In this work, we study the Prigozhin model for growing sandpile with mixed
 boundary conditions and an arbitrary time dependent angle of repose.
 On one part of the boundary the homogeneous Dirichlet boundary condition
 is provided, on the other one the Robin condition is used. Using the
 implicit Euler discretization in time, we prove the existence and uniqueness
 of variational solution of the model and for the numerical analysis we use
 a duality approach