Need help with solving transient heat conduction equation

I have a PDE equation with mixed boundary conditions to solve.I need to plot the transient temperature vs time. Since there are different types of boundary conditions including robin type and neumann type, i am not able to code. My knowledge is constrained to having temperatures as boundary conditions with no complications. I am attaching the equations and the reference research paper for context.

[Torsten]

And what are your expectations that the forum should do for you ?

Read a paper on how to discretize your equation and your boundary conditions. There are dozens of publications on the subject, e.g. the one attached.

If you know Fortran, you can use PDETWO

https://people.sc.fsu.edu/~jburkardt/f77_src/toms565/toms565.html

I wonder why you don't use the PDE Toolbox.