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.

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

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

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 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.

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