  The interior Dirichlet boundary value problem for the diffusion equation in
non-homogeneous media is reduced to a system of Boundary-Domain Integral
Equations (BDIEs) employing the parametrix obtained in (Fresneda-Portillo,
2019) different from (Chkadua et. al 2009). We further extend the results
obtained in (Fresneda-Portillo, 2019) for the mixed problem in a smooth domain
with $L^{2}(\Omega)$ right hand side to Lipschitz domains and source term $f$
in the Sobolev space $H^{-1}(\Omega)$, where neither the classical nor the
canonical co-normal derivatives are well defined. Equivalence between the
system of BDIEs and the original BVP is proved along with their solvability and
solution uniqueness in appropriate Sobolev spaces.
