Spectral solvers based on the FFT are a well stablished approach for microscopic periodic problems due to their very efficient performance for very large degrees of freedom [1]. However, the use of general non-periodic boundary conditions (BCs) becomes ideal in some situations. The current approximation to this problem is the use of discrete sine/cosine transform for function interpolation [2,3,4], but this method presents some limitations in vector-based problems and a relatively slow convergence to the actual solution. In this work we present an alternative direction for spectral solvers with general BCs, a Chebyshev collocation framework, which is particularized for the numerical solution of diffusion equation in bounded domains. The method combines Chebyshev polynomial approximations with fast transform-based operator application, yielding a matrix-free implementation that avoids the explicit assembly of dense differentiation matrices. To improve the practical performance of the iterative solution process on refined grids, a hierarchical refinement strategy based on modal prolongation is also considered. The approach is illustrated on representative benchmark problems including Poisson diffusion, heterogeneous diffusion with discontinuous coefficients, and a nonlinear heat-transfer problem with temperature-dependent boundary effects. The numerical results show that the proposed formulation can accurately approximate non-periodic diffusion problems while retaining the computational advantages of fast transform-based spectral discretizations.