FEM VS section-properties

I apologize in advance for my English, it's not my main language. Do you want to code this exact problem in MATLAB? If you need this specific problem just write the global matrix and global force vector. If you want to implement a full program for heat transfer is not hard either. You must create single element matrices and vectors. Then, using a for loop you can create the global matrix just adding small matrices. Border conditions are very important. You can create a matrix of border conditions. The first column is the node in which the border condition is applied and the second column is the border condition value. You can create both essential, natural and convective border condition matrices. In the computational implementation it is not very useful to remove rows and columns of the global matrix. The reason is that your solution vector size will be different to the number of degree of freedom. That makes the process harder (and slower) (source Reddy's book). There are several methods to assign the border conditions. For example, you can extract the column with the same number of the border condition degree of freedom. Then multiply the whole column by the border condition value and subtract it to the force global vector. Last is to modify the column and row of the global matrix to 0 except the diagonal value which have to be changed to 1. The row of the force vector with the same degree of freedom has to be the border condition value. When you solve the equation system you end with the node solution. You can create graphs with these values or create better graphs using the shape functions. You can create a heat flux graph using shape functions too!! I made a Python FEM package, it has a heat transfer option. It's not Matlab, but maybe you can find it useful. Link https://github.com/ZibraMax/FEM

If you know python, check out this repo on GitHub. It can analyze any cross section, including parametric shapes and DXF shapes. https://github.com/robbievanleeuwen/section-properties

This Python tool: https://github.com/robbievanleeuwen/section-properties