# Finite elements/Model problems

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## Model problems[edit | edit source]

Let us look at a couple of model problems.

### Problem 1: First-order homogeneous ODE[edit | edit source]

The first problem involves a homogeneous first order differential equation and is stated as:

The exact solution is

### Problem 2: Second-order inhomogeneous ODE[edit | edit source]

The second problem involves an inhomogeneous second order differential equation and is stated as:

The exact solution is

### Remarks[edit | edit source]

Both model problems have exact solutions. However, in most situations, such solutions may be impossible to get because:

- No solution exists since the data are not smooth enough.
- Even if a solution exists, it cannot be found in closed form because of the complexity of the problem.

To overcome these problems, we can do the following :

- Reformulate the problem so that it admits
**weaker**conditions on the solution and its derivatives. Thus the equation no longer needs to be satisfied at**all**points in the domain. These are called**weak**or**variational**formulations. - Break the domain up into smaller pieces and require the equation to be satisfied at only a few points within each piece.
- Assume that the approximate solution has a known form.

Finite element methods use a weak (or variational) formulation of the original ** strong** problem.