This problem is straightforward to solve using PDE Toolbox. I suggest you look at this example http://www.mathworks.com/help/pde/examples/heat-distribution-in-a-circular-cylindrical-rod.html for how to handle cylindrical geometry and this example for how to handle multiple layers http://www.mathworks.com/help/pde/examples/deflection-of-a-piezoelectric-actuator.html
You don't show any boundary conditions on the edges where x=constant. If those edges are insulated (i.e. your problem is really one-dimensional), you can use the pdepe function in core MATLAB. pdepe has an option to "automatically" handle the cylindrical coordinate system so it is particularly easy to use for this problem.
Bill
