Finite element solution for 1D Heat Equation Subject to both Neumann and Dirichlet BCs

Question

I am trying to solve the heat equation $$ \frac{\partial u}{\partial t}-\kappa \frac{\partial^{2} u}{\partial x^{2}}=0 $$ with $$ u(0, t)=e^{-16 \pi^{2} \kappa t}, $$ $$ \frac{\partial u}{\partial x}(1, t)=-e^{-\pi^{2} \kappa t} \pi $$ $$ u(x, 0)=\cos (4 \pi x)+\sin (\pi x) $$ using finite element method and I have difficulty deriving mass and stiffness matrices. I start by multiplying both sides of the PDE by test functions $v(x)$ and integrate to make a weak form of the PDE $$ \int_{0}^{1}\left(\frac{\partial u}{\partial t}-\kappa \frac{\partial^{2} u}{\partial x^{2}}\right) v(x) \mathrm{d} x=0 $$ As usual, we apply integration by parts one time to convert the form of the second term on the left $$ \int_{0}^{1}\left(-\kappa \frac{\partial^{2} u}{\partial x^{2}}\right)v(x) \mathrm{d} x=-\kappa\left.\left(\frac{\partial u}{\partial x} v(x)\right)\right|^{1}_{0}+\kappa \int_{0}^{1} \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \mathrm{d} x $$ plugging this into the PDE $$ \int_{0}^{1} \frac{\partial u}{\partial t} v(x) \mathrm d x -\kappa\left(\frac{\partial u}{\partial x}(1, t) v(1)-\frac{\partial u}{\partial x}(0, t) v(0)\right)+\kappa \int_{0}^{1} \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \mathrm{d} x= 0 $$ and using the boundary values $$ \int_{0}^{1} \frac{\partial u}{\partial t} v(x) \mathrm d x -\kappa\left(-e^{-\pi^{2} \kappa t} \pi v(1)-\frac{\partial u}{\partial x}(0) e^{-16 \pi^{2} \kappa t}\right)+\kappa \int_{0}^{1} \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} \mathrm{d} x= 0 $$ I would appreciate any comments on how to proceed to derive mass matrix $M$ and stiffness matrix $K$ and get the equation into the form $$ M \alpha^{\prime}(t)=-K \alpha(t) $$