#### 3.1.1 Variational Form

As a starting point, consider the boundary value problems (BVP) described by

Additionally, define a linear space as:

V = {v: continous functions in [0,1] with v’ piecewise continous and
bounded in [0,1], and v(0)=v(1)=0}
Along this session, it will be shown how the space can be used to reformulate the
problem (3.1). From this new version, a numerical method will be developed based on
the particular definition of , in order to obtain an approximate solution of
(3.1).
To begin, take an element of the space , multiply by (3.1) and integrate over the entire
domain as in

| (3.2) |

The function is known as a test function. Initially, it is not clear how (3.2) can help
to solve (3.1), but it provides a different view of the problem (3.1). Indeed, it is
possible to simplify (3.2) by integrating the left hand side by parts, according
to
| (3.3) |

Subsequently, substituting the results from (3.3) in (3.1) the relation holds
| (3.4) |

(3.4) is known as the variational formulation of the problem (3.1). Variational
formulations can be handled by several numerical methods, and it is usually easier to
prove the existence of the solution of a variational problem in comparison to a
PDE.
Seemingly, the solution of (3.1) solves (3.4), but to be useful, the variational formulation
should work in the opposite direction, where the solution of (3.4) solves (3.1). This is
true for the form described by (3.4), but usually the methods applied to solve
variational problems impose some restriction in the solution, which should also satisfy
(3.1).