Hi! I'm a final year Mechanical Engineering student trying to use Mathematica to solve a transient Conduction problem to plot history of temperature at a point. The question is attached below . 
When using NDSolve , an error pops up saying that boundary and initial conditions are inconsistent. The Code is attached below 
When using NDSolve with the finite element method an error pops up saying that NDSolve::femcscd: The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help. and NDSolve::fembdnl: The dependent variable in (T^(1,0,0))[0,y,t]==0 in the boundary condition DirichletCondition[(T^(1,0,0))[0,y,t]==0,r==0.] needs to be linear. The code is attached below 
I've also attached the mathematica file. Will appreciate any help :)
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