  We develop the general integral transforms (GIT) method for pricing barrier
options in the time-dependent Heston model (also with a time-dependent barrier)
where the option price is represented in a semi-analytical form as a
two-dimensional integral. This integral depends on yet unknown function
$\Phi(t,v)$ which is the gradient of the solution at the moving boundary $S =
L(t)$ and solves a linear mixed Volterra-Fredholm equation of the second kind
also derived in the paper. Thus, we generalize the one-dimensional GIT method,
developed in (Itkin, Lipton, Muravey, Generalized integral transforms in
mathematical finance, WS, 2021) and the corresponding papers, to the
two-dimensional case. In other words, we show that the GIT method can be
extended to stochastic volatility models (two drivers with inhomogeneous
correlation). As such, this 2D approach naturally inherits all advantages of
the corresponding 1D methods, in particular, their speed and accuracy. This
result is new and has various applications not just in finance but also in
physics. Numerical examples illustrate high speed and accuracy of the method as
compared with the finite-difference approach.
