User:Eml5526.s11.team3.sahin/Homework 4

=Problem 4.2 Deriving of the Weak Form for One Dimensional Heat Conduction =

Given: Equations of heat conduction in one dimension
A first Course in Finite Elements, Fish & Belychtschko, John Wiley & Sons Ltd, 2007, p.73, problem 3.5

Strong form for 1D heat conduction problems

Essential boundary conditions, Natural boundary conditions, The condition on the right is a convection condition

Find
Derive the weak form for one dimensional heat conduction.

Solution
We multiply the first equation in the strong form (Eq 2.1) by the weight function and integrate over the domain on $$\Omega$$, which yields

Using Integration by Parts of the first term in (Eq 2.4) gives

Since $$w$$ is arbitrary, we select $$w$$ such that $$w(0)=0$$ to drop out the unknown term at x=0. As we remember we already have essential boundary conditions at x=0 and substitute natural boundary condition (Eq 2.3) into (Eq 2.6). So we get

If we rearrange the equation, we obtain the weak form