I am currently trying to solve it by eigen function expansion method, I have three variables x, r and t. Please provide some helping materials or give suggestions regarding to solving and plotting.
Review the following bibliography with a lot of analytical solutions:
\"Ozisik, N. Necati
{\bf Heat Conduction}, 2$^{nd}$ Edition.
John Wiley \& Sons, 1993.
Carslaw, H. S.; Jaeger, J. C.
{\bf Conduction of Heat in Solids}, Second Edition.
Oxford University Press, 1959.
Arpaci, V. S.
{\bf Conduction Heat Transfer}.
Addison-Wesley, 1966.
If you don't find them send me a message to:
A detailed derivation and outline of the solution for transient heat conduction in a cylinder can be downloaded from my site: https://sites.google.com/site/chemengwithmathematica/home
Go to heat transfer section and you can find what you need. You can download a PDF or the Mathematica notebook that shows how to actually compute the eigenvalues and construct the numerical solution.
The textbooks listed by Andres Granados will also be helpful.
In addition to the previous smart suggestions, consider that you need to have or realize omogeneous equations and boundary conditions so that product solutions work.