Two-dimensional solidification problem of a finite cylinder, in which the liquid phase is initially at the fusion temperature, is solved by using a front fixing approach. The external surfaces of the cylinder are subjected to a temporally or spatially varying temperature below freezing. The method employed is based on one used for the solution of a solidification problem in Cartesian domain. A coordinate transformation is applied in both radial and axial directions to obtain a square computational domain. This transformation results in a computationally intensive grid generation for every time step of solution. Finite difference form of the transformed energy equation is solved for the temperature distribution in the solid phase and the solid-liquid interface energy balance is integrated for the new position of the moving solidification front. The effect of the aspect ratio and spatially varying boundary temperatures on solidification is studied.