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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.

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

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

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

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

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

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

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

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

Mohammad Asadzadeh 2004-08-27