Differential Equations and Boundary Conditions

Two Types of Boundary Conditions

As an example, we have a partial differential equation

\[\frac{d^2u}{dx^2} + f = 0,\]

which describes a 1D problem.

  • Dirichlet boundary condition: specify values for \(u\), such as \(u(0)=u_0\) and \(u(L)=u_L\);
  • Neumann boundary condition: specifiy values for \(u_{,x}\).

If we have only Neumann boundary condition, the solution is not unique. The example for it is tossing a bar, which can have both Neumann BC at both ends but it is moving.

Example Problems

Elasticity Problem

We consider the displacement \(u(x)\) at each space coordinate \(x\) of a elastic bar under some external force. The strain is proportional to \(u_{,x}\). The equation would be

\[-\frac{d\sigma}{dx} = f.\]

Heat Transfer

Define temperature on a bar at each point \(u(x)\). Heat transfer is proportional to the head gradient, \(j= - \kappa u_{,x}\). The quation would be

\[- \frac{dj}{dx} = f.\]

Strong Form and Weak Form of PDE

Strong form of differential equations is basically the original form we write down. Strong form requires each term to be well defined at each point. However, we can derive a weak form that require each part to be well defined through the whole domain only, which is a relaxed requirement.