A new algorithm for the computation of the coefficients of the heat kernel expansion, associated with a second-order non-negative elliptic-symmetric differential operator, defined on an N-dimensional compact riemannian manifold with boundary is presented. Some coefficients are explicitly derived. Using zeta function regularization, an explicit form for the expectation value of the matter stress tensor is given. Corrections to a class of anomalies, due to the presence of the boundary, are found.