User:Yongtao-Li/HW4.2

=Problem 4.2 "Obtain the weak form for the equations of heat conduction"=

Given
The equations of heat conduction

with the boundary conditions and The condition on the right is a convection condition.

Find
The weak form for the equations of heat conduction.

Solution
If we choose an arbitrary function w(x) and multiply the governing equation (4.2.1) and the convection boundary condition (4.2.3), we have

Do integrate by parts with equation (4.2.4), we have

$$ \left. {w\left( x \right)Ak\frac} \right|_0^{10} - \int_0^{10} {\fracAk\frac} dx + \int_0^{10} s dx = 0 $$

Then use the natural boundary condition (4.2.5), we have

where the weighting function satisfy

Therefore the weak form for the equations of heat conduction is to find T such that satisfy (4.2.2) and (4.2.6) with the constrain that satisfy (4.2.7).