  We extend the approach of Carr, Itkin and Muravey, 2021 for getting
semi-analytical prices of barrier options for the time-dependent Heston model
with time-dependent barriers by applying it to the so-called $\lambda$-SABR
stochastic volatility model. In doing so we modify the general integral
transform method (see Itkin, Lipton, Muravey, Generalized integral transforms
in mathematical finance, World Scientific, 2021) and deliver solution of this
problem in the form of Fourier-Bessel series. The weights of this series solve
a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also
derived in the paper. Numerical examples illustrate speed and accuracy of our
method which are comparable with those of the finite-difference approach at
small maturities and outperform them at high maturities even by using a
simplistic implementation of the RBF method for solving the LMVF.
