  We studied the two-loop non-factorizable Feynman diagrams for the $t$-channel
single-top production process in quantum chromodynamics. We present a
systematic computation of master integrals of the two-loop Feynman diagrams
with one internal massive propagator in which a complete uniform transcendental
basis can be built. The master integrals are derived by means of canonical
differential equations and uniform transcendental integrals. The results are
expressed in the form of Goncharov polylogarithm functions, whose variables are
the scalar products of external momenta, as well as the masses of the top quark
and the $W$ boson. We also gave a discussion on the diagrams with potential
elliptic sectors.
