A modified dynamical model of cosmology is derived based on imposing Neumann boundary condition on cosmological perturbation equations. Then, it is shown that a new term appears in the equation of motion which leads to a modified Poisson equation with a new density term. In addition, a modified Hubble parameter due to the presence of this new density term is derived. Moreover, it is proved that, without a cosmological constant, such model has a late time accelerated expansion with an equation of state converging to w<-1. Also, the luminosity distance in the present model is shown to differ from that of the LCDM model at high red shifts. Furthermore, it is found that the adiabatic sound speed squared is positive in radiation dominated era and then converges to zero at later times. We bound the parameters of the model based on Type Ia Supernovae, Hubble parameter data and the age of the oldest stars. Theoretical implications of Neumann boundary condition has been discussed and it is shown that by fixing the value of the conjugate momentum(under certain conditions), one could derive our version of modified dynamics.