\)-dir., 2 in \(y\)-dir., and 2 in \(z\)-dir.

Heat equation is first order in time. Hence one IC needed.