  We study second-order elliptic partial differential operators acting on
sections of vector bundles over a compact manifold with boundary with a
non-scalar positive definite leading symbol. Such operators, called non-Laplace
type operators, appear, in particular, in gauge field theories, string theory
as well as models of non-commutative gravity theories, when instead of a
Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor
that plays the role of a ``non-commutative'' metric. It is well known that
there is a small-time asymptotic expansion of the trace of the corresponding
heat kernel in half-integer powers of time. We initiate the development of a
systematic approach for the explicit calculation of these coefficients,
construct the corresponding parametrix of the heat equation and compute
explicitly the first two heat trace coefficients.
