I am making progress in finite elements (I am studying on my own with my dog Toreto), and I got stuck with the following problem:
In the equation, equipped with the necessary conditions, $$ - u(x)'' = f(x) $$
For the Stiffness Matrix $$ A_{i,j} = \int_I \varphi_j' \varphi_i' dx, \, i,j=1,2, \dots , n-1.$$ with the hat functions $$ \varphi_i(x) = \begin{cases} \frac{x - x_{i}}{h_{i}}, & \text{if } x \in I_i, \\ \frac{x_{i+1} - x}{h_{i+1}}, & \text{if } x \in I_{i+1} \\ 0, & \text{otherwise} \end{cases}$$
we have $$ A_{ii} = \int_I (\varphi_i')^2 dx = \int_{x_i-1}^{x_i}(\varphi_i')^2 dx+\int_{x_i}^{x_i+1}\varphi_i'^2 dx $$
We can interpolate using Simpson's rule and obtain $$ A_{ii}= \frac{1}{h_i}+\frac{1}{h_{i+1}}. $$
However, the case $ -u''' = f(x) $ I arrived at $$ A_{ij} = \int_I \varphi_j'' \varphi_i' dx, \, i,j=1,2, \dots , n-1. $$
It is not possible to use hat functions. What nodal basis functions can I use to be able to integrate? How do I do that?
Note: I managed to learn and implement even the heat equation part using backward Euler from the book: Mats G. Larson, Fredrik Bengzon - The Finite Element Method_ Theory, Implementation, and Applications (2013, Springer).I really wanted to find out how to solve this third-order one.
