Bianchi Type-I Barrow Holographic Dark Energy in f(R) Gravity: Exact Solutions with a Lambert Function Approach

In this paper, we discuss the Bianchi type-I barrow holographic dark energy cosmological model in the frame-work of [Formula: see text] theory of gravity. We consider the model of pressure-less dark matter and barrow holographic dark energy by solving [Formula: see text] theory field equations. Here, we find the exact solution of the field equations, we assume that the time redshift relation follows a Lambert function distribution as [Formula: see text], where [Formula: see text], here [Formula: see text] and [Formula: see text] are positive constants and [Formula: see text] is present age of the Universe. We discuss the cosmological parameters such as energy densities ([Formula: see text] and [Formula: see text]), skewness parameter ([Formula: see text]), equation of state parameter [Formula: see text] and deceleration parameter ([Formula: see text]). Also, we study the statefinder parameters such as [Formula: see text] and [Formula: see text]. We found the present values of the deceleration parameter, which is consistent with recent observational data for the Lambert function distribution. We investigate the thermodynamic quantities and the generalized energy conditions in order to test the viability of our model. Finally, the deceleration parameter evolves with cosmic time from initial deceleration to late time acceleration.