Energy conditions for inhomogeneous EOS and its thermodynamics analysis with the resolution on finite time future singularity problems

In this paper we study two different cases of inhomogeneous EOS the form p_d = \omega(t)\rho_d+w_1 f(H, t) . We derive energy density dark fluid and matter component for both cases. Further calculate evolution density, gravitational constant cosmological constant. also explore finite time singularity thermodynamic stability conditions EOS. Finally, discuss thermodynamics with derivation internal energy, Temperature entropy show that all are satisfied