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We develop the general integral transforms (GIT) method for pricing barrieroptions in the time-dependent Heston model (also with a time-dependent barrier)where the option price is represented in a semi-analytical form as atwo-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 kindalso derived in the paper. Thus, we generalize the one-dimensional GIT method,developed in (Itkin, Lipton, Muravey, Generalized integral transforms inmathematical finance, WS, 2021) and the corresponding papers, to thetwo-dimensional case. In other words, we show that the GIT method can beextended to stochastic volatility models (two drivers with inhomogeneouscorrelation). As such, this 2D approach naturally inherits all advantages ofthe corresponding 1D methods, in particular, their speed and accuracy. Thisresult is new and has various applications not just in finance but also inphysics. Numerical examples illustrate high speed and accuracy of the method ascompared with the finite-difference approach.

Semi-analytical pricing of barrier options in the time-dependent Heston model