I am carrying out a heat transfer analysis on a model using the Galerkin's method of weighted residual (finite element method). I am finding it a challenge to obtain the stiffness matrix for some of the terms in my governing differential equation. I have attached a PDF to briefly show the portion of my work involved. The first term from the governing equation is easy to obtain, but it becomes difficult from the second term onwards (I had to use an approximation which I don't think works). Is there a way to work around this? Is there a step or a mathematical trick I am missing?
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