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In the present research paper, the nonlinear propagation of dust ion acoustic solitary waves in a collisional dusty plasma, which consists of negatively charged small dust grains, positively charged ions and isothermal electrons with background neutral particles, is investigated. The low rates compared to the ion oscillation frequency, of the charge-fluctuation dynamics of the dust grains, the ionization, ion-neutral and dust-neutral collisions (i.e. weak dissipations) are considered. Using the reductive perturbation theory, a damped Korteweg-de Vries (DKdV) equation is derived. On the other hand, the dynamics of solitary waves at a critical phase velocity is governed by a damped modified Korteweg-de Vries (DMKdV) equation. The nonlinear properties of dust ion acoustic waves in the presence of weak dissipations in the two cases are discussed.