Variational Formulation

To obtain a numerical method to compute a solution approximation we start by deriving a variational statement of our problem at hand.

Multiply (1) by a test function $v \in V = \{ \Vert v'\Vert^2 + \Vert v \Vert^2 \leq \infty,  v(0) = 0 \}$ and integrate i.e.

$$\int_0^1 \dot u v dx - \int_0^1 u'' v dx = 0.$$ (4)

As usual we assume $v$ to be smooth enough to allow us to integrate by parts. We get

$$\int_0^1 \dot u v dx - \big[ u'(x)v(x)\big]_0^1 + \int_0^1 u' v' dx = 0.$$ (5)

Since $v(0)=0$ and $u'(1)=0$ we thus obtain

$$\int_0^1 \dot u v dx + \int_0^1 u' v' dx = 0.$$ (6)

Hence, the variational formulation of (1) reads: Find $u(x,t) \in V$ such that for every fixed time $t$

$$\int_0^1 \dot u v dx + \int_0^1 u' v' dx = 0,$$ (7)

for all $v \in V$ and $0 < t < T$.