  The reactions $^{10}$B(p,$\alpha$)$^{7}$Be and $^{11}$B(p,$\alpha$)$^{8}$Be
are studied at thermonuclear energies using DWBA calculations. For both
reactions, transitions to the ground states and first excited states are
investigated. In the case of $^{10}$B(p,$\alpha$)$^{7}$Be, a resonance at
$E_{Res}=10$ keV can be consistently described in the potential model, thereby
allowing the extension of the astrophysical $S$-factor data to very low
energies. Strong interference with a resonance at about $E_{Res}=550$ keV
require a Breit-Wigner description of that resonance and the introduction of an
interference term for the reaction $^{10}$B(p,$\alpha_1$)$^{7}$Be$^*$. Two
isospin $T=1$ resonances (at $E_{Res1}=149$ keV and $E_{Res2}=619$ keV)
observed in the $^{11}$B+p reactions necessitate Breit-Wigner resonance and
interference terms to fit the data of the $^{11}$B(p,$\alpha$)$^{8}$Be
reaction. $S$-factors and thermonuclear reaction rates are given for each
reaction. The present calculation is the first consistent parametrization for
the transition to the ground states and first excited states at low energies.
